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 TannerPower law rheological equation of state. A secondorder nonlinear 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 shearthickening 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.
I. INTRODUCTION
Liquidcrystalline 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).^{2,3} Recent reports have highlighted the benefits of liquid crystallinity in biomedical applications.^{4} These advantages encompass: (i) efficient packaging in solutions containing DNA and viruses,^{5,6} (ii) the ability to guide selfassembly on the cellulose fibers, leading to the development of plantbased biological plywoods, (iii) the facilitation of low viscosity for flow processing as seen in the spinning of spider silk, (iv) the observation of sensor/actuator capabilities in plants and membranes, and (v) mechanical strength, as in cellulosebased plant cell walls, chitinbased exocuticle of insects,^{7–9} and thermodynamic modeling of collagen fibrils in human compact.^{7–9}
This paper presents the derivation of a nonlinear 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.^{1,11} It 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.^{2,12} The 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 thermomechanogeometry theories. This approach extends from simple structures from egg cartons to corrugated surfaces,^{2,13–15} as well as to the liquidcrystalline selfassembly of collagenbased biomaterials.^{1,10,16,17}
A. Flexoelectricity in Outer Hair Cells (OHC)
The operation of outer hair cells (OHC) within the inner ear includes electricfieldinduced periodic curving oscillations of LC membranes^{18,19} and liquid crystal elastomers,^{20} bentcore 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.^{22}
This 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) Nonlinear models.^{25}
The main role of OHC is to enhance amplify sound in the presence of bulkviscous dissipation and to store energy in an elastic membrane through flexoelectricity.^{26}
This sophisticated system addresses neurobiological mechanisms involving cochlear amplification and prestin within the biological system.^{27}
Hence, 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 = −∇Φ_{Ε}.^{28}
The 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.^{5}
As 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.^{29}
The 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).^{5,24,25} Various 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.^{31} Other models include the effects of hair bundles on cochlear OHC and their interaction to minimize fluid–dynamic interactions.^{32} Some 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.^{35} In this context, we have formulated several mathematical methods to describe the curvature of flexoelectric membrane using the framework of electrorheological models, with a focus on modeling devices related to outer hair cells (OHC).^{25}
A notable aspect of this method is the full integration of flexoelectricity, membrane elasticity, and advance viscoelasticity properties of surrounding fluids.^{24} This 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 WoldeKidan^{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.^{22} The 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.^{29} Some mathematical models for the OHC have analyzed the power conversion in the frequency space domain.^{5} These models include current noise membrane capacitance of OHC, and voltage tensiondependent lipid mobility in OHC plasma membranes.^{35}
It is noteworthy that, in this study, we employ a distinctive approach by integrating the Helfrich flexoelectric shape equation with capillary Burgers viscoelastic fluid mechanics.^{23} Through the dynamics of curvature, we arrive at a solution for fluid power delivery.^{5} As demonstrated, a critical biological feature involves the shape and positioning of the power amplification pulse.^{25}
In the proposed flexoelectric model, the nonmonotonic behavior of power linked to its peak (resonance) and trough (antiresonance) is observed and follows three coupling mean effects: (i) elasticmembrane ratio k ∈ (0,1), (ii) the total viscosity ε < Ση < 1 − ε, and (iii) memory of the viscoelastic phase ε < ∏_{λ} < 1/4.^{6}
The 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.^{23}
When 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.^{6}
The behavior of resonance has been examined using an electromechanical model, wherein the nonmonotonic 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 electromagnetorheological 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 nonNewtonian fluid (Jeffrey model), nonuniform 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}
The critical material parameters include the following: (i) viscoelastic relaxation times of both phases, (ii) bulk elasticity of the viscoelastic phases, and (iii) elasticmembrane ratio (k).^{5,6,23–25}
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).^{6}
We focus here on a periodically driven electric field for a hypothetical device that converts shape changes of flexoelectric membranes into rectilinear viscoelastic flows.^{16} We only consider capillary confinements, and as the membrane deforms, it transfers momentum to the surrounding phases.^{5,24,25}
Given the information and observations presented above, the explicit goals of this paper are as follows:

To develop a new dynamic nonlinear model (secondorder ODE) for a flexoelectric membrane attached to a circular capillary tube containing viscoelastic TannerPower law fluids, exposed to a fluctuating smallamplitude electric field of any frequency.^{37}

To compute the oscillatory nonlinear and linear flow response of the electrorheological 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 OldroydB viscoelastic fluid in contractions with abrupt expansions.

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 TannerPower 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 = a and a length z = 2L is depicted in Fig. 2.^{40}
The coordinate origin of the cylindrical coordinate system is located in the left fluid (z = 0), where the vertical zcoordinate is defined. The left viscoelastic TannerPower law fluid occupies the region between the origin and the membrane, with the membrane center at z = L. The right viscoelastic fluid is placed between the membrane and z = 2L.^{40} Both 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 nonlinear viscoelastic model given by Eq. (10), are (a) viscosity functions [η_{L}(II_{L}), η_{R}(II_{R})], (b) relaxation times [λ_{L}, λ_{R}], and (c) elasticflexoelectricmembrane M. The pressure at the left and at the right of the membrane are equal to a constant p(z = 0) = p(z = 2L) = p_{0}. By imposing a unidirectional fluctuating electrical field, E_{z}(t) = E(t) ⋅ e_{z} = E_{0} cos (ωt), 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} = 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, Dρ/Dt = 0, and (iii) describing the rheological behavior using the viscoelastic TannerPower law constitutive equation.^{37}
1. Mass transfer and momentum transfer equations
2. Rheological equation of state: TannerPower law model
The selection of this constitutive equation, compared to other more complex models such as the WhiteMetzner,^{37} Giesekus,^{37} PTT,^{41} GeneralizedPTT,^{41} Structurekinetic,^{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.
3. Velocity field and spatial gradient tensor
4. Stress and shear rate tensors
5. Membrane shape equation
6. Mathematical modeling
B. Left pressure balance
1. Shear stress, axial velocity profile, and volumetric flow rate
2. Axial velocity and volumetric flow rate
3. Pressure profile left side
4. Pressure profile right side
C. Dimensionless variables
III. DERIVED MODELS AND CHARACTERISTIC FLUID PROPERTIES MODES
A. Nonlinear model
Six modes of material properties .  Π_{λ} .  Σ_{η} .  k . 

Low symmetry, low viscosity, and soft membrane (LLS)  10^{–4}  10^{–4}  10^{–4} 
Low symmetry, low viscosity, and rigid membrane (LLR)  10^{–4}  10^{–4}  0.9999 
Low symmetry, high viscosity, and soft membrane (LHS)  10^{–4}  0.9999  10^{–4} 
Low symmetry, high viscosity, and rigid membrane (LHR)  10^{–4}  0.9999  0.9999 
High symmetry, intermediate viscosity, soft membrane (HIS)  0.25  0.5  10^{–4} 
High symmetry, intermediate viscosity, rigid membrane (HIR)  0.25  0.5  0.9999 
Six modes of material properties .  Π_{λ} .  Σ_{η} .  k . 

Low symmetry, low viscosity, and soft membrane (LLS)  10^{–4}  10^{–4}  10^{–4} 
Low symmetry, low viscosity, and rigid membrane (LLR)  10^{–4}  10^{–4}  0.9999 
Low symmetry, high viscosity, and soft membrane (LHS)  10^{–4}  0.9999  10^{–4} 
Low symmetry, high viscosity, and rigid membrane (LHR)  10^{–4}  0.9999  0.9999 
High symmetry, intermediate viscosity, soft membrane (HIS)  0.25  0.5  10^{–4} 
High symmetry, intermediate viscosity, rigid membrane (HIR)  0.25  0.5  0.9999 
B. Linear model
C. Oscillatory flow in nonlinear model
D. Oscillatory flow in linear model
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 shearthinning or shearthickening 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} = n_{R}). At short times, it can be observed that shearthickening or shearthinning 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 shearthickening, curvature change rate also increases. This may be attributed to the shearthickening 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 nonNewtonian inelastic shearthinning liquids in horizontal pipes, it has been noted that such fluids enhance both heat transfer^{44} and mass transfer.^{45}
In Fig. 6, the variation of the absolute value of maximum dimensionless curvature change rate, Max[Q(t_{c})], 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[Q] with n_{L} and n_{R} was observed. It is notable that transitioning from a shearthinning fluid (n_{L} and n_{R} < 1) to a shearthickening 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} = n_{R} = 1). Consequently, if the phases are shearthinning (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 nonstationary dimensionless curvature change rate, Q(t), for both shearthickening phases, n_{L} = n_{R} = 2, as a function of the frequency of the imposed electric field, ω_{0}. A peak of Q(t) is observed at t = 2 and ω_{0} = 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 nonlinear model allows an increase in curvature change through shearthickening 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 sensorlike 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 nonlinear viscoelastic capillary flow model for the contacting phases, we obtained general flowelectrorheological dynamic equation. A numerical and parametric study was performed to identify the conditions that result to a maximum of curvature change rate. Using a robust numerical method, a resonance peak for curvature change rate, at specific imposed electrical field frequency, was found for LLS (low symmetry, low viscosity, and soft membrane) system mode.
Note that the key findings in the simulations regarding the evolution of the average membrane curvature in Tanner fluids obey four coupled effects: (i) maximum dissipation; (ii) reduced membrane storage; (iii) asymmetry in the TannerPower law viscoelastic fluids; (iv) flexoelectric mechanism, which are on par with the bulk elastic fluids; and (v) maximum dissipation is with the shearthickening mechanism induced by the power index of the Tanner–Ostwald–de Waele rheological equation of state. The material parameters of significance in both viscoelastic phases are as follows: (i) Power law parameters (m_{X}, n_{X}); (ii) Maxwell relaxation times (λ_{X}); (iii) elastic moduli (G_{X}); (iv) flexointerfacial parameters associated with bending, torsion, and interfacial tension mechanisms (kc_{1}, kc_{2}, and γ_{o}); (v) geometric characteristic lengths, radial and axial (a, L), and (vi) electric and surface parameters associated with the amplitude of the electrical field, electric charge (E_{0}, Cf), and shape parameter (ℑ = 8/a^{2}), which is a consequence of the electrical field value in a spherical geometry.
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.^{46}
The 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 nonlinear viscoelastic model, which includes the flow behavior index and consistency index of both phases. The secondorder nonlinear differential equation in the average curvature, coupled with the TannerPower law model, allowed us to enhance the curvature change rate by varying the material parameters within a previously reported material space and incorporating the shearthinning and shearthickening 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 nonlinear model were conducted for different shearthinning indices while keeping them equal in each phase (n_{L} = 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 ω_{0}, which increases the curvature change rate by up to 80%. Future extensions include applying this theory to others soft matter system (wormlike micelles), mass and energy transfer induced by external forces,^{44} heat dissipation and nonlinear effects due to high frequencies, compressible systems (density as a function of the pressure drop), and sophisticated mechanogeometric and shape nanowrinkling 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.^{38}
The Tanner–Ostwald–de Waele model captures the essential physics to describe the physiological linear behavior of the OHC. Combining the upperconvected Maxwell equation with the generalized Newtonian model yields a nonlinear model, serving as starting point for understanding thermodynamic–rheological systems. However, this model does not account for:^{41,42}

Thixotropy

Rheopexy

Low and high shear stress values

Instabilities associated with shearbanding forces

Nematic orientation

Electroosmotic forces coupled with mass and energy mechanisms.
Ultimately, the present theory, model, and computational approach contribute to an evolving fundamental understanding of biological membrane shape actuation through electromechanical couplings.^{5–9,13,14}
ACKNOWLEDGMENTS
L.A.R.T. gratefully appreciated the financial support from the Consejo Nacional de Humanidades, Ciencias y Tecnologías (CONAHCyT) through No. CVU 860719. E.E.H.V. acknowledges financial support of PAPIITDGAPA/UNAM Project No. IN102823 and the theoretical discussion of James McGill Professor Alejandro Rey, Chemical Engineering Department, McGill University, Montreal, Quebec, Canada. C.S.C. gratefully appreciated the financial support PAPIITDGAPA/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 financial support of PAPIITDGAPA/UNAM Project No. IT200323. L.A.R.T. acknowledges financial support from División de Posgrado e Investigación, Facultad de Estudios Superiores Zaragoza, Universidad Nacional Autónoma de México, Award No. RP/2320402. E.E.H.V. dedicates this research to the memory of his beloved father Emilio Herrera Caballero “el ave de las tempestades.” L.A.R.T. dedicates this article in memory of his beloved mother, Eulalia Torres Pérez.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Luis Antonio RamírezTorres: Data curation (lead); Investigation (equal); Methodology (equal); Software (lead); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Edtson Emilio HerreraValencia: Conceptualization (equal); Formal analysis (lead); Funding acquisition (lead); Investigation (equal); Supervision (equal); Writing – original draft (equal). Mayra Luz Sánchez Villavicencio: Investigation (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Catalina SorianoCorrea: Investigation (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Vicente Jesus Hernadez Abad: Formal analysis (equal); Methodology (equal); Supervision (equal); Visualization (equal). Fausto Calderas: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Supervision (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
NOMENCLATURE
Letters
 $a$

Radius of the pipe (m)
 [a_{i}, b_{i}]; i = 0,1,2,3,4

Material elasticmembrane, bulk viscous, inertial coefficients (1, s^{−1}, s^{−2}, s^{−3}, s^{−4})
 $ c f$

Membrane flexoelectric coefficient (C)
 G_{R}, G_{L}

Elastic moduli define in the right and the left (Pa)
 h

Height of the spherical dome (m)
 H(t)

Average membrane curvature (m^{−1})
 k_{0}, k_{1}

Membrane bending rigidity and torsion elastic moduli (J)
 L

Characteristic axial length of the capillary (m)
 M

Elasticmembrane parameter (Pa)
 Δp(z,t)

Pressure difference (Pa)
 p_{R}, p_{L}

Pressure defined at the right fluid and the left fluid (Pa)
 p_{0}

Constant pressure in the left and right fluids (Pa)
 Q(t)

Curvature change rate (m^{3}/s)
 Q_{R}, Q_{L}

Right and left curvature change rate (m^{3}/s)
 t

Time coordinate (s)
 V_{z}

Axial velocity (m/s)
Other symbols
Vectors and tensors
Greek vectors and tensors
Greek letters
 ε

Small parameters (1)
 γ_{0}

Interfacial surface tension at zero electric field (F/m)
 η_{R}, η_{L}

Right fluid and left fluid viscosities (Pa s)
 λ_{R}, λ_{L}

Right and left viscoelastic relaxation times (s)
 σ_{rz}

Component rz of the shear stress tensor (Pa)
 ω

Process frequency (rad/s)
 ω_{0}

Electric field frequency (rad/s)
 Σ_{ρ},Σ_{λ},Σ_{G}

Sums of the density, Maxwell relaxation times, bulk elasticity (kg/m^{3}, s, s^{2}, Pa)
 Σ_{η}

Total bulk viscosity (Pa s, Pa s)
 ∏_{λ}

Total Maxwell memory (s^{2}, s^{4})
Dimensionless numbers
Other symbols
Subscript
Superscript
Abbreviators
NOMENCLATURE
Letters
 $a$

Radius of the pipe (m)
 [a_{i}, b_{i}]; i = 0,1,2,3,4

Material elasticmembrane, bulk viscous, inertial coefficients (1, s^{−1}, s^{−2}, s^{−3}, s^{−4})
 $ c f$

Membrane flexoelectric coefficient (C)
 G_{R}, G_{L}

Elastic moduli define in the right and the left (Pa)
 h

Height of the spherical dome (m)
 H(t)

Average membrane curvature (m^{−1})
 k_{0}, k_{1}

Membrane bending rigidity and torsion elastic moduli (J)
 L

Characteristic axial length of the capillary (m)
 M

Elasticmembrane parameter (Pa)
 Δp(z,t)

Pressure difference (Pa)
 p_{R}, p_{L}

Pressure defined at the right fluid and the left fluid (Pa)
 p_{0}

Constant pressure in the left and right fluids (Pa)
 Q(t)

Curvature change rate (m^{3}/s)
 Q_{R}, Q_{L}

Right and left curvature change rate (m^{3}/s)
 t

Time coordinate (s)
 V_{z}

Axial velocity (m/s)
Other symbols
Vectors and tensors
Greek vectors and tensors
Greek letters
 ε

Small parameters (1)
 γ_{0}

Interfacial surface tension at zero electric field (F/m)
 η_{R}, η_{L}

Right fluid and left fluid viscosities (Pa s)
 λ_{R}, λ_{L}

Right and left viscoelastic relaxation times (s)
 σ_{rz}

Component rz of the shear stress tensor (Pa)
 ω

Process frequency (rad/s)
 ω_{0}

Electric field frequency (rad/s)
 Σ_{ρ},Σ_{λ},Σ_{G}

Sums of the density, Maxwell relaxation times, bulk elasticity (kg/m^{3}, s, s^{2}, Pa)
 Σ_{η}

Total bulk viscosity (Pa s, Pa s)
 ∏_{λ}

Total Maxwell memory (s^{2}, s^{4})
Dimensionless numbers
Other symbols
Subscript
Superscript
Abbreviators
APPENDIX: DIMENSIONLESS GROUPS AND VARIABLES
For improved readability and to reduce the use of extensive algebraic procedures, an appendix has been added to this manuscript.
1. Dimensionless variables
2. 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 ∏_{λ}: This is the product of the dimensionless relaxation times of both phases λ_{L}, λ_{R}, which obeys λ_{L} + λ_{R} = 1 and define the elastic asymmetry of the fluids. When ∏_{λ} ≪1 (highly asymmetric case), one of the fluids is nearly inelastic, and when ∏_{λ} = 1/4 (highly symmetric case), both fluids are equally elastic. (ii) Bulk viscous Ση = η_{L} + η_{R} = G_{R}λ_{R} + G_{L}λ_{L}: this represents the total viscosity in the system, where the elastic dimensionless moduli satisfy G_{L} + G_{R} = 1. The numerical value of this dimensionless number is controlled by the product of the two relaxation times ∏_{λ} = λ_{L}λ_{R} and Ση(∏_{λ}) = Ση(λ_{L}λ_{R}). (iii) Elastic ratio k: This is the dimensionless ratio between the membrane and the total system elasticity and falls within the range of k ∈ (0,1). A floppy (soft) and stiff (rigid) membrane corresponds to k ≪ 1 and k ≅ 1, respectively.