In this work, the viscoelastic behavior of a complex structured liquid in a continuous squeeze flow is analyzed. This flow is simulated allowing a continuous flow of liquid into the narrow gap between two circular plates though the lower plate. The complex liquid is characterized by the exponential structure rheological (ESR) constitutive equation, which is a generalized exponential thixotropic-elasto-viscoplastic-banded model, chosen for this study due to its capacity to predict non-Newtonian and complex behavior such as (i) shear-thinning, (ii) shear-thickening, (iii) yield stress, (iv) thixotropy, (iv) rheopexy, and (v) shear banding flow behavior. The exponential rheological equation of state is defined by a class of exponential Phan-Thien–Tanner-type models, which includes specific cases documented in the literature. The viscoelastic, kinetic, and structural mechanisms in the ESR model are characterized by the association of non-dimensional numbers to each mechanism. To solve the set of non-linear partial differential equations, a perturbation scheme is suggested, based on a small parameter that represents the ratio between two characteristic lengths. At zeroth order (neglecting the inertial mechanisms of the momentum equation), it is found that the normal force on the upper disk is directly related to shear dependent viscosity (thixotropy, shear-thinning, shear-thickening, yield stress behavior, and concentration effects). At first order, the normal force is related to the effects of the elasticity, and it is parallel to the first normal stress difference associated with the elasticity of the ESR structured fluid.
I. INTRODUCTION
The analysis of fluid–structure interaction (FSI) is an interdisciplinary field that plays a key role in understanding the complex dynamics between fluid and structure.1
A. Squeeze flow film
One of the most fundamental FSI phenomena in fluid mechanics and rheology is found in the squeeze flow.2 The most common system consists of two parallel plates with a radius of r = a, separated by a distance z = h.3 The liquid is placed on the lower plate, and as the upper plate approaches the fluid, which induces radial flow due to the squeezing mechanism.4 This compression-driven flow system can have various geometric configurations, such as (i) disks with a radius of r = a and separated by a distance h,3 (ii) spheres separated by a distance h,5,6 and (iii) a combination of spheres and disks.3,4 From a rheological perspective, this mechanism is challenging to describe due to the presence of shear strain at the walls, extensional components at the center of the squeeze experiment, vorticity, and inertial mechanisms associated with the acceleration of the upper plate relative to the fluid.7,8 The study of the squeeze flow film raises three important issues that deserve attention from rheologist worldwide: (A) The geometric configurations of the system,7 (B) The experimental and material conditions of the system,9 and (C) The mathematical and physical theoretical approaches, including computational methods, theoretical approximation, asymptotic analysis, and numerical methods.10,11
B. Applications of the squeeze flow film
From an experimental perspective, one of the most important aspects of squeeze flow is the measurement of the material functions.7 In this context, the force required to maintain the flow of the liquid flow between the plates depends on the materials properties of the fluid and the velocity of the plate squeezing the flow.12 Additionally, it has been shown that the inertial mechanisms are minimal, in contrast with the elastic effects associated with the load-bearing capacity of the squeeze flow film.13 The squeeze flow film in isothermal conditions for both Newtonian and non-Newtonian fluids has attracted significant interest due to its numerous applications, including (i) electroosmotic pumps,10 (ii) biorheology,14,15 (iii) rheometry and rheology devices,11 (v) enhanced oil recovery,16 (vi) electrorheological fluids,10 (vii) motors bearing,17 (viii) lubrication,18 (ix) fibers,19 (x) ceramics,20 (xi) thermoplastic composites,21 (xii) dental composites resin,22 (xiii) food,23 and (ix) coagulation system and medicine.24 Remarkable examples of the squeeze flow theory can be found in the formation of foams, where bubble boundaries expand biaxially and shrink in thickness in a similar way as squeezing films (Ref. 7 and references therein). In medicine and biological sciences, the functioning of medical valves, diarthrodial joints, knee prostheses, and biological superfibers, such as spider silk are important.25,26 Interesting transport phenomena and rheology occur during the formation of the collagenous film in tissue engineering.27,28 The compression of food between the tongue and palate can be approximated (for some solid food) as a squeeze flow.29 In biomimetics, the formation of the spider silk in its spinneret can be modeled as a double exponential geometry biological extruder.26 In polymer science processing, squeeze flow phenomena are observed in fabrication operation, such as stamping, injection molding, sheet forming, and rheometric measurements using plastometers that incorporate parallel-disk squeeze flow geometry.30,31
C. Mathematical and physical approaches
From a mathematical perspective, the aforementioned theoretical works involve several mathematical techniques, which include the following: (i) analytical approaches, (ii) regular perturbation techniques, (iii) numerical finite difference schemes, (iv) numerical finite element methods, (v) numerical finite volume methods, (vi) hybrid schemes combining perturbation techniques and numerical approaches, and (vii) computational software (e.g., COMSOL Multiphysics). Phan-Thien32 analyzes the small strain oscillatory squeeze film flow of simple fluids. The fluid is characterized by a functional of the strain history, involving the Cauchy–Green tensor, which reduces to a general integral linear viscoelastic rheological equation of state. The system is solved with a stream function, which reduces the problem to a biharmonic equation. Assuming, non-slip conditions, both analytical and asymptotic solutions were found. Phan-Thien and Tanner18 studied the lubrication mechanisms in a Boger fluid (nearly Newtonian), characterized by an Oldroyd-B constitutive equation. Assuming isothermal conditions and a lubrication approximation, the coupled partial differential equation system was solved using an exact flow kinematic solution and a variant of Galerkin method (One-Galerkin method). Gartling and Phan-Thien33 studied a plastic fluid in a geometric in a parallel-plate plastometer configuration. The system was solved using a programing code based on a Galerking-finite element mesh for the numerical simulation. The in-silico experiments were calibrated with experimental data for complex fluid exhibiting yield stress mechanisms. Phan-Thien and Walsh34 investigated similarity solutions in a squeeze flow film using an Oldroyd-B fluid. Assuming inertia-less flow and that the squeeze velocity varies exponentially with time, a similarity solution exists. Additionally, their results showed that there is a critical Weissenberg number above which at least one component of the stresses unbounded with time. Phan-Thien et al.35 examined the squeeze flow film of an ideal elastic liquid using a constitutive equation similar to Oldroyd-B, and their numerical scheme was compared with rheometric data. According with their results, the non-linear model admits an exact solution. The non-linear case was solved using an explicit second-order finite difference scheme. The elastic mechanisms were well described with this theoretical-computation scheme. Phan-Thien et al.36 studied the effect of a viscoelastic fluid with both solvent and polymer contribution, using a modified Phan-Thien-Tanner model with shear-dependent viscosity (Carreau model). The system was solved using the exact flow kinematic function and a boundary element method (BEM). The algebraic equations were solved using Gaussian elimination and predictor-corrector Rung–Kutta numerical methods. The squeeze flow film problem was addressed with a fourth order Runge–Kutta scheme. According to their simulations, load enhancements are related to the overshoots and the materials properties of the system. Phan-Thien and Low37 investigated the squeeze flow of a viscoelastic fluid using a lubrication model. The system was modeled by splitting the solvent and polymer contribution, which allows for a Carreau-type viscosity and stress overshoot a startup of shear flow. This research compared the full numerical solution obtained with boundary element method, to the classical lubrication approximation, described by a fourth-order coupled partial differential equation in time and one spatial coordinate. The lubrication squeeze flow problem was solved using a finite difference scheme, and the results showed that the lubrication is a good alternative to full numerical analysis. Phan-Thien38 studied the oscillatory squeeze flow film of a viscoelastic solid, using a neo-Hookean rubber-like finite deformation and the upper convected Maxwell models. The research, demonstrated that the load can exhibit a significant degree of asymmetry, which is largely due to the rubber-like elasticity in the response, which results in a larger force during the downward phase of the displacement. The system admits an analytical solution, and the general scheme was implemented using a central difference numerical scheme. The non-linear coupled differential equations were solved using explicit leap-frog and Euler methods. Phan-Thien et al.39 analyzed large-amplitude oscillatory squeeze flow data for a complex biological material. The system was characterized by two rheological models: a bi-viscosity Newtonian model and a non-linear Maxwell model. The non-linear Maxwell model better matched the experimental data. The non-linear response of the material in large-amplitude oscillatory flows is primarily attributed to shear thinning.
D. Continuous squeeze flow film
In contrast to traditional squeeze flow systems, various approaches have been investigated to avoid the inertia of the upper plate approaching the fluid.40,41 Oliver et al.42 proposed a new system called “continuous squeeze flow film.” In this system, the fluid is continuously injected at a constant rate into the gap between two parallel circular plates in close proximity, through a uniform array of the holes in the lower plate.43 This system has the advantage of not considering the inertia of the plates, allowing the system to be treated as a steady-state problem. Oliver and Shahidullah44 studied both normal and reverse squeeze flow in a continuous flow system with Newtonian and Non-Newtonian liquids.45 The fluid is injected through the lower plate, with neither of the plates moving, and the liquid exudes from 1580 uniformly distributed holes in the plate surface. All tests were performed at a temperature of 24 °C.
Waters and Gooden46 analyzed the flow in continuous-flow squeeze film from an analytical perspective. The inelastic liquid is characterized by the power-law model, and the system was solved using perturbation techniques at zero and first orders. They found that the force at zeroth and first orders depends on the volumetric flow and material
Properties, with inertial mechanisms contributing to the normal force on the upper plate. Waters and Gooden47 studied the flow of a viscoelastic fluid in a continuous squeeze flow film. The liquid is characterized by a non-linear equation Oldroyd-B rheological equation of state.48,49 The system was solved using a perturbation series at zeroth and first orders, and the normal force was obtained using a numerical scheme. At zeroth order, viscous mechanisms are relevant and depend on the material parameters of the constitutive equation. At first order, elastic (storage) mechanisms are significant, while inertial mechanisms do not contribute to the normal force.
E. Complex fluids
Structure fluids are complex systems that exhibit intriguing steady and unsteady behavior under flow.50 These systems are found in various industries and are foundational in biomimetics.51,52 The physics of these systems is a powerful tool for explaining biological phenomena, such as (i) polypeptides, (ii) cellulose, (iii) blood, (iv) DNA, (v) natural fibers, and (vi) synthetic materials (e.g., molten polymer nanocomposites, surfactant solutions, liquid crystals polymers, bio-composites). These materials often exhibit crystalline order, anisotropy, defects, viscoelasticity, and flow aligning structures.53–55 Additionally, the mechanical response of the human ear has been studied through flexoelectric membranes embedded in non-Newtonian fluids,56 and the effects of the solvent depletion on electrokinetic energy conversion in non-linear viscoelastic fluids.57 Thixo-elasto-visco-plastic fluids can be described using constitutive equations that take into account build-up and break-down kinetics by flow.56 These models incorporate sequences of self-assembly, structural changes, kinetic processes under flow, and mass transfer.54 The characterization and modeling of these phenomena have been extensively studied by various authors using sophisticated rheological equations of state, such as (i) the Pom–Pom model,58 (ii) the extended Pom-Pom model by Verbeeten et al.,59 which incorporates a non-zero second normal stress difference in simple shear flow to avoid discontinuities at steady-state elongation and the unboundedness of the orientation equation at high strain rates. A sophisticated non-linear model introduced by Giesekus, accounts for mobility mechanisms associated with anisotropic Brownian motion and/or anisotropic hydrodynamic drag of the polymer molecules.60 The Phan-Thien Tanner (PTT) and extended-generalized Phan-Thien–Tanner equation (g-PTT) have been studied in simple shear flow and classic inhomogeneous Poiseuille flow by Ferras and Afonso.61 These models serve as the basis for computer simulations based on relaxation time spectra or transient-network theory.61,62 Boek et al.63 proposed a model, which separates the contributions of solvent and polymer, consisting of a coupled system of non-linear equations, including the upper-convected Maxwell equation and a kinetic equation for structural breakdown. Building on the Phan-Thien–Tanner (PTT) formalism, Herrera-Valencia et al.64 developed a new model based on structural energy changes the induced by flow. This model utilizes predator-prey dynamics and population balance principles biomathematics and bioengineering.65 The kernel function is based on structural activation energy to alter microstructure, dependent on thermodynamic viscoelastic dissipation work that breaks down microscopic structures.66
F. Wormlike micellar structures
Viscoelastic surfactants are characterized by an entangled network of large, worm-like micelle structures.67 These structures break and re-form during flow, exhibiting complex rheological behavior.68 Predicting the flow behavior of viscoelastic surfactants using constitutive equations has been a challenging issue.53 These systems display Maxwell-type behavior in small–amplitude oscillatory shear flow and shear stress saturation in steady simple shear, which leads thixotropy and shear banding flow.69 In the non-linear viscoelastic regime, elongated micellar solutions also exhibit notable features, such as a stress plateau in steady shear rate accompanied by slow transients to reach steady state70 and normal stresses upon inception of flow.71 The plateau, where the stress becomes independent of shear rate has been interpreted as a purely mechanical instability leading to shear banding.72 Viscoelastic surfactants have been used as rheological modifiers in coating process73 and in enhanced oil recovery operations,74 particularly in the fracturing of subterranean formations.75 Additionaly, oil extraction can be achieved by hydraulically inducing fractures in rock formations.76 Water-based fracturing fluids have typically used high molecular weight water-soluble polymers.77 Recently, polymer-free fracturing fluids, based on viscoelastic surfactants, have been developed for fracturing underground formations.78 Fluids composed of viscoelastic surfactants can enhance fracture conductivity compared to polymer-based fluids.79 Hydraulic fractures are characterized by having one dimension the width being very small compared to the lateral dimensions, height, and length.80 This characteristic aligns with the lubrication, approximation, which assumes that flow is well approximated by a locally uniform flow between parallel plates separated by the local fracture width.81
Despite their significant technological applications, it is surprising that the rheological modeling of complex fluids, such as viscoelastic surfactants (e.g., worm-like micellar aqueous solutions), and their complex behavior (e.g., thixotropy, rheopexy, shear-banding flow) in lubrication squeeze films have been scarcely studied in the current literature. These rheological phenomena represent a test for new constitutive equations and enhancing rheometric techniques, which motivates the present investigation.56,61–64,70,82–84
In this regard, the main objectives of this work are
-
To predict the flow behavior of a continuous squeeze film of a complex liquid modeled by the exponential structure rheological (ESR) constitutive equation of state.64
-
To analyze the non-Newtonian mechanisms and their interactions with the material properties associated with the thixotropic-elasto-viscoplastic mechanisms of the fluid, through dimensionless groups that represent the physical characteristic in the system.
-
To study the effect of viscoelastic surfactant concentration by using rheometric data of an aqueous worm-like micellar solution (cetyl trimethyl ammonium tosylate) to predict the flow behavior for various micellar concentrations.83
This study is organized as illustrated in Fig. 1.
Section I provides an introduction to the problem and a review of previous work. Section II covers continuity, momentum transport, and rheological equation of state (ESR model). Section III addresses the physical constraints and problem formulation. Section IV presents the scaling rules and dimensionless groups. Section V presents the regular perturbation technique applied to the ratio between axial and radial characteristic lengths, up to zero and first orders. In Sec. VI, analytical expressions and numerical solutions for radial and axial velocities, volumetric flow rate, pressure gradient, and normal force at the upper plate are provided, as a function of the shear-thinning/thickening, thixotropy/rheopexy, yield-stress, wormlike micellar data. Finally, Sec. VII includes concluding remarks and future work (Figs. 2–4).
II. THEORETICAL EQUATIONS
A. Mass and momentum transfer equations
B. Exponential structural rheological constitutive equation (ESR)
C. Exponential kernel viscoelastic dissipation function
D. Particular cases
1. Small deformations of ESR model
2. Moderate and high shear strain deformations
III. PROBLEM FORMULATION
The physical system consists of two disks with a radius r = a and a gap z = h, as shown schematically in Fig. 1.46,47 The lower disk has an array of holes uniformly distributed. The structured fluid (dilute worm-like micellar solution) flows from the lower to the upper plate, followed by a radial flow. The key objective is to calculate the normal force by the complex fluid on the upper plate.
To model this problem, the following assumptions are made:
-
It is assumed that the system is in a stationary state, meaning that the time derivatives of the kinematic and dynamic variables are set to zero, i.e., ∂v/∂t = ∂D/∂t = ∂σ/∂t = 0.
-
Gravitational forces are neglected compared to other forces, i.e., g ≅ 0.
-
The structured fluid is considered incompressible, i.e. the density does not vary with position or time. As a consequence, the material derivative of the density is zero, i.e., Dρ/Dt = 0⇒ ∇ ⋅ v = 0.
-
To describe the geometry of the flow system, cylindrical coordinates (r, θ, z) are chosen, defined with respect to an origin at the center of the lower disk.
-
Angular symmetry is assumed, meaning the angular derivatives of all kinematics and dynamic variables are set to zero, i.e., ∂v/∂θ = ∂D/∂θ = ∂σ/∂θ = 0.
-
Relaxation mechanisms are small in dilute and semi-dilute complex liquids, i.e., 1 + λD f(Tr (σ ⋅ D))∂/ ∂t ≅ 1
-
There is not mass-flux contribution. i.e., J = 0.
-
There are not instabilities associated to the shear banding flow, λB = 0
-
The liquid is characterized by the linearized ESR rheological equation of state.
The kinematic and dynamic tensors in the system are (i) spatial field vector, (ii) spatial velocity gradient tensor, and (iii) viscoelastic stress tensor. These mathematical objects can be represented in the following matrix form:
IV. SCALING RULES
A. Non-dimensional variables
To simplify the momentum and rheological equations, the following dimensionless variables for the axial and radial coordinates, radial and axial velocities, components of shear-stress, pressure gradient, total stress, fluidity, and second invariant of the shear-strain are introduced as follows:
B. Mass conservation
C. Non-dimensional groups
In terms of structure, shear thinning behavior can be conceived as a structure breakdown process, where the system transitions from higher to lower structure states, while shear thickening is the opposite mechanism. The three dimensionless groups [Jr∞, We∞, φr] form a material parametric 3D space.
D. Continuity and momentum components
E. Total stress component
F. Rheological components
V. ASYMPTOTIC ANALYSIS
A. Regular perturbation scheme
B. Taylor series and linear approximation
It is important to highlight the following points:
-
The zeroth order corresponds to the steady-state and Poiseuille flow.
-
The first order in the parameter α1 represents the local linear approximation, incorporating the elastic mechanisms through the first normal stress difference.
C. Zeroth-order theory O (α0)
1. Material functions
2. Radial component of the field velocity
3. Volumetric flow rate
4. Axial component of the field velocity
5. Pressure gradient
-
Choosing a numerical value for the dimensionless numbers {We∞, Jr∞, φr} and m related to the non-homogeneity of the system.
- Choosing the analytical form of the radial velocity distribution u(r), associated with the distribution of the holes in the lower plates. The continuous function u (r) must satisfy the normalized Eq. (21), so the mathematical u (r) function is given by
The homogenous case appears when m = 0, i.e., uH = u (r = 0) = 2. Physically, this means that the volume of the complex non-Newtonian fluid entering through the orifices of the lower plate is the same, and there is no distribution in the system.
-
Solve the non-analytical expression for the pressure gradient dp0/dr. Given the steps A-B, and chosen and numerical value of r ε [0,1]⊆ R, the system can be solved with an iterative procedure such as the Newton–Raphson method.47
- The starting value, based on the Newtonian solution, is given by Eq. (41), convergence is found to be very rapid in this case.
6. Normal force
a. Lower shear strain
b. High shear strain
7. Yield stress
D. First-order theory O (α1)
1. Radial component of the field velocity
a. Low shear rate X
b. High shear rate
2. Axial component of the field velocity to first order
a. Low shear rate
b. High shear rate
3. Pressure gradient
a. Low shear rate
b. High shear rate
4. Normal force
a. Low shear rate
b. High shear rate
VI. SIMULATIONS AND RESULTS
In this section, the simulations of the ESR constitutive equation in a continuous squeeze flow film at zeroth and first orders are presented. The following mechanisms are explored: (i) shear thinning, (ii) shear thickening, (iii) thixotropy, (iv) rheopexy, (v) yields stress, and (vi) the weight of the worm-like micelles. The dimensionless numbers used in the simulations are given by: {We∞, φr, Jr∞}. The analytical and numerical equations that are used in the simulations are (i) the axial velocity [Eq. (56)], (ii) the normal force to zeroth order in α (67), and (iii) the normal force to first order in α (67). The axial velocity is determined analytically, while the normal forces are computed numerically. The non-linear pressure gradient is calculated using the Newton–Raphson method, and the numerical integral is evaluated using a quadratic Gaussian technique. The results were programed in Wolfram Mathematica using institutional license codes. The numerical values of the material properties are taken from Herrera-Valencia et al. (2017).83 The dimensionless numbers as a function of the weight concentration of the worm-like micelles solution are given in Table I. In what follows, the first analysis will be the zeroth-order in alfa, and the second analysis will be the first-order contribution. To facilitate the calculation, the normal force will be normalized by the Newtonian value, i.e., F0/FN∞ and F1/FN∞, where FN∞ =3πa3⟨V⟩(φ∞)−1/h2.
Material properties vs mechanisms . | Irreversible thermodynamical viscoelastic work constant β0 (Pa−1 s) . | Interfusion structure-momentum relaxation time λD (s) . | Elastic moduli J0 =1/G0 (a−1) . | Fluidity at low shear rate φ0 =1/η0 (Pa−1 s−1) . | Fluidity at high shear rate φ∞ =1/η∞ (Pa−1 s−1) . |
---|---|---|---|---|---|
Shear-thinning | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 0.20 |
1 | |||||
Newtonian | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 0.0053 |
Shear-thickening | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 0.0002 |
0.002 | |||||
Yield stress | 5.46 × 10−6 | 0.14 | 0.0054 | 0 | 10.5 |
0.0001 | |||||
0.001 | |||||
0.0053 | |||||
1 | |||||
10.5 | |||||
Thixotropy | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 1 |
3.9 × 10−6 | 0.10 | ||||
3.9 × 10−7 | 0.01 | ||||
3.9 × 10−8 | 0.001 | ||||
Concentration CTAT (wt. %) | |||||
5% | 30 | 0.12 | 0.0240 | 0.0275 | 19.8 |
10% | 10 | 0.33 | 0.0057 | 0.0061 | 15.0 |
15% | 4 | 0.38 | 0.0072 | 0.0050 | 12.6 |
20% | 1.8 | 0.42 | 0.0016 | 0.0042 | 12.0 |
Material properties vs mechanisms . | Irreversible thermodynamical viscoelastic work constant β0 (Pa−1 s) . | Interfusion structure-momentum relaxation time λD (s) . | Elastic moduli J0 =1/G0 (a−1) . | Fluidity at low shear rate φ0 =1/η0 (Pa−1 s−1) . | Fluidity at high shear rate φ∞ =1/η∞ (Pa−1 s−1) . |
---|---|---|---|---|---|
Shear-thinning | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 0.20 |
1 | |||||
Newtonian | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 0.0053 |
Shear-thickening | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 0.0002 |
0.002 | |||||
Yield stress | 5.46 × 10−6 | 0.14 | 0.0054 | 0 | 10.5 |
0.0001 | |||||
0.001 | |||||
0.0053 | |||||
1 | |||||
10.5 | |||||
Thixotropy | 5.46 × 10−6 | 0.14 | 0.0054 | 0.0053 | 1 |
3.9 × 10−6 | 0.10 | ||||
3.9 × 10−7 | 0.01 | ||||
3.9 × 10−8 | 0.001 | ||||
Concentration CTAT (wt. %) | |||||
5% | 30 | 0.12 | 0.0240 | 0.0275 | 19.8 |
10% | 10 | 0.33 | 0.0057 | 0.0061 | 15.0 |
15% | 4 | 0.38 | 0.0072 | 0.0050 | 12.6 |
20% | 1.8 | 0.42 | 0.0016 | 0.0042 | 12.0 |
A. Zeroth order mechanisms: O (α0)
1. Shear-thinning/thickening
Figure 5 shows the normalized force to zeroth order F0/FN∞ at the upper plate in the continuous squeeze flow to zero order in α vs the Weissenberg number We∞, as a function of the dimensionless number φr for values: = (i) 0.001, (ii) 0.01, (1), (5), (10). The other dimensionless number used in the simulation is Jr∞ =1. It is interesting to note that the force is proportional to the viscosity of the system, so the shape of the curves observed in Fig. 5 is the same as the viscosity of the system (shear thinning, Newtonian, and shear thickening) as follows:
-
At low Weissenberg number We∞, all the curves show a constant plateau independent of the Weissenberg number (Newtonian zone), for φr equal to 1, the force remains constant independent of the Weissenberg number We∞ (Newtonian case).
-
At intermediate Weissenberg number We∞, a power law region is shown for for φr > 1 (shear thinning) and φr < 1 (shear thickening). In this power law region, the systems experience constant structural changes, with flow and rheology dominated by the dimensionless numbers {Jr∞,φr}, which are associated with the yield stress mechanism, bulk-elastic shear forces, and the fluidities at low and high infinity Weissenberg numbers. From a mathematical perspective, in the shear thinning mechanism, the system displays a monotonically decreasing behavior, while in the shear thickening forces, a monotonically increasing behavior is observed.
-
At large Weissenberg number We∞ ≫ 1, the normal force (F0/FN∞) becomes constant and independent of the Weissenberg number (second Newtonian plateau). All the simulations (a-e) illustrate different material conditions, demonstrating rheological transitions from a constant structure to a more compact structure, influenced by the flow and rheology mechanisms through the dimensionless numbers. The partial conclusions from this plot are summarized as follows:
-
The normal force (F0/FN∞) is proportional to the viscosity function in the case of viscometric flow, specifically in the case of homogeneous flow (m = 0).
-
The shear-thinning/thickening mechanisms are characterized by the dimensionless groups {Jr∞, φr}.
-
2. Pseudo thixotropy/rheopexy
Figure 6 shows the normalized force F0/FN∞ to zeroth order vs Weissenberg number We∞, as a function of the viscoelastic dissipation function at high shear rates and the bulk elastic mechanisms through the dimensionless number Jr∞. The other parameter fixed in the simulation is φr = 1.
It is evident the competition of all these mechanisms causes the complex fluid not to recover after a period of deformation F0/FN∞. At zeroth order, different deformation histories are shown by the loops and they correspond to different relaxation times, as these curves are obtained in steady state, we have decided to call them pseudo thixotropy/rheopexy as the shape of the curves is similar to the unsteady case. Mathematically, Fig. 6 display two plateaus at small and high Weissenberg numbers and intermediate power low region with a slope close to unity. Unsteady state simulations may require more powerful numerical methods and they will render non analytical solutions which are outside the objectives of the present work and will be studied as future work.
The following important points are summarized as follows:
-
The normal force F0/FN as a function of the Weissenberg number We∞ [i.e., F0/FN = F0/FN(We∞)] is proportional to the apparent shear viscosity in a parallel plate system sheared by a pressure gradient dp0/dr.
-
The normalized force and apparent shear stress loops are controlled by the two dimensionless numbers [Jr∞, φr], respectively.
-
At small and large Weissenberg numbers, both systems are independent of Jr∞, and its values is determined by the fluidity ratio φr.
-
The normalized plateau can be extended when the compliance associated to the viscoelastic irreversible work is small compared to bulk-elastic compliance. As Jr∞ → 0, the plateau extends, and the structured fluid require more energy to change the microscopic entanglements, bonds or links. In contrast, when Jr∞→ ∞, the plateau changes with less energy.
3. Yield stress
Figure 7 shows the normal force and the apparent shear stress vs Weissenberg number We∞ as a function of the fluidity ratio φr = φ0/φ∞. The simulation (a) corresponds to the case when the system is Newtonian, and the fluidity ratio is equal to the unity, i.e., φr = 1. Here, the system is independent of the value of Jr∞ and the structure is completely disrupted, and it is equivalent to the Newtonian solvent viscosity. The simulations (b)–(d) show a constant behavior at small and moderate Weissenberg number We∞ = J0⟨V⟩/hφ∞. At a critical value of the We∞, the activation of the viscoelastic dissipation structure reaches its maximum, and the compact structure shows a monotonically decreasing behavior, which can be considered a small shear-thinning region followed by plateau behavior. Finally, the last corresponds to the highest value of the normal force at small and moderate We∞ numbers (φr → 0). The following summary finding are
-
The dimensionless value of φr controls the first plateau when the structure is constant and independent of the Weissenberg number (We∞).
-
When φr → 0, the first plateau decreases drastically, and the system reaches the maximum compact structure, which its value determined by the force at zeroth Weissenberg value.
- The value of the normal force in the maximum fluid structure is related with a simplified equation, given by the following expression:
This equation can be linearized to theoretically obtain the yield stress value.
4. Wormlike micellar solutions data
Figure 8 shows the effect of CTAT content on the normal force and apparent shear stress as a function of the Weissenberg number. The material properties used in the simulations are provided in Table I. In both simulations, the effect of the concentration induces greater structural-interactions. Generally, Fig. 8 exhibits the same mathematical behavior as shown in the shear-thinning mechanisms (Fig. 5). At small and high Weissenberg numbers, the system displays two plateaus, while at moderate Weissenberg numbers, there is a monotonically decreasing and increasing behavior for the normal force and apparent shear stress. The main effects of the concentration on the load capacity of the continuous squeeze flow film are summarized as follow:
-
Increasing the CTAT content decreases the load-bearing capacity of the continuous squeeze flow film.
-
At high concentration (20 wt. %), the force is at a minimum compared to the other sample concentrations. Physically, as the concentration increases, the system exhibits more entanglement, links, or bonds, resulting in a more compact liquid structure.
B. Velocity profiles
In this section, the mechanisms of shear thinning/thickening, yield stress, and sample weight concentration are explored through the axial velocity profile. The material properties are obtained directly from Table I. In all cases, the system exhibits a parabolic profile, where the velocity is zero at the wall (non-slip boundary condition), and the maximum value of the velocity vr0max = vr0 (r, z =1/2) is reached and can be calculated using Eqs. (74a) and (74b). The following important points are highlighted in Fig. 9:
-
Newtonian profile (red color) is observed when φr = 1, and is independent of Jr∞, which represents the ratio between the yield stress compliance and bulk elastic mechanisms.
-
The shear thinning mechanisms display a larger parabolic profile compared to the Newtonian case (constant structure, simulations a and b). Here, the system is controlled by the dimensionless number φr ≪ 1.
-
The shear thickening mechanism shows a significant decrease in the parabolic profile (simulations d and e). Physically, the system exhibits a more compact structure due to internal mechanisms. Here, the dimensionless number φr is greater than one, i.e., φr ≫1.
-
Yield stress mechanisms are explained in Fig. 9(b). It is evident that the system shows a substantial decrease in the maximum velocity value. This behavior is observed when the φr → 0, i.e., φ0 → 0 (infinite viscosity). In this case, the structured fluid has a highly compact structure and requires significant structural activation to flow approximating to plug flow (flat plateau, letters d and e).
-
Micellar concentration samples [Fig. 9(c)] show a parabolic velocity profile. As the concentration increases, the number or entanglements or links increases, and the velocity profile decreases considerably. The system is dominated by the two dimensionless numbers {Jr∞, φr}.
C. Non-homogeneous flow
Figure 10 shows the effect of the homogeneity through the parameter m on the normalized force at zeroth order. It is evident that the system exhibits the same rheological behavior as the viscosity shear-thinning mechanism (first Newtonian plateau, then power law zone, and finally second Newtonian plateau). The effect of the non-homogeneity, related to the parameter m, shifts the curve to lower values of the normalized force; however, the mathematical structure remains the same. Additionally, the non-homogeneity, represented by the parameter m, depends on the distribution of the holes in the lower plate and is completely determined by the experimental conditions.73,74
D. First order mechanisms: O (α1)
1. Elastic-shear-thinning/thickening
Figure 11(a) shows the force at first order in the parameter α vs the Weissenberg number (We∞) as a function of the shear-thinning/thickening mechanisms through the dimensionless number φr. The first normalized force is related to the elastic contributions. For both mechanisms (shear thinning or thickening forces), the system exhibits three regions: (i) At low Weissenberg number, the system shows linear behavior (Power law). (ii) At intermediate Weissenberg numbers, the system shows a smooth, monotonically increasing behavior for the shear-thinning forces, while the shear-thickening mechanisms exhibit a pronounced transition. (iii) Finally, at high Weissenberg number, the system exhibits linear behavior again (Power law). Note that, in power law regions, the system undergoes constant changes in the elastic structure, and the flow and rheology are dominated by the numbers {Jr∞, φr}.
2. Elastic-pseudo thixotropy/rheopexy
Figure 11(b) shows the effect of the number Jr∞ on the force at first order in the parameter α. It is clear that this number is related to the structural activation energy associated with the breakdown and buildup mechanisms through the shear-thinning/thickening mechanisms. At low Weissenberg numbers (We∞ ≪ 1), the system shows linear behavior with a slope close the one. When 1 ≤ We∞ < 102, it exhibits a smooth, monotonic transition, followed by linear behavior when 102 ≤ We∞ < 104. The elastic shear-thickening/thickening behavior shows a power law zone with an abrupt change in the internal microstructure, and for a critical Weissenberg number, it follows a linear behavior. These simulations show that elastic-shear-thinning/thickening mechanisms are coupled trough the pressure gradient to first order.
3. Elastic-yield/stress
Figure 11(c) shows the normal force to first order (F1/FN∞) vs Weissenberg number (We∞) as a function of the material conditions associated with the yield stress forces. It is clear that, in the elastic-yield stress zone, the force at first order is independent of the Weissenberg number. At a critical value of the Weissenberg number, the system exhibits a linear behavior associated with the fluid zone (simulation a). The simulations (b-d) exhibit a mathematical behavior similar to shear-thinning mechanisms, with all these numerical results being controlled with the dimensionless number φr. The final case resembles Newtonian behavior.
4. Elastic-concentration wormlike micellar data
In Fig. 11(d) shows the effect of the force at first order vs the Weissenberg number as a function of the worm-like sample concentration in the system (see Table II). It is clear that the force at first order exhibits three basic zones. The first one, 0 < We∞ < 10°, is associated with linear behavior. The second zone, 10° < We∞ < 102, shows a monotonically increasing behavior, while the last zone shows a second linear behavior at high Weissenberg numbers. The simulations display the same mathematical behavior; however, as the worm-like micellar concentration increases, the elastic transition zone at moderate Weissenberg numbers decreases due to the increased structural interactions and elasticity.
CTAT (wt. %) . | Ca = J0/Jm . | We∞ = J0/Jv∞ . | φr = φ0/φ∞ . | Jr = Jp/J0 . | α = h/a . |
---|---|---|---|---|---|
5% | 0.076 | 720 | 0.000 105 | 3. 97 × 10−6 | 0.018 |
10% | 0.015 | 2459 | 6.1 × 10−6 | 8.83 × 10−7 | |
15% | 0.0055 | 2520 | 2.18 × 10−6 | 7.23 × 10−7 | |
20% | 0.0024 | 2857 | 8.04 × 10−7 | 6.07 × 10−7 |
CTAT (wt. %) . | Ca = J0/Jm . | We∞ = J0/Jv∞ . | φr = φ0/φ∞ . | Jr = Jp/J0 . | α = h/a . |
---|---|---|---|---|---|
5% | 0.076 | 720 | 0.000 105 | 3. 97 × 10−6 | 0.018 |
10% | 0.015 | 2459 | 6.1 × 10−6 | 8.83 × 10−7 | |
15% | 0.0055 | 2520 | 2.18 × 10−6 | 7.23 × 10−7 | |
20% | 0.0024 | 2857 | 8.04 × 10−7 | 6.07 × 10−7 |
VII. SUMMARY AND CONCLUSIONS
The merits of this research focus on the study of structure–fluid interactions in a continuous squeeze-film flow of a complex liquid. The flow was simulated in a conventional squeeze film fixture by continuously injecting fluid into the narrow gap between two circular plates through the lower plate. The viscoelastic fluid is characterized by a new model based in the Phan-Thien–Tanner formalism.64 The ESR non-linear equation of state describes the structure interactions through the activation energy associated with the minimal irreversible work to alter the structure under the influence of flow. This model incorporates five material properties related to shear-thinning/thickening, viscoelasticity, pseudo thixotropy/rheopexy, yield-stress, and banding mechanisms. The continuity, momentum, and rheology equations were scaled using characteristic variables, and the microscopic mechanisms are described by five dimensionless groups: (i) infinity Weissenberg number We∞ = J0/φ∞h ⟨V⟩−1 (viscoelastic mechanisms), (ii) fluidity ratio φr = φ0/φ∞ (shear thinning and thickening mechanisms), (iii) compliance ratio Jr∞ = Jp/J0 (yield stress and bulk-elastic mechanisms), (iv) structure-interdiffusion Weissenberg WeλD = λD⟨V⟩/h (structure-interdiffusion and flow mechanisms), and finally (v) banding-flow Weissenberg WeλD = λB ⟨V⟩/h (flow instabilities). The scaled non-linear partial differential equations were solved with a regular perturbation-numerical technique to zeroth (viscosity forces) and first (elastic forces) orders, respectively. The study leads to the following conclusions.
A. Force to zeroth order: O (α0)
- The force to zeroth order measures the effects of structure and is governed by two physical groups associated with the shear-thinning/thickening and yield/elastic-bulk/concentration mechanisms through the dimensionless groups (Jr∞, φr). The structure mechanisms can be evaluated through the numerical integral as follows:
Its magnitude is proportional to the variation in viscosity (inverse of the fluidity) with the rate of shear.
-
Shear thinning/thickening forces: The effects of the level of structure vs Weissenberg number can be measure through the dimensionless number φr = φ0/φ∞. The shear-thinning mechanisms are given by the following condition: φr = φ0/φ∞ < 1, and shear thickening mechanism: φr = φ0/φ∞ > 1. The Newtonian behavior is reached when φr = φ0/φ∞ = 1 (constant structure).
-
Pseudo thixotropy/rheopexy forces: The effect of the pseudo thixotropy and rheopexy (steady state, fully developed flow) can be evaluated using the dimensionless numbers (Jr∞, φr). This means that the competition between the yield-stress and bulk-elastic forces is related to the viscoelastic-irreversible work required to change the structure. When Jr∞→0, the plateau is extended, and the microstructure requires more structural activation energy to change. For very large Jr∞ values, i.e., Jr∞→∞, the first plateau decreases as a function of viscoelastic properties because the viscoelastic structural activation energy decreases, and the structure breaks down at lower Weissenberg numbers.
-
Yield-stress forces: Yield stress is maximized when the highest number of entanglements, links, or bonds. This effect is obtained when φr → 0, i.e, φ0 → 0 (η0 →∞). The yield stress force depends on the dimensionless numbers (Jr∞, We∞), and the scaling law is given by Fy-stress ≈ Jr∞−1 We∞−1.
-
Weight concentration samples: The effect of different weight sample concentrations in the system is directly related to the numbers (Jr∞, We∞). At low sample concentrations, the system promotes shear-thinning and pseudo thixotropy mechanisms. High micellar concentration induces more interactions through links, bonds or entanglements. The entire system is fully coupled, promoting the shear thickening/pseudo rheopexy/yield stress forces.
B. Force to first order: O (α1)
- Normal stress difference: The force to first order in the perturbation parameter α (F1) measures the first normal stress difference, i.e., N1 = σ(rr)0-σ(zz)0, and depends on the dimensionless numbers (Jr∞, φr, We∞). It can be calculated through the following numerical integral F1/FN∞, where g (r) is a function of the radial coordinate:
-
The valor of F1/FN∞ is negative, i.e., F1/FN∞ < 0. Physically, this is a consequence of the normal forces induced by the radial flow.
-
Small Weissenberg number We∞: At small Weissenberg numbers We∞, the system admits analytical solutions. The effects of elastic mechanisms decrease until a maximum value, determinate by a coupling effect between the dimensional groups. This demonstrates that although the overall effect is small and negative for small Weissenberg numbers, it is possible for normal stress effects to occur.
-
Load enhancement: The effects of the viscoelastic-thixotropic-elasto-plasto mechanisms demonstrate the marked load enhancement associate to normal-stress effects N1 = σ(rr)0-σ(zz)0
-
Material properties: The material property (β0, J0, φr, λD, λD) associated with structure-activation energy are numerically fitted parameters of the models discussed. They implicitly depend on the underlying microscopic mechanism of micellar scissions and fusion.
-
Lubrication approximation: This system is equivalent to the traditional squeeze flow film, where the lubrication approximation is used, and dh(t)/dt is replaced by a dh(t)/dt = a <V>/2h, with a being the constant approach velocity of the disk in the conventional squeeze film.
It would be worthwhile to compare the theoretical predictions of the effect of pseudo thixotropy with experimental observations, for example, by using viscoelastic surfactants such as CTAT, EHAC, liquid crystalline suspensions or associative polymers. One of the most interesting effects of complex fluids is shear-banding flow, where mechanical and thermodynamic instabilities cause the system to separate into regions of different viscosities. The drastic shear-thinning behavior may lead to a significant enhancement in the context of lubrication theory.
ACKNOWLEDGMENTS
E.E.H.V. acknowledges the financial support of PAPIIT-DGAPA/UNAM project No. IN102823, PAPIME-DGAPA/UNAM project No. PE106224, and the theoretical discussion of James McGill Professor Alejandro Rey, chemical engineering department McGill University, Montreal, Quebec, Canada. L.A.R.T. gratefully appreciated the financial support from the Consejo Nacional de Ciencia y Tecnología (CONAHCyT) through CVU 860719. C.S.C. gratefully appreciated the financial support PAPIIT-DGAPA/UNAM project No. IN210023 and the Dirección General de Cómputo y de Tecnologías de Información y Comunicación (DGTIC) of the UNAM for allocation of computer time on the Miztli Supercomputer. V.J.H.A. acknowledges the financial support of PAPIIT-DGAPA/UNAM project No. IT-200323. This research is dedicated to the memory of my beloved father Emilio Herrera Caballero “el ave de las tempestades.”
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
L. A. Ramirez-Torres: Funding acquisition (equal); Methodology (equal); Software (lead); Visualization (equal). E. E. Herrera-Valencia: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). C. Soriano-Correa: Conceptualization (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal). M. L. Sánchez-Villavicencio: Data curation (equal); Project administration (equal); Software (equal); Validation (equal). L. Campos-Fernández: Formal analysis (equal); Software (equal); Visualization (equal). G. Ascanio: Formal analysis (equal); Software (equal); Supervision (equal); Validation (equal). V. J. Hernández-Abad: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). F. Calderas: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.