Non-linear electro-rheological model of a membrane immersed in Tanner-Power law fluids applied to outer hair cells: Shear-thinning mechanisms

Flexoelectric actuation employs an applied electric field to induce membrane curvature, which is the mechanism utilized by the outer hair cells (OHC) present in the inner ear. The model developed for this study, representing the OHC, integrates two key components: (i) an approximation of the flexoelectric membrane shape equation for circular membranes attached to the inner surface of a circular capillary, and (ii) the coupled capillary flow of contacting liquid viscoelastic phases characterized by the Tanner-Power law rheological equation of state. A second-order non-linear differential equation for average curvature has been derived, and a robust numerical method has been programmed. This model simplifies to a linear model used previously. The main challenge involves identifying and describing the enhancement in curvature change rate. It was observed that low symmetry, low viscosity, and soft membrane and shear-thickening behavior of the phases enhance the curvature change rate. Additionally, there exists a critical electric field frequency value that maximizes the curvature change rate (resonance effect). The current theory, model, and computational simulations add to the ongoing development comprehension of how biological membrane shape actuation through electromechanical couplings.

Non-linear electro-rheological model of a membrane immersed in Tanner-Power law fluids applied to outer hair cells: Shear-thinning mechanisms

I. INTRODUCTION
Liquid-crystalline organization, structure, and properties are observed in many biological materials, such as proteins and carbohydrates, whose synthesis is specified by the genetic code, resulting in precise chemical sequences, spatial conformations, and molecular weights. 1 The understanding of cell membrane mechanical dynamics, muscle functionalities, and tissue morphogenesis relies on comprehending the fluidity, orientational order, and collective behavior of liquid crystals (LC). 2,3Recent reports have highlighted the benefits of liquid crystallinity in biomedical applications. 4][9] This paper presents the derivation of a non-linear mathematical model and its computational simulation for a physiological actuator device that relies on the distinctive electromechanical and rheological properties of mesophases, 10,11 with a focus on exploring resonance responses to enhance curvature change rate and maximize power dissipation. 1,11It is important to note that this mathematical framework represents the starting point in soft matter research, especially in the study of thermal fluctuations in viscoelastic semiflexible filaments and polymers. 2,12The shape and dissipative evolution of biological membranes depend on the geometry and structure of liquid crystals, enabling the description of wrinkled liquid crystal surfaces, drops, and biological membranes through sophisticated thermo-mechano-geometry theories.This approach extends from simple structures from egg cartons to corrugated surfaces, 2,[13][14][15] as well as to the liquid-crystalline self-assembly of collagen-based biomaterials. 1,10,16,17 Flexoelectricity in Outer Hair Cells (OHC) The operation of outer hair cells (OHC) within the inner ear includes electric-field-induced periodic curving oscillations of LC membranes 18,19 and liquid crystal elastomers, 20 bent-core nematic liquid crystals elastomers, 21 which, through their bending and oscillating motion, transfer momentum, entropic forces, and generate flow to the contacting bulk viscoelastic fluids. 22his important phenomenon has been studied with several mechanical and mathematical modeling approaches using different rheological constitutive equations such as (i) Maxwell, 23,24 (ii) Jeffreys, 6 (iii) Burgers, 24 and (iv) Non-linear models. 25he main role of OHC is to enhance amplify sound in the presence of bulk-viscous dissipation and to store energy in an elastic membrane through flexoelectricity. 26his sophisticated system addresses neurobiological mechanisms involving cochlear amplification and prestin within the biological system. 27ence, a thorough explanation and comprehension of OHC operation should include the behavior of flexoelectric membranes within viscoelastic media when subjected to an oscillating electric field E ¼ À$U E . 28he oscillating input electric field, E, via the flexoelectric effect, an electromechanical phenomenon, causes oscillations in curvature in the elastic membrane of OHC, which is surrounded by viscoelastic materials. 5s a result, the oscillating elastic membrane moves the surrounding viscoelastic liquids due to the effects of mechanical viscoelastic and power viscoelastic dissipation phenomena 5 and electromotility. 29he integrated electromechanical energy conversion is facilitated through the integration of the flexoelectric effect (imposing an electrical field, E, on a flexoelectric membrane) and the mechanical effect (involving the elasticity of the membrane along with the viscoelastic bulk fluid flow of the interacting phases). 5,24,25Various analytical approaches have been used to model changes in the average membrane curvature of a flexoelectric membrane in response of E. 30 These models incorporate the mechanical response of individual cochlear outer hair cells and viscoelastic relaxation dynamical response of curvature through different rheological tests, including SAOS (smallamplitude oscillatory shear) and creep flow. 31Other models include the effects of hair bundles on cochlear OHC and their interaction to minimize fluid-dynamic interactions. 32Some models have employed mechanical systems to analyze the response of isolated cochlear OHC, 33 a kinetic membrane model of outer hair cells, 34 and mathematical and physical approaches to understand flexoelectricity in lipid bilayer membranes induced by external charge and dipolar contributions. 35In this context, we have formulated several mathematical methods to describe the curvature of flexoelectric membrane using the framework of electro-rheological models, with a focus on modeling devices related to outer hair cells (OHC). 25 notable aspect of this method is the full integration of flexoelectricity, membrane elasticity, and advance viscoelasticity properties of surrounding fluids. 24This involves coupling the shape equations of flexoelectric membranes with linear momentum equations that describe complex viscoelastic flows. 5,24,25

B. Potential biological applications
As Ciganovic and Wolde-Kidan 32 and Brownell et al. 33 have reviewed, the OHC of the ear utilize electromotility to overcome viscosity, sharpen the resonance effect, and enhance the required frequency resolution.Receptor potential generates active oscillations of the cell body in OHC. 22The mechanical response to the electric input drives the oscillations in the cell's length, which occur at the frequency of the incoming sound, providing mechanical amplification. 29Some mathematical models for the OHC have analyzed the power conversion in the frequency space domain. 5These models include current noise membrane capacitance of OHC, and voltage tension-dependent lipid mobility in OHC plasma membranes. 35t is noteworthy that, in this study, we employ a distinctive approach by integrating the Helfrich flexoelectric shape equation with capillary Burgers viscoelastic fluid mechanics. 23Through the dynamics of curvature, we arrive at a solution for fluid power delivery. 5As demonstrated, a critical biological feature involves the shape and positioning of the power amplification pulse. 25n the proposed flexoelectric model, the non-monotonic behavior of power linked to its peak (resonance) and trough (anti-resonance) is observed and follows three coupling mean effects: (i) elasticmembrane ratio k 2 (0,1), (ii) the total viscosity e < Rg < 1 À e, and (iii) memory of the viscoelastic phase e < Q k < 1/4. 6he resonance behavior is achieved when there is rheological asymmetry between the viscoelastic phases, if one phase is nearly Newtonian, the other should be highly viscoelastic. 23hen the inertia mechanism is introduced via the Mach number, all the models exhibit multiple secondary peaks and a significant dominant peak, particularly when fluid memory is also introduced. 6he behavior of resonance has been examined using an electromechanical model, wherein the non-monotonic trends depend on the material properties within the system and the length of OHC. 29 In blood flow, Ponalagusamy and Manchi 36 conducted a review of the applications of modeling a hydrodynamic electro-magneto-rheological biphasic flow (solid-liquid) with a pulsatile pressure gradient in a capillary.Additionally, they reported the combined effects of a magnetic field, electric field, rheology of a non-Newtonian fluid (Jeffrey model), non-uniform geometry, and pulsatile blood flow.Through numerical computational analysis, they concluded that at specific values of electric and magnetic field intensity, blood flow is substantially influenced; in fact, they observed that increasing the field intensity increases blood flow.Ponalagusamy and Manchi 36 also studied the effect of other physical properties on blood flow in a stenosis, among which periodic body force, pulsatility, and rheological parameters stand out.
From a biological and physiological perspective, the most significant aspects in OHC are the width, maximum value, and minimum value of the power dissipation peak. 6,23][25] Based on the rheological equation of state utilized in the momentum transfer equation, the power resonance form can exhibit multiple curves under conditions of small inertia and zero inertia (Deborah or Mach numbers equal to zero). 6e focus here on a periodically driven electric field for a hypothetical device that converts shape changes of flexoelectric membranes into rectilinear viscoelastic flows. 16We only consider capillary confinements, and as the membrane deforms, it transfers momentum to the surrounding phases. 5,24,25iven the information and observations presented above, the explicit goals of this paper are as follows: (1) To develop a new dynamic non-linear model (second-order ODE) for a flexoelectric membrane attached to a circular capillary tube containing viscoelastic Tanner-Power law fluids, exposed to a fluctuating small-amplitude electric field of any frequency. 372) To compute the oscillatory non-linear and linear flow response of the electro-rheological device, considering the viscoelastic nature of the interacting fluids.We will prioritize obtaining an ordinary differential equation that describes our system due to the inherent issues in numerical methods, especially finite element methods, known for their straightforward computational scheme and costeffective algorithmic structure, in solving a viscoelastic fluid flow problem.This is mentioned by Ponnalagarsamy and Kawahara 38 in their research, where they model an Oldroyd-B viscoelastic fluid in contractions with abrupt expansions.(3) To identify the material properties that enhance the curvature change rate and, consequently, the electromechanical power dissipation of OHC.
This study is organized as illustrated in Fig. 1.Sections I and II introduce the key components of the electrorheological model embedded in Tanner-Power law viscoelastic fluids.The balance equations 39 are formulated by combining (i) the flexoelectric membrane shape equation, applied to a circular membrane attached to the inner surface of a circular capillary, and (ii) the capillary flow of the interacting viscoelastic phases.Section III presents the derived model and characteristic modes used in simulations.Section IV presents chosen representative numerical results of simulations involving dimensionless average curvature and dimensionless curvature change rate at different material properties.Finally, Sec.V presents the conclusions and outlines future work.

II. SCHEMATIC DIAGRAM OF OHC
The physical setup of the flexoelectric membranes tethered to a cylinder with a radius of r ¼ a and a length z ¼ 2L is depicted in Fig. 2. 40 The coordinate origin of the cylindrical coordinate system is located in the left fluid (z ¼ 0), where the vertical z-coordinate is defined.The left viscoelastic Tanner-Power law fluid occupies the region between the origin and the membrane, with the membrane center at z ¼ L. The right viscoelastic fluid is placed between the membrane and z ¼ 2L. 40Both fluids are appraised incompressible, and the height of the horizontal fluid column is 2L.Effects of small membrane stretching area are neglected.The material parameters, defined in the non-linear viscoelastic model given by Eq. (10), are (a) viscosity functions [g L (II L ), g R (II R )], (b) relaxation times [k L , k R ], and (c) elastic-flexoelectric-membrane M. The pressure at the left and at the right of the membrane are equal to a constant p(z ¼ 0) ¼ p(z ¼ 2L) ¼ p 0 .By imposing a unidirectional fluctuating electrical field, E z (t) ¼ E(t) Á e z ¼ E 0 cos (xt), the membrane oscillates and displaces the left and right incompressible viscoelastic fluids.It is important to emphasize that this type of flow, where Q L ¼ Q R , is generated by the flexoelectric effect due to the imposed E z (t) field, and there is no external pressure gradient in the axial direction.Additionally, the width of membrane is neglected, ensuring flow continuity.

A. Flexoelectric device process modeling
To solve the problem showed in Fig. 2, the following assumptions were employed: (i) Neglecting inertial and external forces (such as gravitational forces) when compared to other mechanisms (surface forces), (ii) assuming incompressibility, i.e., constant density, Dq/Dt ¼ 0, and (iii) describing the rheological behavior using the viscoelastic Tanner-Power law constitutive equation. 37

Mass transfer and momentum transfer equations
The governing equation, which describes transient mass distribution in both viscoelastic fluids considering no chemical reaction, can be expressed as follows: 37 @q X @t þ r Á q X V X ¼ 0; X ¼ fL; Rg: The subscript X ¼ [L, R] in Eq. ( 1) represents the fluid parameter on the left and right sides.The mathematical description of Newton's second law for a continuum viscoelastic fluid is 37 In Eq. ( 2), f Bx is the body force in the phase X, and T x is the total stress tensor in the phase x.The total stress tensor can be expressed as follows: In Eq. ( 3), p x is the isotropic pressure in every viscoelastic phase.For example, for the left side, the pressure p I satisfies the following inequality p 0 p I p L , and for the right viscoelastic liquid phase, the pressure inequality is given by p R p II p 0 , V x V x is the dyadic product of the velocity field, and r x is the viscoelastic (deviatoric) stress tensor of the phase x.The unit tensor I is given by the following matrix form: Therefore, the momentum equation of the phase X takes the following form: In Eq. ( 5), V X is the phase velocity, p is the pressure gradient in the system, serving as the starting point for balancing pressure in each phase, r X is the viscoelastic stress tensor in each phase, and q X is the density in every viscoelastic liquid phase.The motion of the flexoelectric membrane contributes to the momentum balance equation through the pressure gradient, 5,6,[23][24][25] rpðtÞ, and this is explained in Sec.II A 2.

Rheological equation of state: Tanner-Power law model
Most of the complex biological liquids exhibit elastic properties and shows shear-thinning and shear-thickening behavior under flow. 10,16,17One of the simplest constitutive equations that considers alterations in viscosities, caused by applied shear rate, is the viscoelastic rheological equation of state Tanner-Power law model. 37This model is a non-linear viscoelastic model that contains a constant relaxation time and a shear rate viscosity function where k X is the relaxation time of the phase X, g X (II D ) is the viscosity function of the phase X, II D is the second invariant of shear rate tensor, and D is the shear rate tensor.Accordingly, to the power law, the viscosity function can be expressed as follows: Here, m x and n x are the flow consistency index and flow behavior index for phase X, respectively.Note that if n x < 1, the phase X behaves as shear-thinning fluid.On the other hand, the upper-convected derivative for the stress tensor is given by The selection of this constitutive equation, compared to other more complex models such as the White-Metzner, 37 Giesekus, 37 PTT, 41 Generalized-PTT, 41 Structure-kinetic, 42 and Exponentialstructure models, 5 is due to its ability to yield analytical results for simple shear flow.These results include (i) velocity profile, (ii) volumetric flow rate, and (iii) pressure balance in the two viscoelastic phases.

Velocity field and spatial gradient tensor
The velocity field, V X , of the two viscoelastic fluid phases is represented as V X ¼ ð0; 0; Vz X ðr; tÞÞ, and the spatial velocity tensor gradient rV X is given by the following analytical expression: In Eq. ( 9), the scalar shear strain tensor is defined by

Stress and shear rate tensors
The shear stress r X , rate of deformation D X and vorticity W X tensors, applicable to schematic shown in Fig. 2, reported elsewhere, 5,25,36,38,43 are given by

Membrane shape equation
The equation for membrane shape is derived from a pressure balance within the flexoelectric membrane, which accounts for the electric field, bulk stress jump, and the Hookean elastic properties of the membrane.This equation can be expressed analytically as shown in the following form: 5,10,11 where the elastic-parameter M is expressed as follows: In Eq. ( 12), cf. is the membrane flexoelectric coefficient, I ¼ 8/a 2 is a geometric factor, 23 E z (t) is the axial component of the electrical field, k is the outer unit tensor, DTb is the change of the curvature tensor, 23 c o is the interfacial surface tension at zero electric field, 23 k 1 and k 2 are the membrane bending rigidity and torsion elastic moduli, 23 and H(t) is the average curvature and M can be interpreted as an elasticmembrane parameter. 5,6The bulk stress jump (discontinuity in the thermodynamic system) is given by 25 kk : By combining Eqs. ( 12) and ( 13), the following expression is obtained: In Eq. ( 14), rzp(t) ¼ @p(t)/@z.The curvature change rate in a capillary tube with radius r ¼ a is given by On the other hand, the curvature change rate produced by the membrane is calculated by the time derivative of the mean curvature in the electro-rheological system

Mathematical modeling
In this section, the mathematical deduction of the non-linear model associated with the evolution of the average curvature is presented.Once the velocity vector, spatial velocity gradient tensor, and shear rate tensor are substituted into the continuity, momentum, and constitutive equations, the following analytical expressions are obtained: The continuity equation indicates that the velocity vector is independent of the axial coordinate, z [Eq.(17)].This equation satisfies every liquid phase and is the starting point for our analysis.
B. Left pressure balance

Shear stress, axial velocity profile, and volumetric flow rate
Once the inertia and gravitational forces are neglected, the relevant component of the shear stress tensor in the momentum equation is as follows: By integrating the above equation with respect to the r spatial coordinate and applying the non-slip boundary condition at r ¼ 0 (where the shear stress is finite), we obtain Equation ( 25) is general and applicable to both left and right Tanner-Power law liquids, as it does not depend on the material properties of the liquid.

Axial velocity and volumetric flow rate
Replacing the shear stress [Eq.(25)] into the rz component of the Tanner-Power law model, we obtain Using the non-slip condition in Eq. ( 26) and integrating with respect to radial coordinate, the following expression for the axial velocity is found: Hence, the curvature change rate can be obtained by integrating with respect to the cross-sectional area, and it is given by the following analytical expression: By performing the integration of Eq. ( 28), we arrive at (29a)

Pressure profile left side
The curvature change rate can be expressed through the negative of the time derivative of the average curvature, i.e., By combining Eqs.29(a) and 29(b), the following expression, which involves the average curvature as a function of the material properties and the evolution of the pressure gradient, is obtained: Isolating the pressure gradient yields In Eq. ( 31), the following material parameter b nL has been defined as follows: Equation ( 31) is a linear differential equation, which can be solved using standard techniques to find the pressure gradient In Eq. ( 32), C 1 is a constant of integration, which satisfies the following initial condition: So, the value of the constant is C 1 ¼ 0, then Equation ( 34) can be integrated with respect to the axial coordinate z as In Eq. ( 35), the parameter e corresponds to the width of the flexoelectric membrane.The left pressure can be expressed in the following analytical form:

Pressure profile right side
In the same fashion of Eq. ( 34), it is possible to obtain the pressure gradient of the right side By integrating Eq. ( 37), the following expression is obtained in terms of the left pressure: Then, the general solution for the right pressure is given by the following analytical mathematical expression: The pressure difference, Dp ¼ p L -p R , is given by Making all the algebraic steps and taking the limit when the width of the flexoelectric membrane goes to zero, the pressure gradient is obtained, i.e., ÀDp/L ¼ Lim e!0 (p R À p L )/L.
The curvature change in terms of the curvature rate takes the following form: Once Eq. ( 42) is substituted into Eq.( 40), the pressure gradient is obtained as follows: Using the driving force equation in terms of the viscosity and interfacial elasticity, we have which can be rewritten in the following analytical form: Substitution of Eq. ( 43) into Eq.( 45) renders ) Taking the time derivation in both sides of Eq. ( 47), we have Multiplying by the negative exponential e Taking a second time derivative of the above expression, we have Changing R by L and L by R Adding both equations (summarizing the model to eliminate any bias), we have Simplifying e t/kL þ e t/kR 6 ¼ 0, and multiplying by the product between the Maxwell relaxation times, we obtain where the generalized elastic moduli are given by In the above equation, the constants m were used.Finally, in dimensionless form, Eq. ( 53) takes the following analytical form: where the non-linear coefficients are given by b 2n M; n i ; G in ; P k ; dH dt In Eq. (59), R k and P k are the total viscoelasticity and memory of the complex fluids.Note when the power indexes are equal to the unit, i.e., n R ¼ n L ¼ 1, the Abou-Dakka et al.'s model is recovered 23 In the following section, the dimensionless expression of the general non-linear model given by Eqs. ( 58) and ( 59) is obtained in terms of the dimensionless numbers associated with the macroscopic flexoelectricity, viscoelasticity, memory, shear-thinning/thickening, and elasticity of the flexoelectric membrane.Note that Eq. 59(d) is the bulk-viscous mechanisms in the thermodynamic liquid phases.

C. Dimensionless variables
In this section, the dimensionless variables of the flexo-viscoelasticelectro-rheological system are presented.The average membrane curvature, H, is made dimensionless with the radial length of the capillary a, the time process t is made dimensionless with the total relaxation time R k , the shear elastic bulk of every phase [G L , G R ] is made dimensionless with the total bulk elasticity of the fluids, i.e., R G .In the same way, every relaxation Maxwell time of every liquid phase is made dimensionless with the total viscoelasticity of the fluids, i.e., R k .The total bulk viscosity R g is made dimensionless with the product between the bulk elasticity and the total relaxation viscoelastic time R G R k , the flexoelectric membrane M becomes dimensionless with the total bulk elasticity, i.e., R G , the frequency x is dimensionless with the total relaxation time R k .The oscillatory electrical field E(t) is bounded with its amplitude E 0, and finally, the volumetric rate flow Q is dimensionless with a characteristic curvature change rate, i.e., pa 3 /R k .This selection led us to compare it with the previous model developed in flexoelectricity From now on, we will exclude the Ã for the sake of simplicity.

III. DERIVED MODELS AND CHARACTERISTIC FLUID PROPERTIES MODES A. Non-linear model
Once the dimensionless variables [Eq.( 60)] are substituted into Eqs.( 57)-(59), we derive the subsequent dimensionless, non-linear second-order differential equation based on the material properties of the system 6,10 b 2n The non-linear coefficients are calculated by the following expressions: b 2n k; n i ; G ix ; dH dt With the following boundary conditions: In Eqs. ( 61)-( 64), we define the following variables associated with (i) memory, (ii) elastic ratio of the membrane, k, and (iii) flexoelectric mechanisms.
The elastic moduli were defined as follows: To assess the flexibility of the membrane, both in its floppy and rigid states, it is recommended to implement the following parameter modification: When M ! 1, k! 1 (solid membrane), whereas M !0, k ! e ¼ 10 À4 (floppy membrane).Table I summarizes all material properties (as described Abou-Dakka et al. 23 and Herrera-Valencia and Rey 5,6,23 ) under various conditions of viscoelastic phases and membranes used in simulations.

B. Linear model
Assuming linear viscoelasticity, i.e., small strain and constant viscosity and with n L ¼ n R , Eqs. ( 61 The coefficients of the model are given by Equation ( 71) provides the expression for the dimensionless total bulk viscosity In the following section, we show the prediction of the non-linear model in oscillatory flow conditions using a numerical method programmed in Wolfram Mathematica 13 (institutional license).

C. Oscillatory flow in non-linear model
In this section, the non-linear model is examined in a rheological oscillatory flow experiment.The solution is constructed based on an initial condition, allowing the non-linear model to be analyzed through a specific electrical field E(t).In particular, for simplicity, the input force is defined as follows: So, the non-linear differential equation can be expressed in the following form: In order to find the evolution of H(t) with time t, it is necessary to specify the numerical value of the dimensionless numbers: (i) elastic ratio k, (ii) memory P k ¼ k L k R , (iii) flexo-electric a 0 , (iv) bulk elasticity (G Ln , G Rn ), (v) shear-thinning index (n L , n R ), and (vi) frequency x.

D. Oscillatory flow in linear model
The second-order linear non-homogeneous ordinary differential equation has an analytical solution, and this can be computed using a direct mathematical expression.
The solution of the homogeneous part where eigenvalues are given by Thus, the general solution can be expressed in the subsequent analytical form Applying the boundary conditions [see Eq. ( 64)], the constants C 3 and C 4 have the following analytical form:

IV. SIMULATIONS AND RESULTS
Figure 3 depicts the evolution of the average membrane curvature from the linear model as a function of the dimensionless time in an oscillatory flow experiment for various system material properties modes presented in Table I.At short times, membrane deformation remains independent of the material properties of the phases.Past a critical dimensionless time, the average membrane deformation increases at different rates for each mode.In fact, the LLS mode generates the most significant deformation, whereas the HIR mode yields the least deformation.Additionally, note that LHR and HIR exhibit peak deformations at different times, with LHR reaching its maximum at a later time.Consequently, lower phase symmetry, lower phase viscosity, and a soft membrane (LLS) all promote flexoelectric membrane deformation.
In Fig. 4, the absolute value of curvature change rate as a function of time for all modes in linear model is displayed.In contrast to what was observed in Fig. 3, Fig. 4 reveals that, at short dimensionless time, the curvature change rate varies among all system modes.However, Fig. 4 displays two minima corresponding to LHR and HIR, and three maxima in LLS, LHS, and HIS, with LLS achieving the highest curvature change rate.It is worth mentioning that the maxima for LLS, LHS, and HIS occur at different dimensionless times, with HIS reaching its peak earliest.Figures 3 and 4 suggest that the LLS mode maximizes both membrane deformation and curvature change rate.Subsequent sections will elucidate the impact of shear-thinning or shear-thickening mechanisms of both phases on the curvature change rate in OHC.
Figure 5 displays the absolute value of curvature change rate as a function of dimensionless time for the LLS mode at varying flow behavior indices but equal between phases (n L ¼ n R ).At short times, it can be observed that shear-thickening or shear-thinning mechanisms do not significantly impact the curvature change rate.However, as the dimensionless time increases, this effect becomes noticeable.Indeed, in all cases, a peak in curvature change rate can be observed at approximately the same dimensionless time (t c % 0.7853).The increase in curvature change rate with the flow index appears to follow a linear trend (see Fig. 6).Additionally, as the flow index increases, i.e., the viscoelastic phases become shear-thickening, curvature change rate also increases.This may be attributed to the shear-thickening effect of viscoelastic phases, which generates a longer contact time with the flexoelectric membrane, promoting better interaction between the membrane and fluids and consequently leading to an increased curvature change rate.This effect is surprising because, typically, in flow of non-Newtonian inelastic shear-thinning liquids in horizontal pipes, it has been noted that such fluids enhance both heat transfer 44 and mass transfer. 45n Fig. 6, the variation of the absolute value of maximum dimensionless curvature change rate, Max[jQ(t c )j], at a critical dimensionless time, t c % 0.7853, at different values of flow behavior indices for both phases, n L and n R .A linear increase in Max[jQj] with n L and n R was observed.It is notable that transitioning from a shear-thinning fluid (n L and n R < 1) to a shear-thickening one (n L and n R > 1) results in an approximate 10% increase in curvature change rate compared to the scenario where the phases behave in a Newtonian manner (linear model, n L ¼ n R ¼ 1).Consequently, if the phases are shear-thinning (n L and n R < 1), according to our simulations, the curvature change rate generated by the flexoelectric membrane will be lower.Considering that the flow behavior index and the consistency index can be related to the chemical composition of the viscoelastic phases, it is expected that a phase rich in biopolymers, ions, and other solutes would increase both parameters.Consequently, altering the composition of viscoelastic phases in OHC will result in an increased curvature change rate (thus greater dissipated power in the OHC 23,30 ) The simulations of the model proposed in this work indicate that there is another way to increase curvature change rate through other rheological properties of the viscoelastic fluids surrounding the OHC membrane.
Figure 7 shows the variation of non-stationary dimensionless curvature change rate, jQ(t)j, for both shear-thickening phases, n L ¼ n R ¼ 2, as a function of the frequency of the imposed electric field, x 0 .A peak of jQ(t)j is observed at t ¼ 2 and x 0 ¼ 1, indicating a resonance effect between the imposed electric field and shearthickening mechanisms.Previous studies 23,30 have demonstrated a similar effect, albeit in relation to the linear viscoelastic material properties of the phases, where the resonance frequency of the process varied using a Fourier transform within the linear model and a linear viscoelastic constitutive equation.In this work, it was shown that the proposed non-linear model allows an increase in curvature change through shear-thickening mechanisms, in addition to a resonant effect induced by the electric field.

V. DISCUSSION
Flexoelectricity in membranes signifies a novel electromechanical coupling phenomenon noticed in materials that can polarize due changes in their geometric curvature.The sensor-like effect happens by inducing electric polarization when the membrane bends, while the reverse action occurs as the membrane curves in response to an applied electric field.The flexoelectricity of the membrane is crucial in how OHC function as amplifiers to counteract viscous dissipation through mechanic transduction, thereby enabling the process of hearing.Understanding the interplay between the oscillatory flexoelectric actuation and the viscoelastic behaviors of the fluids in contact with the oscillating membrane stands as a significant challenge.Using the flexoelectric shape equation in conjunction with a non-linear viscoelastic capillary flow model for the contacting phases, we obtained general These parameters can be adjusted by altering the chemistry composition of the system through changes in concentration and the molecular weight distribution of dissolved biopolymer chains. 46he advantage of this model is that all parameters can be calculated from independent rheological experiments in both steady and unsteady states.For example, the material parameters of power law can be obtained from a steady flow experiment, while other viscoelastic materials properties can be determined through SAOS, stress relaxation, and fast Fourier transformation tests. 47

VI. SUMMARY AND CONCLUSIONS
The merit of this research is to extend the material parametric space in order to include a non-linear viscoelastic model, which includes the flow behavior index and consistency index of both phases.The second-order non-linear differential equation in the average curvature, coupled with the Tanner-Power law model, allowed us to enhance the curvature change rate by varying the material parameters within a previously reported material space and incorporating the shear-thinning and shear-thickening mechanisms of the fluid.
A quantitative evaluation of the present model predictions, based on oscillatory flow profile, indicates that exists a material condition (LLS, low symmetry, low viscosity, and soft membrane) that maximizes the deformation of the average curvature.Additionally, since membrane curvature and curvature change are related, it was found that the LLS mode also exhibits a maximum curvature change rate at a given dimensionless process time (t c % 0.7853).With this, simulations of the non-linear model were conducted for different shear-thinning indices while keeping them equal in each phase (n L ¼ n R ).It was discovered that as the fluid becomes more thickening, the curvature change increases.In fact, it was observed that the maximum curvature change increase linearly with n L and n R .Finally, a thickening fluid under LLS conditions exhibits a resonant peak at the induced electrical field frequency x 0 , which increases the curvature change rate by up to 80%.Future extensions include applying this theory to others soft matter system (worm-like micelles), mass and energy transfer induced by external forces, 44 heat dissipation and non-linear effects due to high frequencies, compressible systems (density as a function of the pressure drop), and sophisticated mechano-geometric and shape nano-wrinkling theories are employed to understand the effect of curvature on the biophysiological properties of auditory mammals and complex human systems.In this context, the use of finite element methods and highperformance scientific computation is essential for solving the coupled phenomena, 48 following the approach of Ponnalagarsamy and Kawahara. 38he Tanner-Ostwald-de Waele model captures the essential physics to describe the physiological linear behavior of the OHC.Combining the upper-convected Maxwell equation with the generalized Newtonian model yields a non-linear model, serving as starting point for understanding thermodynamic-rheological systems.][9]13,14 x 0 Electric field frequency (rad/s) R q ,R k ,R G Sums of the density, Maxwell relaxation times, bulk elasticity (kg/m 3 , s, s 2 , Pa) R g Total bulk viscosity (Pa s, Pa s) Q k Total Maxwell memory (s 2 , s 4 )

De Deborah k Elastic ratio
Other symbols (r,/,z) Cylindrical coordinates (m,1m) p Pi constant (1) @V z /@r Shear strain (1/s) [@/@z, @/@t] Spatial and temporal time derivatives (m À1 ,s À1 For improved readability and to reduce the use of extensive algebraic procedures, an appendix has been added to this manuscript.

Dimensionless variables
In order to simplify the analytical results given in Eq. (57), the following dimensionless variables were proposed for radial coordinate, relaxation time, process time, pressure difference, average membrane curvature, viscosity operator, elastic membrane, left and right fluid elastic moduli, Poiseuille and membrane curvature change rates, power dissipation, and electrical field.
The material parameters are Notice that for equations showed in (A2), the following restrictions are satisfied: The choice of these defining variables includes (i) total viscosity, (ii) total relaxation times, and (iii) total bulk fluid elasticity.These specific variables enable us to tune internal (inertial, viscous, and viscoelastic) and external characteristic mechanisms (frequency and electrical field).

Parametric material space
The aim of this appendix is to provide the essential information needed for conducting an analysis of the dimensionless numbers in the presented model [Eqs.(57)-( 59)].
The governing equations (A1) and (A2) incorporate four dimensionless numbers that are associated with nine distinct mechanisms: (i) Memory Q k : This is the product of the dimensionless relaxation times of both phases k L , k R , which obeys k L þ k R ¼ 1 and define the elastic asymmetry of the fluids.When Q k (1 (highly asymmetric case), one of the fluids is nearly inelastic, and when Q k ¼ 1/4 (highly symmetric case), both fluids are equally elastic.(ii) Bulk viscous k L : this represents the total viscosity in the system, where the elastic dimensionless moduli satisfy G L þ G R ¼ 1.The numerical value of this dimensionless number is controlled by the product of the two relaxation times (iii) Elastic ratio k: This is the dimensionless ratio between the membrane and the total system elasticity and falls within the range of k 2 (0,1).A floppy (soft) and stiff (rigid) membrane corresponds to k ( 1 and k ffi 1, respectively.

FIG. 2 .
FIG. 2. Schematic of an idealized flexoelectric device utilizing an externally applied electric field.(a) Physical setup of the flexoelectric device representing the OHC, (b) Schematic of the spherical cup deformation of the membrane.

FIG. 3 .
FIG. 3. Dimensionless average curvature vs dimensionless time of oscillatory flow experiment for all modes of the system.

FIG. 5 .
FIG. 5. Dimensionless curvature change rate vs dimensionless time for different and symmetric flow behavior indices for LLS mode.
Low and high shear stress values (D) Instabilities associated with shear-banding forces (E) Nematic orientation (F) Electro-osmotic forces coupled with mass and energy mechanisms.
AbbreviatorsLC Liquid crystal NLC Nematic liquid crystal OHC Outer hair cellAPPENDIX: DIMENSIONLESS GROUPS AND VARIABLES . 1. Flow chart of the paper organization.
V C Author(s) 2024