The non-Newtonian rheology of a dilute suspension of hydrodynamically interacting colloids is studied theoretically via active, nonlinear microrheology. In this Stokes-flow regime, a Brownian probe is driven through the suspension by a fixed external force; its motion distorts the configuration of background bath particles, which in turn alters probe motion. This interplay was utilized in our recent article to obtain the normal elements of the suspension stress via the nonequilibrium statistical mechanics theory [Chu and Zia, J. Rheol. 60(4), 755–781 (2016)]. In the present article, we focus on the normal stress differences N1 and N2, osmotic pressure Π, and their evolution with the strength of the probe force and strength of hydrodynamic interactions. As hydrodynamic interactions grow from weak to strong, the influence of couplings between the stress and the entrained motion on N1 changes with the strength of flow. When flow is strong, hydrodynamic interactions suppress the magnitude of N1, owing to collision shielding that preserves structural symmetry. In contrast, when flow is weak, hydrodynamic interactions enhance disparity in normal stresses and, in turn, increase the magnitude of N1. The first normal stress difference changes sign as flow strength increases from weak to strong, due strictly to the influence of elastic interparticle forces. Regardless of the strength of flow and hydrodynamic interactions, the second normal stress difference is identically zero owing to the axisymmetry of the microstructure around the probe. Hydrodynamic forces act to suppress the osmotic pressure for any strength of flow; when the forcing is strong, this effect is qualitative, reducing the flow-strength dependence from linear to sublinear as hydrodynamic interactions grow from weak to strong. Non-Newtonian rheology persists as long as entropic forces play a role, i.e., in the presence of particle roughness or even very weak Brownian motion, but vanishes entirely in the pure-hydrodynamic limit.
The non-Newtonian rheology of colloidal dispersions and other complex fluids is set by the asymmetry of the arrangement of the constituent particles in the embedding medium. In dilute suspensions of smooth noncolloids, structural symmetry under strong flow is preserved by the reversibility of relative Stokes-flow trajectories, giving Newtonian rheology. In the opposite limit of an equilibrium suspension of colloids, Brownian motion acts, on average, to preserve symmetry, thus preserving Newtonian rheology. Between these two limits, a rich non-Newtonian rheology emerges in tandem with the development of an asymmetric microstructure, as Brownian motion destroys the symmetry of Stokes-flow trajectories. The presence of other entropic forces such as those introduced by surface asperities gives rise to similar non-Newtonian phenomenology. Many previous studies have demonstrated that the strength of Brownian and hydrodynamic forces evolves with flow strength, producing well-known and familiar behaviors, such as flow thinning, flow thickening, and normal stress differences. The detailed study of the connection between the evolving balance of microscopic forces, and the formation of asymmetric microstructure has been carried out in focused flow regimes in a prior work. A more comprehensive study aimed toward understanding the role played by entropic forces is the focus of the present work.
The experimental study of flow-induced non-Newtonian rheology of suspensions of noncolloidal spherical particles dates back to the work of Bagnold , in which he reported the appearance of normal stresses during shear flow. He hypothesized the presence of a particle microstructure, and that shear-flow induced changes in its shape were the origin of non-Newtonian rheology. Following this pioneering work, several studies convincingly inferred the presence of a shear-induced structure in concentrated noncolloidal suspensions [2,3]. Direct visualization of the distorted structure soon followed in a systematic study of the pair distribution function of the suspended particles by Gadala-Maria and co-workers [4,5] who, via video imaging, revealed an accumulation of particle pairs along the compressional axes and depletion in the extensional quadrants. In a landmark study of shear-induced migration under structural gradients, Leighton and Acrivos  attributed the origin of such a structural asymmetry to three-body and higher-order interactions influenced by particle roughness. Numerous subsequent investigations were conducted for suspensions of a wider range of concentration [7–9], and also in suspensions comprising particles with controlled roughness , all confirming that structural asymmetry is necessary to produce non-Newtonian rheology. However, the influence of flow strength and entropic forces on the suspension stress had not been fully explored experimentally owing to two primary challenges: First, the dominance of Brownian noise under (weak) flow makes measurements of weak non-Brownian signals difficult in the low-Pe regime. Second, to interrogate entropic effects, the thermodynamic sizes of particles are required, but such measurement can be challenging. During the same timeframe, development of the nonequilibrium statistical mechanics theory for dilute colloidal dispersions provided a guiding insight.
In a series of seminal papers [11–13], Batchelor confirmed that microstructural asymmetry, not just distortion, is necessary to generate non-Newtonian rheology and, in so doing, paved the path on which the pair-level origin of much non-Newtonian rheology was discovered. He developed the nonequilibrium Smoluchowski framework that governs the evolution of a flowing microstructure under the influence of thermodynamic and hydrodynamic forces, along with expressions for the average suspension stress in dilute colloidal dispersions undergoing flow, and examined them specifically in the pure-hydrodynamic limit under strong flow, . Here, the Péclet number, Pe, is a measure of the strength of advection, which distorts the suspension, relative to the strength of diffusion that acts to recover the equilibrium microstructure. In the pair limit, the microstructure is fore-aft symmetric and, utilized as a weighting function to compute the average suspension stress, reveals that the symmetric structure induces no non-Newtonian behavior. Batchelor predicted, however, that the presence of (even weak) Brownian motion or particle roughness would destroy reversible Stokes' flow trajectories and lead to non-Newtonian rheology. Brady and Vicic  validated Batchelor's hypothesis in a study of a weakly sheared suspension in the presence of strong Brownian motion, showing that an asymmetric structure arises from flow disturbance, and indeed produces nonzero normal stress differences and a nonequilibrium osmotic pressure. These studies focused, however, only on the limit of strong hydrodynamics. The significance of and the interplay between entropic and hydrodynamic forces in the nonequilibrium stress, as well as the evolution of the stress with flow strength, remained unexplored.
The influence of hydrodynamic interactions on nonequilibrium rheology was later studied by Brady and Morris , who employed an excluded-annulus model  to study a strongly sheared suspension in the limits of asymptotically strong and weak hydrodynamics. Their results revealed that both normal stress differences decrease as hydrodynamic interactions become weak, for two limits: and . In a subsequent work, Bergenholtz et al.  investigated the evolution of the suspension stress with arbitrary flow strength by solving the Smoluchowski equation numerically for a range of strengths of hydrodynamic interactions. They recovered the high-Pe behavior of the normal stress differences found by Brady and Morris, and further demonstrated that, in the presence of hydrodynamic interactions, the first normal stress difference changes sign from positive to negative, in contrast to the positive first normal stress difference shown for the entire range of flow strength in the absence of hydrodynamics, thus revealing their complex evolution with flow strength. To understand the influence of individual microscopic forces, they computed the hydrodynamic, Brownian and interparticle contributions to the total normal stress differences, and asserted that the interparticle force makes no contribution in the limit of strong hydrodynamics, consistent with the dogma that lubrication interactions prevent particle contact in this limit. However, Chu and Zia  extracted the normal stresses from the data of Bergenholtz et al. and discovered that the interparticle normal stresses are not zero even in the limit of strong hydrodynamics. The idea that the interparticle force makes no contribution is unphysical, and was an artifact of the authors' choice to move part of the trace of the interparticle stress to the Brownian stress. This ad hoc grouping of stress contributions smears out the roles played by the Brownian and the interparticle forces, obscuring the sought-after connection of hydrodynamic interactions and other microscopic forces to strength of suspension stress.
In an effort to elucidate the dependence of the nonequilibrium stress on the range of interparticle repulsion and on flow strength, Chu and Zia undertook the study of the suspension stress in a dilute colloidal dispersion of hydrodynamically interacting spheres, utilizing the framework of active microrheology rather than traditional shear macrorheology, to leverage the connections between the Stokes-Einstein relation, single-particle motion, and fluctuation dissipation [18,19]. Seeking a detailed understanding of the role played by interparticle forces in the evolution of the suspension stress, Chu and Zia examined the evolution of the normal stresses with the range of interparticle repulsion (equivalently, the strength of hydrodynamic interactions) and strength of external probe forcing (equivalently, flow strength), finding that the nonequilibrium stress grows with increasing flow strength at fixed interparticle repulsion range, whether a suspension is at or far from equilibrium. They connected this evolution of the rheological behavior to the trajectories of particle pairs. This was utilized to understand the mechanisms by which hydrodynamic and entropic forces generate stress, and to relate the nonequilibrium stress to the entropic energy storage. The next step carried out here is to examine the normal stress differences and osmotic pressure.
In a prior study, Zia and Brady  inspected the normal stress differences and the osmotic pressure for suspensions of nonhydrodynamically interacting colloids. They found that the second normal stress difference is identically zero due to the axisymmetry of the perturbed microstructure around the probe, and the evolution of the first normal stress difference and the osmotic pressure depends on the strength of the external force: and under weak forcing, and and under strong forcing. These results were obtained via three approaches: A traditional micromechanical framework that requires the statistics of particle distribution as described above; Brownian dynamics simulations; and a new phenomenological theory describing the nonequilibrium fluctuation and dissipation that produces stress, a nonequilibrium Stokes-Einstein relation. But without an understanding of the role played by hydrodynamic interactions, which are important in many physical systems, a comprehensive understanding of how evolution of such forces with flow strength changes non-Newtonian rheology was still unknown. In a separate work, Yurkovetsky and Morris  utilized Stokesian dynamics simulations to study the nonequilibrium particle pressure in a sheared concentrated Brownian suspension, showing the expected exchange of the dominance of the Brownian force for the hydrodynamic force when a suspension is driven from equilibrium. However, the Stokesian dynamics framework that they employed only considered suspensions interacting with strong hydrodynamic interactions, leaving unexplored the effect of interparticle force on osmotic pressure. Overall, prior studies have left open the question of how radial interparticle forces and hydrodynamic interactions evolve in tandem to influence the normal stress differences and osmotic pressure. In a recent article, we conducted a theoretical study of the particle phase stress in a colloidal dispersion, and studied the dependence of the nonequilibrium normal stresses on flow strength and on arbitrary strengths of hydrodynamic interactions . The richness of the discussion of the individual stress elements was sufficient to warrant separate treatment of the normal stress differences and osmotic pressure, which is presented here.
In the present work, we utilize the framework of active microrheology to study the influence of the relative strengths of entropic and hydrodynamic forces on normal stress differences and the osmotic pressure. This mathematical model is applicable to dilute, unbound colloidal suspensions in the Stokes' flow regime, for arbitrary strength of flow and hydrodynamic interactions. Careful interpretation of the thermodynamic stresses and their evolving influence on the microstructure provides new insight into a sign change in the normal stress differences. While it was shown in our previous work that entropic forces enhance suspension stress for the entire range of flow strength, we examine if this behavior is preserved in the normal stress differences or the nonequilibrium pressure.
The remainder of this paper is organized as follows: In Sec. II, we begin with a brief review of the physical model for the evolution of the structure and the suspension stress, followed by a physical description of the suspension stress in Sec. III. Next, in Sec. IV, we recapitulate the expressions for the total average material stress and its contributions from the external, Brownian and interparticle forces derived by Chu and Zia , with which we use to compute normal stress difference and osmotic pressure in this article. Results are presented in Sec. V, starting with analysis of the asymptotic behaviors of the normal stress differences and osmotic pressure in the limits of weak and strong probe forcing, and weak and strong hydrodynamic interactions. Next, the same quantities are computed numerically for the full range of strength of hydrodynamic interactions and forcing. To gain insight into the evolution of the total normal stress differences and osmotic pressure, their underlying external, Brownian and interparticle contributions are investigated. The study is concluded in Sec. VI with a summary.
II. MODEL SYSTEM
We consider a suspension of neutrally buoyant, colloidal hard spheres of hydrodynamic radius a, dispersed in an incompressible Newtonian solvent of dynamic viscosity η and density ρ. A fixed external force, , drives one of the particles, the “probe,” through the suspension. The strength of fluid inertia relative to viscous forces defines a Reynolds number, , where U is the characteristic velocity of the probe. For micron-sized particles, Re is much less than unity, allowing neglect of inertial forces; Fluid motion is thus governed by Stokes' equations. The probe number density, na, is much smaller than the number density of bath particles, nb. Probe motion deforms the microstructure while Brownian motion of the bath particles acts to recover their equilibrium configuration. The degree of microstructural deformation, and its influence of probe motion, is thus set by the strength of the probe forcing, F0, relative to the Brownian restoring force, , where k is Boltzmann's constant, T is the absolute temperature, and ath is the thermodynamic size of particles, defining the Péclet number: . Interactions between the probe and the surrounding microstructure slow the mean motion of the probe [21–24] and induce a diffusive spread of the probe's trajectory [25–28]. These changes can be utilized to infer suspension properties.
Particles can interact via various forces, e.g., hydrodynamic, Brownian, and interparticle forces. Here, we employ an excluded-annulus model  to account for short- and long-range nature of these interactions (Fig. 1). The hydrodynamic size of the particles, a, is set by the surface at which the no-slip condition is obeyed. In contrast, surface features of the particles, for instance, electrostatic double layers or surface asperities, may also be present and set the strength of direct particle interactions, determining the minimum-approach distance rmin with which two particles can approach one another. The minimum-approach distance, in turn, sets the thermodynamic size of particles, ath, where the no-flux condition is met and for equally sized particles. In general, the no-flux surface can extend beyond the no-slip surface, .
An interparticle force derivable from a potential V(r) between particles serves as a simplified model for electrostatic or steric repulsion from surface roughness or surface modifications to promote dispersion stability. Here, we adopt a hard-sphere model, where particles exert no force on one another until their no-flux surfaces touch, , at which an infinite potential prevents them from overlapping
The strength of hydrodynamic interactions is determined by how closely two particles can approach one another, characterized by a dimensionless interparticle repulsion range
In the limit of strong hydrodynamics, , particles can approach each other closely enough to experience lubrication interactions. As κ grows, , hydrodynamic interactions are weakened, but their impacts on the evolution of the microstructure and rheology persist over long distances. In the limit of weak hydrodynamics, , the long-range interparticle repulsion keeps particles sufficiently separated such that even long-range hydrodynamic interactions are negligible.
III. PHYSICAL DESCRIPTION OF THE STRESS
The particle-phase stress in colloids is frequently viewed as a model for the stress generated by a molecular gas on its enclosing volume . The isotropic osmotic pressure in equilibrium suspensions arises due to particle diffusion, where thermal fluctuations of the solvent drive particle collisions with “walls” of a fictitious enclosing container, giving rise to the pressure. In an equilibrium colloidal dispersion, the osmotic pressure is a measure of the tendency of the particles to push outward on the walls of the fictitious container, causing it to expand—similar to the way a drop of dye diffuses outward in a liquid . The sign of the osmotic pressure, defined as the negative one-third of the trace of the stress tensor, , reflects this diffusion or expansion outward of the particle phase when . In contrast, when , the particle phase is condensing, contracting inward, rather than diffusing outward. The minus sign signifies the opposite sense of stress felt by the fictitious boundary: As the particle-phase pushes outward, the “container” would push inward on the particle phase, as sketched in Fig. 2. This can be expressed compactly via the normal components of the stress tensor
That is, osmotic pressure gives the isotropic normal stress felt by the “box,” whereas reflects the stress felt by the particle phase as they encounter the box.
To focus on the nonequilibrium rheology, we defined the nonequilibrium stress, , as
where is the equilibrium suspension stress in the absence of external force. The three-dimensional stress tensor comprises six independent elements in general. In active microrheology, they reduce to three nonzero diagonal elements, the normal stresses Σxx, Σyy and Σzz, owing to the axisymmetry of the microstructure around the probe. This structural axisymmetry also yields identical normal stresses along the orthogonal axes, , leaving only two relevant quantities: The normal stresses acting parallel and perpendicular to the direction of the external force, and . They are obtained by projecting the stress tensor as
where and are the unit vectors in the direction parallel and perpendicular to the external force, respectively.
Normal stresses were examined in detail in our recent article . In this work, we focus on the non-Newtonian rheology revealed by the normal stress differences and nonequilibrium osmotic pressure, a hallmark of non-Newtonian behavior in complex fluids. The first and the second normal stress differences are defined as
The distortion of microstructure can also bring forth a nonequilibrium osmotic pressure, giving rise to isotropic expansion or contraction of the particle phase. The osmotic pressure is defined as the negative one-third of the trace of the stress tensor
Away from equilibrium, Zia and Brady  studied nonhydrodynamically interacting suspensions utilizing an active microrheology framework, and showed that a nonequilibrium osmotic pressure contributes to particle stress. In the present study, we will examine the nonequilibrium osmotic pressure in hydrodynamically interacting suspensions arising from the underlying external, Brownian and interparticle forces.
IV. THEORETICAL FRAMEWORK
The theoretical framework for a micromechanical analysis of the evolving structure and its connection to rheology requires three elements: The details of the microscopic forces between the particles; a description of the particle configuration set by microscopic forces and imposed fields or flow; and a means by which to relate the two to macroscopic rheological quantities. The first of these, microscopic forces, was described in Sec. II. The latter two—formulation of a Smoluchowski equation governing microstructural evolution, and its use in calculating an average, probe-phase stress—were fully developed in our recent article in which we studied the individual elements of the normal stresses. Here, we shall briefly recapitulate the main aspects of the theory and present the final governing equations from which solutions are utilized to compute the normal stress differences and osmotic pressure; the reader interested in the detail of the full derivation is referred to Chu and Zia .
A. Micromechanical description of the stress
In this section, we briefly recapitulate our recent development of the stress tensor for a dilute bath driven from equilibrium by an externally forced probe . The particle-phase stress, , can be divided into nonhydrodynamic and hydrodynamic contributions following Batchelor [11,13] as
where I is the isotropic unit tensor, and the angle brackets denote an ensemble average over all permissible positions of the bath particles. The approach we shall take here is to consider a material through which numerous probe particles move in response to an external force. The probes are taken to be dilute and thus do not interact with one another. A detailed discussion of the averaging, following Batchelor's program (1970), is given by Chu and Zia .
We highlight the importance of a careful analysis of the individual contributions from interparticle, external, Brownian and hydrodynamic forces. The first term in Eq. (10), , is the ideal osmotic pressure associated with the equilibrium thermal energy of the Brownian particles. The second term, , is the nonhydrodynamic interparticle stress and originates from interparticle elastic collisions to which we also refer as the elastic stress, . The remaining term, , is the hydrodynamic stress induced by the combined influence of external probe forcing, Brownian motion and interparticle force via hydrodynamic coupling. It is the stress exerted by the fluid on the particles
The superscript P, dis refers to the dissipative part of the interparticle stress: When an interactive force between particles causes relative motion, if they are hydrodynamically coupled, the motion will produce a stresslet . The average stresses in Eq. (11) can be expressed in terms of the corresponding stresslets, viz., , , , and , such that,
The stresslets correspond to the symmetric part of the first moment of the stress tensor and are induced by disturbance flows created by particle motion. These are in turn given by the hydrodynamic couplings between particle motion and hydrodynamic tractions on the particle surfaces 
where and are the translational and rotational velocities of a particle due to the external force applied to the probe, and and are the translational and rotational velocities of a particle due to interparticle forces between particles. The hydrodynamic resistance and mobility tensors , and couple surface tractions on one particle to its own motion and the motion of other particles. (A brief discussion of the hydrodynamic resistance and mobility functions is given in Appendix B.) The sum of the elastic and dissipative interparticle stresslets is the (total) interparticle stresslet 
The external-force induced stresslet, , the Brownian stresslet, , and the dissipative interparticle stresslet, , all vanish in the absence of hydrodynamic interactions, as do their sum, the hydrodynamic stresslet . In contrast, the elastic interparticle stresslet, , is always present regardless of the strength of hydrodynamic interactions.
Before moving on to expressions for the average stresses, we highlight two important aspects of the stress tensor defined in our recent work. First, we define the hydrodynamic stress as the sum of the external-force induced stress, the Brownian stress, and the dissipative interparticle stress [cf. Eqs. (11) and (12)], rather than the historical definition, [13,17]. The latter definition was developed by Batchelor  who, in the first part of his study, defined the particle stress due solely to external straining flow (analogous to our external-force induced stress) in a noncolloidal suspension as the hydrodynamic stress. In another part of the study, he considered a homogeneous suspension of colloids at equilibrium to obtain expressions for the thermodynamic (Brownian and interparticle) stresses, and proposed that the total stress in a flowing colloidal suspension is a superposition of the hydrodynamic and thermodynamic stresses. The carry-over of the nomenclature of the hydrodynamic stress from a noncolloidal to a colloidal system has produced ambiguous expressions—while the hydrodynamic stress equals the external-force induced stress in a noncolloidal suspension, the hydrodynamic stress in general comprises the external-force induced stress, the Brownian stress and the dissipative interparticle stress in a colloidal suspension, [cf. Eqs. (11) and (12)].
Second, prior studies traditionally combine the trace of the dissipative interparticle stresslet with the Brownian stresslet (see, e.g., ). We chose not to combine these stresses and, in so doing, showed that the trace of the dissipative interparticle stress must reside in the interparticle stress, counteracting the elastic interparticle stress to recover the net zero contribution from the hard-sphere interparticle force to the suspension stress in the limit of strong hydrodynamics, the concept at the heart of the dogma that lubrication forces replace hard-sphere forces at contact. In Sec. V, we will show that the same grouping of stresses is necessary to properly account for the physics via the normal stress differences and the osmotic pressure in the limit of strong hydrodynamic interactions.
In the dilute limit, we found previously the average external-force induced, Brownian, and interparticle stresses by integrating the stresslets, Eqs. (12)–(16), over all admissible pair configurations
Here, the volume fraction of bath particles and is the unit vector pointing in the direction of the external force. The pair distribution function, , describes the spatial distribution of bath particles around a probe particle and has been studied for all Pe and κ previously [18,21,22,25]; these results are briefly recapitulated in Appendix A. In each of the three equations above, the last term represents the trace of the stresslet, associated with an isotropic nonequilibrium pressure. All remaining terms are traceless, except the elastic stress in Eq. (19). The isotropic and traceless contributions are identified here because, historically, the stress tensor was computed as a traceless quantity with corresponding hydrodynamic functions [29,30]. However, a subsequent work explored the trace of stress tensor in order to quantify the osmotic pressure and, in so doing, generated a distinct, corresponding set of hydrodynamic functions . The components of the hydrodynamic resistance and mobility functions Xαβ, Y αβ, xαβ, yαβ, G, and H are defined following the conventional notations [29–34]. They depend only on the relative separation between a pair of particles, r, and the dimensionless repulsion range, κ. Further details can be found in Appendix B. We note that the integrals above are absolutely convergent, as their integrands all scale as , where .
The quantities of interest of this work, the normal stress differences and osmotic pressure, are obtained from Eqs. (17)–(19) via Eqs. (7)–(9). In particular, the normal stress differences are associated with the traceless portion of the parallel and the perpendicular normal stresses, where the two normal stresses were studied in detail in prior work . To facilitate the discussion of the normal stress differences in Sec. V, we recapitulate the asymptotic results of the two normal stresses as follows.
For the asymptotic limit of weak probe force, , the external-force induced, Brownian, and interparticle contributions to the nonequilibrium parallel normal stresses are given in Appendix C. In the dual limits of weak probe force and weak hydrodynamic interactions, and , the hydrodynamic stress vanishes and only the nonhydrodynamic elastic interparticle stress remains, and to are given by [18,19]
where the coefficients and correspond to the elastic stress and are given in Appendix D. The functions and give the contact values of the structure corresponding to monopolar and quadrupolar distortions, respectively (cf. Appendix A).
In the dual limits of weak probe force and strong hydrodynamic interactions, and , the interparticle stress vanishes while the Brownian and external-force induced stress remain, and to are 
where the coefficients and correspond to the traceless and the isotropic parts of the stress tensor, respectively, and are given in Appendix D. The function gives the radial dependence of the structure corresponding to dipolar distortion (cf. Appendix A).
In the opposite limit of a strong probe force, , the external-force induced, Brownian, and interparticle contributions to the nonequilibrium parallel and perpendicular normal stresses are given in Appendix C. In the dual limit of strong probe force and weak hydrodynamic interactions, and , the two normal stresses are given, to O(Pe), as [18,19]
where for compactness we have defined .
In the dual limits of strong probe force and strong hydrodynamic interactions, and , the external-force induced stress dominates. The hydrodynamic functions in Eqs. (17)–(19) take on their lubrication-limit values and, to leading order, the high-Pe asymptotes read
where the radial integration is carried out over the stretched radial coordinate and . The coefficients and are associated with the traceless and isotropic hydrodynamic functions, respectively, and are given in Appendix D. Inspection of Eqs. (23a)–(23b) suggests divergent behavior; following our prior work, the O(1) angular diffusion was neglected in simplifying the boundary layer equation to obtain the analytical microstructural solution, [Eqs. (A11)–(A12)], as originally shown by Khair and Brady . Angular advection and diffusion act to close the wake formed by the detaching boundary layer, but the diffusive contribution is most important in a small region , where . In consequence, neglecting it near θ = 0 causes the solution to diverge. To resolve this artifact while fully accounting for downstream behavior, a value of ϵ is selected to meet two criteria: Large enough to avoid the divergent region at and very near θ = 0, but small enough to avoid enclosing a significant portion of the detaching boundary layer structure, where , i.e., to primarily enclose the low-particle density wake where . A value of was selected to meet both criteria. Further reduction of this value showed negligible improvement in accuracy of the computation.
The connection between microscopic forces and the evolution of particle configuration built in Sec. IV can be utilized to evaluate the particle stress in a dilute colloidal suspension. The individual normal stresses require a detailed interpretation and discussion, and were presented in a separate work . The seemingly simple calculation of the normal stress difference and osmotic pressure reveals sufficiently rich behavior to warrant separate examination, presented here. We divide the results into two parts: First, the asymptotic limits of weak and strong forcing; and second, evolution of normal stress differences and osmotic pressure for arbitrary strength of forcing and strength of hydrodynamic interactions. The second normal stress difference is identically zero regardless of the strength of flow and hydrodynamics, owing to the axisymmetry of the perturbed structure around the probe (cf. Sec. IV A). We begin the discussion with behavior of the first normal stress difference, followed by the study of the osmotic pressure.
A. Normal stress differences
To obtain the general expressions for the first normal stress difference, the expressions for the parallel and perpendicular normal stresses are first obtained by projecting the stress tensor, Eqs. (17)–(19), onto two orthogonal subspaces corresponding to planes normal and parallel to the external force, [cf. Eqs. (5)–(6)], then subtracting the perpendicular from the parallel normal stress. Only the traceless portion of the normal stresses contributes. The external-force induced, Brownian, and interparticle contributions to the first normal stress difference are
where the components of the hydrodynamic resistance and mobility functions Xαβ, Yαβ, xαβ, yαβ, G, and H are given in Appendix B. We begin the investigation of the first normal stress difference by studying the limit of weak probe forcing.
1. Weak probe force,
To compute the first normal stress difference under weak flow, the asymptotic results for the weakly distorted microstructure, [Eqs. (A9) and (A10)], are inserted into expressions (24)–(26). To leading order, the external-force induced, Brownian, and interparticle contributions to the first normal stress difference are obtained as
where , and are compact expressions for the hydrodynamic functions and are given in Appendix D. The coefficients are associated with the traceless part of the stress tensor, and the terms associated with the coefficients and correspond to the elastic stress.
In the absence of hydrodynamic interactions, , the hydrodynamic stress vanishes. Only the nonhydrodynamic elastic interparticle stress survives. The low-Pe behavior is given, to , as
recovering the result of Zia and Brady . In this limit, normal stresses are generated solely by direct elastic collisions between the probe and surrounding bath particles, which are equally likely in the parallel and perpendicular directions at equilibrium. Thus, even though the individual normal stresses grow as Pe2, their scalar coefficients are identical [cf. Eqs. (20a) and (20b)] and cancel one another precisely, reflecting the weak anisotropy of the structure. This behavior is plotted in Fig. 3(a).
In the opposite limit of strong hydrodynamic interactions, the external-force induced stress and the Brownian stress dominate, while the interparticle stress vanishes. The low-Pe, strong hydrodynamics behavior is given, to , as
which, when combined, grows as , shown by the solid asymptote in Fig. 3(a).
To examine how the strength of the first normal stress difference changes as entropic and hydrodynamic forces evolve from the limit to the limit, each contribution [Eqs. (27)–(29)] is plotted over five decades of κ in Fig. 3(b). The external force-induced contribution is negative for all ranges of repulsion κ, because is larger in magnitude than , for all κ. The uniformly negative value of accompanies the tendency of the externally imposed flow to distort the structure. A stronger distortion along the flow produces the asymmetry required for non-Newtonian rheology, and owes its origin to the more rapid decay of relative radial motion [G(r)] compared to relative transverse motion [H(r)]. For any relative pair trajectory, each of the two orthogonal hydrodynamic couplings, G(r) and H(r), plays an evolving role throughout the pair encounter. The upstream and downstream portions of the trajectory are dominated by G(r) and thus occupy the majority of the encounter. At the middle region of the encounter, H(r), a weaker coupling permits faster relative motion. The upstream and downstream interactions contribute primarily to the parallel stress while the midpoint interactions contribute primarily to the perpendicular stress. In consequence, the resultant structural asymmetry produces a negative normal stress difference.
In contrast, the Brownian and interparticle contributions are both positive for the entire range of κ because, for each, the underlying normal stress components of and are all negative, with larger magnitudes in the perpendicular direction. The difference is weak in this near-equilibrium regime, but nonzero. Both and are O(1), differing only by 15% in magnitude owing to the strength of Brownian motion. The sign of these two entropic contributions is opposite that of the external force-induced hydrodynamic contribution since, at low Pe, entropic forces resist the relative particle motion induced by the flow, i.e., tend to make uniform and symmetric the structure. This relation reflects the tendency of Brownian motion to act against the flow, restoring structural symmetry and thus making zero the total first normal stress difference. Physically, Brownian drift always acts to drive particles to regions of higher mobility. Along the azimuthal angle, this always separates a pair and reduces stress. In the fore and aft regions, the same force drives the pair apart upstream but drives them together downstream.
Similarly, entropic scattering from elastic collisions also tends to make zero the total normal stress difference by enhancing the perpendicular stress. Physically, when bath particles pass by the probe, elastic collisions in the direction perpendicular to the external force drive a pair apart to give an O(1) contribution . However, in the direction parallel to the external force, while weak advective motion of the probe enhances collisions on the upstream face of the probe, once a bath particle passes by and resolves the encounter, there is low probability of a downstream elastic collision . This results in a weaker compared to , and hence .
Overall, to have non-Newtonian rheology, structural asymmetry must be present. However, when flow is weak and Brownian motion is strong, such asymmetry cannot be maintained longer than the time scale of thermal fluctuations—it is relaxed quickly by Brownian motion. Thus, the only way to sustain a finite normal stress difference is to slow down this relaxation. In the limit of strong hydrodynamics, [the left end of the axis of Fig. 3(b)], despite the presence of strong Brownian motion, hydrodynamic interactions slow down the approach and the separation of particles, resulting in a finite first normal stress, recovering the asymptotic coefficient, 0.13. The interparticle normal stress difference is identically zero in this limit because the pair collisions required to produce structural asymmetry are prevented entirely by hydrodynamic deflection. As the repulsion range κ grows, both the external force-induced and Brownian contributions to stress decay to zero approximately as and , respectively, as the no-slip surfaces of particles are kept widely separated. The growth of the effective thermodynamic particle size also permits more frequent hard-sphere collisions, with a longitudinal bias owing to remaining weak hydrodynamic coupling. The elastic stress reaches a maximum near , and then decays rapidly on approaching the limit of weak hydrodynamics, , the right end of the axis.
2. Strong probe force,
When probe forcing is strong, the region upstream of the probe comprises two domains: An outer region where only advection matters, and an -thin boundary layer at the probe surface where diffusion balances advection. That is, the presence of even very weak Brownian motion (or surface roughness) destroys Stokes-flow symmetry of pair trajectories and, as predicted by Batchelor , the Newtonian rheology associated with spherically symmetric structure is lost. The nonequilibrium stress is determined primarily by the dynamics of the boundary layer, and the contribution from the outer region is negligible . The high-Pe first normal stress difference is computed by inserting the boundary-layer solutions for the distorted microstructure, Eqs. (A10) and (A11), for the limits of weak and strong hydrodynamic interactions, respectively, into Eqs. (17)–(19).
The subtraction of the perpendicular normal stress from its parallel counterpart automatically eliminates the isotropic stress; only the traceless portion of the normal stresses contributes to the normal stress difference. In the absence of hydrodynamic interactions, only the interparticle stress matters, giving
where for compactness we have defined . In this dual limit of strong forcing and weak hydrodynamic interactions, the first normal stress difference scales linearly in the external forcing strength Pe, recovering the result of .
In the opposite limit of strong hydrodynamic interactions, the external force, Brownian motion, and interparticle forces all play a role. In this dual limit of and , the external-force induced stress dominates because Brownian motion is weak when flow is strong, and strong hydrodynamic interactions shield pairs from elastic collisions. The hydrodynamic functions in Eqs. (17)–(19) take on their lubrication forms, and the high-Pe asymptotic forms of the interparticle, Brownian, and external-force induced contributions are
where the radial integration is carried out over the stretched radial coordinate , and . For compactness, we have introduced the coefficients and associated with the traceless hydrodynamic functions, and and for the elastic stresslet . The detailed expressions for the , and coefficients are given in Appendix D. Before discussing the scaling of Eqs. (33)–(35), we note again the apparently divergent behavior noted in Sec. IV A for Eqs. (23a)–(23b) where again the region of angular integration carefully avoids an ϵ-small region to maintain convergent behavior and accurate modeling of the physics.
Let us inspect the scaling in Pe of each of these expressions, beginning with the interparticle contribution. The interparticle contribution to the high-Pe first normal stress difference, Eq. (33), scales as . However, the terms corresponding to the dissipative and the elastic stress cancel precisely, giving an overall zero contribution as lubrication forces prevent particle collisions. The Brownian contribution, Eq. (34), is much weaker, . Finally, the external-force induced contribution, Eq. (35), comprises an and an term. Of these two terms, the makes the least contribution because the coefficient ; physically, the stress generated by the relative rotational motion exactly balances (cancels) that generated by relative transverse translational motion . Thus, the term dominates the high-Pe behavior. We summarize these scalings as follows:
The three contributions are combined to give the total first normal stress difference, , plotted in Fig. 4 as a function of the external force, Pe. As shown by the asymptotes for and , hydrodynamic interactions qualitatively change the influence of flow strength on the first normal stress difference. As either hydrodynamic interactions or flow strength grow stronger, entropic contributions weaken, because at small κ interparticle repulsion is weak, and at high Pe Brownian motion is weak. As discussed in Sec. V A 1 for weak probe force, hydrodynamic interactions act to enhance the small Pe first normal stress difference by amplifying the disparity in the duration of longitudinal versus transverse particle encounters, and the disparity in Brownian drift in the two orthogonal directions. In contrast, hydrodynamic interactions act to suppress the first normal stress difference when Pe is large, owing to a deflection of collisions that preserves microstructural symmetry.
3. Arbitrary strength of probe force and hydrodynamics
To explore the regime where the external probe forcing Pe and strength of hydrodynamic interactions κ take on arbitrary values, the first normal stress difference is computed numerically via a finite difference discretization of the Smoluchowski equation (A5)–(A7) (see  for details of this method) and inserting the microstructural solution into Eqs. (17)–(19). Figure 5 shows the total first normal stress difference , along with the external force-induced, Brownian and interparticle contributions, as a function of Pe for four different strengths of hydrodynamic interactions, κ, going from weak (a) to strong (d). The stress is made dimensionless by the ideal (Brownian) osmotic pressure nakT and volume fraction of bath particles , and is scaled by Pe to give an advective scaling of the stress 
In the presence of weak hydrodynamic interactions [κ = 1, panel (a)], the first normal stress difference is primarily set by elastic interparticle collisions, and linear in Pe in the strong-flow limit, because elastic collisions destroy Stokes-flow symmetry and thus produce non-Newtonian rheology, even in the limit , for .
The same elastic interparticle collisions are responsible for a (weak) change in sign of the first normal stress difference, where hydrodynamic interactions are just weak enough to permit collisions, but still strong enough to create the disparity in parallel versus perpendicular hydrodynamic forces that produces a normal stress difference. This behavior is highlighted in the inset of panel (a), where the first normal stress difference (a solid line) is shown alongside the parallel (a dashed line) and perpendicular normal (a dashed-dotted line) components. In previous studies, the sign change was attributed to a transition from dominant Brownian forces to dominant hydrodynamic forces [17,36]. However, the results in the present study suggested a new mechanism causing such a sign change: The interparticle force also produces a sign change. When Pe < 1, elastic collisions are more numerous in the direction perpendicular to the line of the external force (a dashed-dotted line) compared to the parallel direction (a dashed line), resulting in a stronger perpendicular normal stress and thus a positive normal stress difference (a solid line). As flow strength increases, Pe > 1, the microstructure becomes increasingly asymmetric with accumulation of bath particles in front of the probe, giving rise to more frequent collisions in the direction along the line of the external force. The parallel normal stress overtakes the perpendicular, changing the sign of the normal stress difference from positive to negative. In fact, for , the interparticle contribution is at least an order of magnitude larger than the Brownian and the external force-induced components, for the entire range of Pe. Thus, the interparticle force alone can lead to a sign change in the first normal stress difference.
Physically, a sign change in the normal stress differences affects the anisotropic expansion or contraction of the particle phase (cf. Fig. 2). We recall from the normal stresses that both parallel and perpendicular components are negative for the entire range of Pe, representing the tendency of the particle phase to expand in the corresponding direction . A negative first normal stress difference indicates that the parallel stress is more negative (larger in magnitude) than the perpendicular stress, signifying an expansion of the particle phase along the line of the external force, accompanied by a contraction in the perpendicular direction. In contrast, a positive first normal stress difference corresponds to a contraction in the parallel direction and an expansion in the perpendicular direction. While these results may be challenging to measure in hard-sphere suspensions in the low-Pe regime, a suspension of deformable particles may show a more pronounced prolate-to-oblate shape transition.
In panel (b), the interparticle repulsion range shrinks, , and the growing importance of hydrodynamic coupling permits the emergence of the dissipative part of the interparticle stress, . For all Pe, its sign is opposite that of the elastic interparticle stress, , which itself decreases in magnitude as κ shrinks, overall leading to a smaller interparticle stress. The reduction of the total interparticle stress is accompanied by a growth of the external and Brownian stresses. The Brownian contribution is positive over the entire range of Pe. In contrast, the external force-induced contribution is negative. In this competition between the external and Brownian forces, the former acts to expand the particle phase along the line of the probe forcing and contract it in the perpendicular directions, whereas the latter counteracts the motion, acting to restore the equilibrium configuration. Despite the opposite signs of these components, the Brownian contribution is never strong enough to dominate the external force-induced contribution, even in the low-Pe regime, which can be understood as follows. As discussed in the section for weak probe force, when Pe is small, the majority of the Brownian contribution to the normal stresses lies in the trace, and computation of the first normal stress difference automatically eliminates it. That is, for , the evolution of the total first normal stress difference follows that of its external-force induced component, and there is no change of sign in the quantity with Pe.
The interparticle contribution continues to diminish when the repulsion range, κ, decreases [panel (c)], as hydrodynamic and Brownian contributions grow. When hydrodynamic interactions are strong [panel (d)], only the external and Brownian forces matter. At high Pe, the suppressive effect of hydrodynamic interactions becomes clear, as the growth of the normal stress difference decays toward an eventual Newtonian plateau that can be reached when and κ = 0. In this pure-hydrodynamic limit, the microstructure is spherically symmetric and the first normal stress difference vanishes, and the suspension behaves as a Newtonian fluid as predicted by Batchelor . This is consistent with the idea that structural asymmetry is required for non-Newtonian response in suspensions of smooth hard-spheres.
Surprisingly, the interparticle force still plays a prominent role when hydrodynamic interactions are strong: In panel (d), when , the interparticle contribution to the first normal stress difference is indeed negligible, but only because the elastic and dissipative parts are of identical strength and opposite sign. Each is nonzero, meaning that the interparticle force still plays a role in preventing particle overlaps, whereas the external-force induced stress is responsible for preventing particle touching via lubrication interactions. In fact, a pair-trajectory analysis shows that the elastic and the dissipative interparticle stresses act against each other—the former drives particles apart and the latter pulls particles together . Prior studies of normal stress differences in sheared suspensions did not identify the role of interparticle force in the strong hydrodynamics regime, owing to a choice to combine the trace of the dissipative interparticle stresslet with the Brownian stresslet. The interparticle force then appeared to contribute nothing to the first normal stress difference as, by definition, the first normal stress difference comes only from the traceless part of the stresslet. However, with such a grouping, the interparticle portion of the two normal stresses would then no longer be zero, contradicting the dogma that the interparticle force is zero and “replaced” by lubrication forces. Indeed, both must play a role, even in the pure-hydrodynamic limit .
The evolution of the total first normal stress difference is summarized in Fig. 6, where the sign change arising with changes in structural symmetry is shown for several values of κ. When flow is weak, the numerical results recover the weak- and strong-hydrodynamics asymptotic solutions. The first normal stress difference is positive when entropic forces dominate, , and becomes strongly negative when entropic forces are suppressed, . When flow is strong, hydrodynamic interactions suppress the growth of the first normal stress difference, from linear in Pe to sublinear. The approach to the asymptotic strong-flow limit, , grows as , becoming perceptible only when , owing to the influence of the residual Brownian force. When the flow goes from weak to strong, changes sign, but only if hydrodynamic interactions are weak enough to permit interparticle collisions and, in this case, the dominance of perpendicular normal stress over parallel normal stress reverses, owing to an increase in force-aft asymmetry from O(1) at small Pe to at large Pe.
B. Osmotic pressure
The particle-phase osmotic pressure, given by one-third of the negative of the trace of the stress tensor, describes the tendency of particle motion to isotropically expand or contract. In this section, we examine the role of flow and microscopic forces in producing nonequilibrium osmotic pressure. The three contributions to the osmotic pressure found by taking the trace of Eqs. (17)–(19)
where the components of the hydrodynamic resistance and mobility functions Xαβ, xαβ, and G are given in Appendix B. We begin the investigation of the nonequilibrium osmotic pressure by studying the limit of weak probe forcing.
1. Weak probe force,
In the limit of a weak external force, the nonequilibrium osmotic pressure is set by the weakly deformed microstructure, [Eqs. (A9) and (A10)]. Insertion into expressions (40)–(42) gives the external-force induced, Brownian and interparticle contributions to the osmotic pressure
where , and are compact expressions for the hydrodynamic functions, and are given in Appendix D. The coefficients are associated with the isotropic part of the stress tensor, and the term associated with the constant corresponds to the elastic stress. The first-order nonequilibrium osmotic pressure is weakly nonlinear, as indicated by the Pe2 scaling in Eqs. (43)–(45).
In the absence of hydrodynamic interactions, , the external-force induced and Brownian stresses vanish, and only the elastic interparticle stress survives
recovering the result of Zia and Brady .
In the opposite limit of strong hydrodynamic interactions, the external-force induced stress and the Brownian stress dominate, and the interparticle stress vanishes. The low-Pe asymptotes reveal that the first effect of flow is weakly nonlinear
to give a total nonequilibrium osmotic pressure . The total osmotic pressure is plotted in Fig. 7(a), as a function of the strength of the external probe forcing, Pe. Two asymptotes are shown, corresponding to the limit of weak () and strong () hydrodynamic interactions, giving in both limits of κ. Overall, hydrodynamic interactions suppress the osmotic pressure. From an entropic prospective, shrinking the effective particle size increases the available free volume, resulting in a decrease in the osmotic pressure.
To examine how osmotic pressure evolves with thermodynamic size and the strength of hydrodynamic coupling, Eqs. (43)–(45) are plotted for several values of κ in Fig. 7(b). On the left end of the horizontal axis, the repulsion range is short and hydrodynamic interactions are strong (). Hydrodynamic shielding prevents most interparticle collisions, leading to negligible interparticle osmotic pressure. Disturbance flows from Brownian motion and the external force drive the osmotic pressure, with the former dominant in this regime of weak advection. The total pressure asymptotes to . Toward the right end of the horizontal axis, , hydrodynamic interactions are weak and the interparticle repulsion range is long. The growth in effective particle size keeps particles so far apart that the disturbance flows decay over distances much smaller than the thermodynamic size. Thus, as the effective size of particles increases, the importance of both external-force induced and Brownian osmotic pressure decreases, accompanied by a dramatic enhancement in entropic exclusion that permits frequent and numerous particle collisions. Because the increase in entropic stress is more pronounced than the reduction in the external-force induced and Brownian stress, the overall nonequilibrium osmotic pressure is higher in suspensions with weak hydrodynamic interactions, reaching the value of 8/3, recovering the asymptotic result of Eq. (46). Overall, hydrodynamic interactions suppress small-Pe osmotic pressure.
2. Strong probe force,
Here, we examine the nonequilibrium osmotic pressure for asymptotically strong probe forcing. In the limit of weak hydrodynamic interactions, both external and Brownian forces are negligible, and only the interparticle force matters. Inserting the boundary-layer solution for the distorted microstructure , Eq. (A10), into Eq. (19), the high-Pe asymptote of the osmotic pressure obtained here reads
which demonstrates that the osmotic pressure scales linearly in Pe with a prefactor of unity, recovering the result of .
Osmotic pressure in the opposite limit of strong hydrodynamic interactions can be obtained by inserting the solution for the distorted microstructure , Eq. (A11), into Eqs. (17)–(19), and utilizing the lubrication forms of the hydrodynamic functions to obtain
where . Here, , and are compact expressions for the hydrodynamic functions, and are given in Appendix D. The coefficients are associated with the isotropic stress, and corresponds to the elastic stress. The excluded region of angular integration, ϵ, is identical to that discussed in the computation of high-Pe first normal stress difference (cf. Sec. V A).
The scaling of each contribution can be written compactly as
The total nonequilibrium osmotic pressure for both and is plotted in Fig. 8 as a function of the external forcing strength, Pe, which reveals that hydrodynamic interactions suppress nonequilibrium osmotic pressure in the high-Pe limit as well. When , the pressure decays from a linear growth in Pe when hydrodynamic interactions are weak, , to when hydrodynamic interactions are strong, . The latter limit reflects both a direct and an indirect effect of hydrodynamic interactions. The direct effect is an O(1) hydrodynamic coupling represented by the hydrodynamic function in Eq. (49). Hydrodynamic interactions also reduce bath particle accumulation in the boundary layer from for to for , indirectly decreasing the osmotic pressure. Each suppresses the osmotic pressure.
In summary, entropic forces enhance the osmotic pressure in the two asymptotic limits of Pe or, equivalently, hydrodynamic interactions shield particles from collisions, preserving microstructural symmetry, and suppressing the asymmetry required for non-Newtonian rheology. To discover whether this trend persists for all Pe, we next numerically compute the osmotic pressure for arbitrary κ and Pe.
3. Arbitrary strength of probe force and hydrodynamics
To analyze the evolution of the nonequilibrium osmotic pressure for arbitrary strength of external force Pe and hydrodynamic interactions κ, the total nonequilibrium osmotic pressure is computed numerically via the finite difference scheme utilized in Sec. V A. The results are plotted in Fig. 9(a) as a function of Pe, for several values of κ. When probe forcing is weak or strong, the numerical results recover the asymptotic theory in the limits of weak and strong hydrodynamic interactions. Between these two limits, the total osmotic pressure rises monotonically with increasing Pe, as advective distortion of the structure outpaces Brownian smoothing. In Secs. V B 1 and V B 2, we saw that hydrodynamic interactions suppress the osmotic pressure for asymptotically weak and strong forcing. Figure 9(a) shows that hydrodynamic interactions suppress the osmotic pressure for all values of Pe. The data from Fig. 9(a) are scaled by Pe, and this advectively scaled osmotic pressure is plotted in panel (b), where the suppressive effect of hydrodynamic interactions on the nonequilibrium osmotic pressure shows an approach to Newtonian rheology in the pure-hydrodynamic limit. To understand the underlying mechanisms for the evolution of the osmotic pressure, we study the individual contributions arising from various microscopic forces.
In Fig. 10, the external-force induced, Brownian and interparticle contributions of the total nonequilibrium osmotic pressure, , are plotted as a function of Pe for four different values of κ, ranging from (a) weak to (d) strong hydrodynamic interactions. In the case of weak hydrodynamic interactions, plot (a), only the elastic interparticle stress contributes to the osmotic pressure. When scaled diffusively as shown in the inset, the quadratic and linear growth in Pe in the low- and high-Pe limits are recovered. The advective scaling of the data in the main plot of (a) effectively scales out the O(Pe) frequency of particle collisions, giving a high-Pe plateau in the pressure. From the rheological perspective, the nonzero constant high-Pe pressure indicates that non-Newtonian rheology persists in a nonhydrodynamically interacting suspension, even at , owing to the destruction of Stokes flow symmetry by particle collisions.
When hydrodynamic interactions begin to play a role, in plot (b), hydrodynamic shielding results in fewer particle collisions, as indicated by a weaker elastic contribution ; but now the dissipative part of the interparticle stress, , grows. The overall interparticle contribution weakens but still dominates the external-force induced and Brownian contributions over the entire range of Pe.
As repulsion range continues to shrink, in plot (c), the elastic and dissipative parts of the interparticle stress become nearly equal but opposite in sign, leading to a negligibly small interparticle stress. But this entropically induced stress is replaced by the Brownian stress, which dominates the total stress for , until the external force contribution dominates. In the high-Pe regime, the Pe-dependence of the osmotic pressure weakens to a sublinear scaling, owing to the reduction in particle encounters by hydrodynamic shielding. Upon further shrinking of the repulsion range, the limit of strong hydrodynamic interactions is reached [plot (d)]. The interparticle osmotic pressure becomes negligible for the entire range of Pe which, as was the case for the normal stresses  and the first normal stress difference, is due to the precise counter-balance between the elastic and dissipative interparticle stresses. In the pure-hydrodynamic limit, κ = 0 and , the high-Pe asymmetric boundary layer and wake structure becomes fore-aft symmetric. The nonequilibrium osmotic pressure vanishes, recovering Newtonian rheology.
In summary, we find that the low- and high-Pe growth of the nonequilibrium osmotic pressure scale as and , respectively, where δ ranges from 0.799 to 1 from strong to weak hydrodynamics. Hydrodynamic interactions suppress the total osmotic pressure for all values of Pe, signaling a weaker tendency of the particle phase to expand (equivalently, weaker collisions with a fictitious boundary ). In the advective frame of reference, hydrodynamic interactions were shown to weaken the non-Newtonian response far from equilibrium. The weakening effect becomes more prominent as Pe increases, resulting in a vanishing nonequilibrium osmotic pressure in the pure-hydrodynamic limit. Only in a nonhydrodynamically interacting suspension can one measure a nonzero nonequilibrium osmotic pressure at .
We have developed a theoretical model of the non-Newtonian rheology of semidilute dispersions of hydrodynamically interacting colloidal spheres, forming a connection between microscopic forces and microstructural evolution to normal stress difference and osmotic pressure, utilizing active microrheology. The influence of the strength of entropic and hydrodynamic forces on non-Newtonian rheology was studied over six decades of flow strength, Pe, and eight decades of hydrodynamic strength (interparticle repulsion range), κ, alongside limiting behaviors for asymptotically weak and strong hydrodynamic interactions and flow. The first normal stress difference, , and nonequilibrium osmotic pressure, , both evolve in magnitude with changes in strength of flow or hydrodynamic coupling. In addition, exhibits a sign change with growing flow strength Pe, if hydrodynamic interactions are only moderately strong and permit interparticle collisions.
When flow is strong, , a short repulsion range permits particles to experience strong hydrodynamic interactions that shield them from collisions, preserving structural symmetry and suppressing the growth of the first normal stress difference. When flow is weak, , decreasing the repulsion range, in contrast, enhances the first normal stress difference by amplifying both the disparity in the duration of longitudinal versus transverse particle encounters, and the disparity in Brownian drift in the two orthogonal directions. Between the weak- and strong-flow limits, the first normal stress difference exhibits a sign change, but only if hydrodynamic interactions are weakened by surface roughness or other repulsive forces. In this regime, the first normal stress difference is dominated by elastic interparticle interactions; when flow is weak, the stress normal to the flow is stronger than that along it, , but this reverses as flow strength grows and interparticle collisions along the line of forcing grow in frequency.
While in prior work the sign change in the first normal stress difference was ascribed solely to the growing dominance of hydrodynamic over Brownian forces [17,36], here we identified a novel mechanism that interparticle repulsion alone can lead to such sign change. Interrogation of this behavior revealed that, even when hydrodynamic interactions are very strong, the hard-sphere interparticle force still plays a role in the first normal stress difference. The vanishing of the interparticle stress in this limit results from the precise balance between its elastic and dissipative contributions; this physical description cannot be recovered from prior approaches that extract and move the trace of the dissipative interparticle stress to the Brownian stress.
The present work also revealed that entropic repulsion enhances the nonequilibrium osmotic pressure; that is, hydrodynamic interactions act to suppress osmotic pressure, regardless of the strength of flow. The effect is quantitative when flow is weak but, as flow strength increases, O(1) structural asymmetry becomes , leading to a corresponding amplification of the pressure. The suppressive influence of hydrodynamics is qualitative when forcing is strong: The growth of the pressure changes from in the limit of weak hydrodynamics to when hydrodynamic interactions are strong.
This theoretical framework can provide guidance for the interpretation of experimental measurements of non-Newtonian rheology of colloidal dispersions. For example, it can provide a means by which to detect the relative strength of hydrodynamic interactions, a quantity that can be difficult to measure. One would expect a high-Pe plateau of the advectively scaled osmotic pressure and first normal stress difference, if hydrodynamic interactions are weak. For dispersion in which hydrodynamic interactions are strong, this non-Newtonian plateau will give way to a slow, decay, where the slope of the decay can be utilized to infer the interparticle repulsion range.
Many interesting questions remain. While the present model is fully generalizable for arbitrary probe- to bath-particle size ratio, the details of this behavior were not explored here. In practice, active microrheological studies often utilize probes with a range of sizes relative to the bath particles [37,38], and recent studies have shown that the size ratio of the probe and bath particles exerts a pronounced influence on rheological measurements [28,39–41]. In an active microrheology system with small or large probe, size ratio exerts only a quantitative effect on viscosity , force-induced diffusion , and suspension stress . However, it was recently shown by Hoh and Zia  that when hydrodynamic interactions matter, changes in probe size relative to bath particles reveal a pronounced, qualitative change in force-induced diffusion. Future studies of suspension stress should interrogate this dependence.
The assumption of semidiluteness is useful for making analytical progress, but three-body and higher-order interactions importance in dense suspensions lead to concentration-dependent effects on rheology. Our recent results obtained from Stokesian dynamics simulations reveal the qualitative influence of such interactions on structural symmetry and non-Newtonian rheology [42,43]. Recently developed hydrodynamic functions for dense suspensions  may provide a bridge to concentrated theory.
Finally, the present work motivates a connection to the sign change in the first normal stress difference observed in sheared suspensions. In the plane transverse to forcing in active microrheology, the axisymmetric external force results in particle accumulation and depletion that lie along the line of the forcing. In the corresponding flow/flow-gradient plane of a shear flow, particles accumulate and deplete along the compressional and extensional axis, respectively, giving an antisymmetric structure. Despite the difference in the structures, Zia and Brady  showed that the first normal stress difference (as well as the normal stresses and osmotic pressure) between two flows demonstrate qualitatively similar evolution in the absence of hydrodynamic interactions. As pointed out in that work, a rotation of quadrants makes structures between two flows nearly identical. It is possible that a “normal stress difference” defined by coordinates aligned to the compressional and extensional axes in a shear flow might reveal, more clearly, behavior common in both flows. Reconciling the difference in structures between two flows will provide insight into devising a unified means to control the evolution of normal stress differences and other rheological behaviors.
This work was supported in part by Office of Naval Research Young Investigator Award (N00014-14-1-0744). The authors thank Christian Aponte-Rivera and Dr. Nicholas J. Hoh for many helpful discussions.
APPENDIX A: EVOLVING MICROSTRUCTURE
In this section, we present the formulation of the governing equation of the pair distribution function, and recapitulate the results in the asymptotic limits of weak and strong probe forcing and hydrodynamic interactions [18,21,22,25]. Readers are referred to, e.g.,  and , for a detailed study of the microstructure for arbitrary strength of forcing and hydrodynamics.
The Smoluchowski equation governs the spatio-temporal evolution of the N-particle probability distribution function. In the dilute limit, where only pair interactions are important, the N-particle probability density becomes the pair probability density of finding the probe and a bath particle centered at and , where subscripts 1 and 2 denote quantities associated with the probe and bath particle, respectively. To analyze the change in the microstructure around the probe, we adopt a frame of reference moving with the probe centered at , where bath particle positions are now identified relative to probe position, (cf. Fig. 1). The steady-state pair-Smoluchowski equation in this reference frame governs the configuration of the bath particles around the probe as it evolves with advective and diffusive flux [21,22]
where is the pair distribution function defined as ; and the relative translational velocity and diffusion tensor are given by and . There is no relative flux at contact for hard spheres and no long-range order
Here, is the unit vector pointing along the line of centers from the probe to a bath particle. The equations are made dimensionless as
The governing equations in their dimensionless form are thus
where the subscripts on , and are dropped for brevity.
To interrogate the nonequilibrium response of suspensions, we define the distortion to the microstructure, , as where geq is the equilibrium microstructure. Solutions of Eqs. (A5)–(A7) obtained originally by Squires and Brady  and Khair and Brady  were utilized to compute the microviscosity, and by Zia and Brady  to compute force-induced diffusion. We recovered these results in the present work in order to utilize them in the computation of the stress tensor. We briefly recapitulate the asymptotic results for ease of reference.
1. Weak probe force, , and weak hydrodynamics,
In the limit of weak external force and weak hydrodynamic interactions, we follow the procedure established by Squires and Brady  and performed a regular perturbation expansion in Pe to obtain the leading order structural disturbance as , where . To obtain higher order disturbances, we followed the approach of Khair and Brady , who recognized the singular nature of the problem and derived a matched asymptotic solution for the structure as
2. Weak probe force, and arbitrary strength of hydrodynamics, κ
In the limit of weak external force, the singular nature of the governing equation requires a singular perturbation expansion in Pe, as originally shown by Khair and Brady . Following this procedure, we obtain the structural disturbance to for arbitrary strength of hydrodynamic interactions as
where , and are governed by a set of ordinary differential equations, solved numerically, and give the radial dependence of the structure corresponding to monopolar, dipolar and quadrupolar distortions, respectively. The resulting microstructure is then utilized for in the expressions for the stress derived in the present work for the linear-response rheology in Sec. V.
3. Strong probe force, and weak hydrodynamics,
In the limit of strong external force, a scaling analysis of the Smoluchowski equation revealed that downstream of the probe there is a depletion wake with O(1) particle accumulation; while the buildup of particles in the boundary layer upstream of the probe depends on the strength of hydrodynamic interactions: , where as the influence of hydrodynamics evolves from weak (, δ = 1) to strong (). In the limit of weak hydrodynamic interactions, we recover the result of 
4. Strong probe force, and strong hydrodynamics,
In the limit of strong external force and strong hydrodynamic interactions, we follow the procedure established by Khair and Brady  to obtain the microstructure:
Here, f0 is a constant determined by matching the inner and outer solutions, Γ is the gamma function, and is the first confluent hypergeometric function (Kummer's function). The functions , and are the leading order expansions of the hydrodynamic mobility functions G(r), H(r), and , respectively, at contact . The polar-angle variation in boundary-layer thickness is set by the parameter . The result is inserted into the expressions for the stress derived in this work.
APPENDIX B: HYDRODYNAMIC RESISTANCE AND MOBILITY FUNCTIONS
In this section, the hydrodynamic functions required to obtain the stresslets and the average stresses [cf. Eqs. (13)–(19)] in Sec. IV A are presented. These functions depend only on the relative separation between a pair of particles, r, and the dimensionless repulsion range, κ. The hydrodynamic resistance function that couples the translational velocity of particle β to the induced stresslet on particle α, ; and the hydrodynamic resistance function that couples the rotational velocity of particles β to the induced stresslet on particle α, , are defined, respectively, as [30,31],
where ri is the unit vector along the line of centers of particles α and β, δij is the identity tensor, ϵijk is the Levi-Civita tensor, and Xαβ and Y αβ govern the motion of particle α and β along and transverse to their line of centers, respectively. Two remarks are made regarding the above resistance functions. First, gives the complete relation between particle kinematics and the induced stresslet since the stresslet is not restricted to be traceless. The hydrodynamic function , which is associated with the trace of the stresslet or osmotic pressure, has been grouped into . Second, is traceless, signifying that rotations of spheres do not contribute to the trace or osmotic pressure .
The hydrodynamic mobility function that couples the force on particle β to the induced translational velocity of particle α, ; and the hydrodynamic mobility function that couples the force on particle β to the induced rotational velocity on particle α, , are defined, respectively, as 
where xαβ and yαβ govern the motion of particle α and β along and transverse to their line of centers, respectively.
APPENDIX C: LOW- AND HIGH-Pe ASYMPTOTES OF THE PARALLEL AND PERPENDICULAR NORMAL STRESSES
Here recapitulate the low- and high-Pe asymptotes of the nonequilibrium parallel and perpendicular normal stresses obtained by Chu and Zia . In the asymptotic limit of weak probe forcing, , the external-force induced, Brownian and interparticle contributions to the nonequilibrium parallel normal stresses were obtained as 
and for the perpendicular normal stress
where 's and 's are derived from the traceless and the isotropic parts of the stress tensor, respectively, and the term associated with the constants and correspond to the elastic stress. The detailed expressions for 's, 's, , and are given in Appendix D.
In the opposite limit of strong probe force, , the hydrodynamic functions in Eqs. (17)–(19) take on their lubrication-limit values; the expressions for the external-force induced, Brownian, and interparticle components of the nonequilibrium parallel normal stress were obtained as 
and for the perpendicular normal stress
where the radial integration is carried out over the stretched radial coordinate and . The coefficients 's and are associated with the traceless hydrodynamic functions, 's with the isotropic ones, and and for the elastic stresslet . The detailed expressions for the 's, 's, , and are given in Appendix D.
APPENDIX D: COEFFICIENTS OF THE LOW- AND HIGH-Pe ASYMPTOTES OF THE FIRST NORMAL STRESS DIFFERENCE AND THE OSMOTIC PRESSURE
In this section, the coefficients comprise the expressions of the low- and high-Pe asymptotes of the nonequilibrium normal stresses, first normal stress difference and the osmotic pressure are presented. The coefficients in the low-Pe asymptotes are composed of hydrodynamics functions, and thus only have radial dependence. With reference to Eqs. (27)–(29), (43)–(45), and (C1a)–(C2c), the coefficients associated with the low-Pe asymptotes read
where 's and 's are associated with the traceless and isotropic parts of the stress, respectively, and , and correspond to the elastic stresslet .
The coefficients in the high-Pe asymptotes are form the lubrication forms of the hydrodynamics functions, and they are constants with no radial and angular dependence. With reference to Eqs. (32)–(33), (48)–(51) and (C3a)–(C4c), the coefficients associated with the high-Pe asymptotes read
where 's and are associated with the traceless part of the stress, 's with the isotropic part, and , and correspond to the elastic stresslet. In Eqs. (D10)–(D19), are the coefficients of the lubrication expressions of the hydrodynamic functions defined in, e.g., [30,31,33,34], where with “11” representing single-particle (self) interaction and “12” representing two-particle (probe-bath particle) interaction; indicates the order of the coefficient in the lubrication expressions; , indicates the longitudinal or the transverse component of a hydrodynamic function; and are the notations of the hydrodynamic resistance functions defined in Sec. IV A.