Fluctuating Flexoelectric Membranes in Asymmetric Viscoelastic Media: Power Spectrum through Mechanical Network and Transfer Function Models

Abstract

Flexoelectric liquid crystalline membranes immersed in asymmetric viscoelastic media is a material system model with physiological applications such as outer hair cells (OHCs), where membrane oscillations generate bulk flow. Motivated by this physiological process, here we extend our previous work by characterizing the force transmission output of our model in addition to viscoelastic fluid flow, since solid–fluid interactions are an essential feature of confined physiological flow and flow in immersed elastic structures. In this work, the rigidity of the confinement results in a passive force reception, while more complete solid–fluid interactions will be considered in the future. A significant contribution of this work is a new asymmetry linear viscoelastic electro-rheological model and the obtained implicit relation between force transmission and flow generation and how this relation is modulated by electric field frequency and the material properties of the device. Maximal force and flow are found at resonant frequencies of asymmetry viscoelastic bulk phases, flexoelectric and dispersion mechanisms through the elastic and Womersley numbers.

1. Introduction

Liquid crystalline organization, structure, and properties are found in a variety of biological systems, whose behavior is specified by complex processes [1,2]. The science of liquid crystal has been used in biological systems such as: (i) plants, (ii) insects, and (iii) membranes and living matter physics [3]. This paper presents a new asymmetric rheological model to describe the complex behavior of a thermodynamical system consisting of two viscoelastic liquid phases and a flexoelectric solid system, which can be applied to the modeling of the outer hair cells of mammalians [4,5,6]. It has been demonstrated that the rheology of the OHC is the key issue in the amplification of sound and its relationship with the natural electrophysiological field in the human body [4,5,6].

Mathematical theories have been used to characterize the nonlinear physics in the OHC [6]. The most remarkable physiological property in the OHC is called flexoelectricity [7,8,9]. The shape of the electrical field, material properties of the liquids, and the solid membrane are the most important issues of the power viscoelastic dissipation and their impact on the ability to tune sound information [4,5,6]. Figure 1 shows sensor (top) and actuator (bottom) modes of flexoelectricity, including energy harvesters, stress sensors, fluid pumps, and force generators; we note that the OHC, which is the topic of this paper, corresponds to the fluid pump and force generator device (bottom left in Figure 1 [9,10]).

In

Table 1. Summary of relevant mechanisms/materials and devices related to the present paper.

Mechanisms/Materials	Device/Process	Source
Membrane flexoelectricity	Force pump actuation device and mechano-electric transduction	Herrera-Valencia and Rey (2014, 2018) [3,4]
Flexoelectric membranes	Nonlinear actuator	Rey, A.D. (2008) [5]
Flexoelectric liquid crystalline membranes	Stress sensor devices (signal detecting stress)	Rey, A.D. et al. (2014) [9]
Flexoelectric rods	Energy harvesting	Rey, A.D. et al. (2013) [10]
Membrane dissipation	Linear viscoelasticity for bending and torsion	Rey, A.D. (2008) [11]
Flexo-dissipation	Curvature dissipation	Aguilar-Gutierrez et al. (2017) [12]
Flexoelectric viscoelastic semiflexible filaments and polymers	Thermal fluctuation spectrum	Wang, Z. et al. (2022) [13]

, we summarized directly related theoretical applications of the flexoelectric membranes, involved in the functioning of the sensor and actuator devices shown in Figure 1. Many of these devices involve dissipative bulk fluid flows (first, second rows) or membrane/filament dissipation (third–fifth rows), and the latter require new couplings between geometry and dissipation (sixth row), which is a topic of emerging research [3,4,5,9,10,11,12,13] and central to transient flexoelectric devices (seventh row).

Next, we focus on the aspects of OHCs directly related to this paper [3,4]. A number of analytical approaches have been employed to simulate the changes in the average membrane curvature of a flexoelectric membrane as a function of the electrical field [3,4] (and references therein). Rabbits et al. 2009 [14] analyzed the power efficiency of outer hair cell somatic electromotility with an electro-mechanical model based on first principles. Their results showed a frequency dependence of the power resonance, which is a function of the material parameters in the system [14]. Furthermore, some authors have employed electro-mechanical models to describe the electro-motility, and the stress of the membrane capacitance of the auditory cell in the outer hair cell has been studied at length with a mechanical viscoelastic model [15,16]. In addition, the frequency voltage and tension dependent lipid mobility in the OHC plasma membrane has been investigated using an electro-mechanical model [14,15,16,17]. Electro-motility in the cochlea system and the stereociliary columns in the mammalian cochlear hair have been studied at length using complex mathematical models [15,16,17,18,19,20]. From a mechanic-electrodynamic point of view, the mechanism effects of the electric and magnetic fields on cells have been analyzed [21]. In addition, Lahlou et al. (2023) [22] have been working on the genetics of auditory hair cells to develop new stem cell therapies. These models include the mechanical response of isolated cochlear OHCs, viscoelastic forces, start-up mechanisms, and linear viscoelasticity [23,24]. Furthermore, other researchers have developed the description of lipid mobility, plasma membrane in terms of electro-mechanical approaches, liquid crystal theories, and transport momentum and rheology constitutive equations [24,25]. A distinguishing feature of the approach is the full integration of flexoelectricity, membrane elasticity, and high-order viscoelasticity of surrounding fluids, coupling flexoelectric membrane shape equations with linear momentum equations that describe complex viscoelastic flows [3,4]. Several aspects of design/control/prediction application in the flexoelectric energy conversion device deserve discussion [9,10]. Based on our previous work [3,4,5], it has been established that the mechanical network analogy in terms of the elasticity of the flexoelectric membrane operating in parallel with viscoelastic elements provides the necessary dynamic structure to produce extrema in the viscous power spectrum of the surrounding media [3,4,9,10].

Since these viscoelastic structural elements represent the surrounding viscoelastic media and have a strong impact on power resonance, a robust, generic, and well-characterized model will be useful [26,27].

One of the main objectives of the aforementioned research is to find the flow rate and power dissipated in the surrounding viscoelastic fluids induced by the changes in the membrane curvature as a consequence of the imposed time-periodic electric field [3,4,5]. In order to include the physical nature of the contacting and surrounding fluids, several rheological constitutive equations have been employed to describe the momentum transfer [3,4,5,10], including Euler (inviscid fluid) [10], Newton [5], and variants of the Maxwell models [3,4,5]. In these previous works, the average membrane curvature vs. applied electrical was investigated [3,4,5]. Another important issue considered in the present work is to establish a wider understanding of membrane-based flexoelectric outputs and to find the relationships between the driven electrical force (input), volumetric flow, and inertial-viscoelastic wall stress (outputs) [3,4]. Figure 2 shows a schematic of a fluctuating membrane that is attached to an elastic substructure (pillars, base filaments; see ref. [12]). Hence, understanding and characterizing force transmission from the fluid flow generated by the oscillating membrane onto the elastic supporting structures are essential [26,27]. The interaction between the liquid viscoelastic phases and solid wall has been found in nonlinear biomaterials [5,26,27]. Notwithstanding, and despite all the theoretical and experimental works, there are no studies that relate the two transfer functions: (i) electric field-inertial-viscoelastic volumetric flow rate transfer function T_F and (ii) electric field-inertial-viscoelastic wall stress transfer function T_S, which is the main focus of this work and can be synthetized as an implicitly coupling stress-flow relation F:

$$\underset{\mathrm{Stress}-\mathrm{Transfer}-\mathrm{Function}}{\underbrace{{\mathrm{T}}_{\mathrm{S}}\left(\mathsf{\omega }\right)}}=\mathrm{F}\left(\underset{\mathrm{Flow}-\mathrm{Transfer}-\mathrm{Function}}{\underbrace{{\mathrm{T}}_{\mathrm{F}}\left(\mathsf{\omega }\right)}}\right)$$

where F is an important function in flexoelectric devices, and whether it is unique or multivalued. Figure 3 succinctly summarizes the methodology we developed to find the function F (Figure 3III). We first formulate and solve the device model using a control-type transfer function formalism, indicated in Figure 3II, that can be summarized as:

$$\frac{\mathrm{O}\left(\mathsf{\omega }\right)}{\mathrm{E}\left(\mathsf{\omega }\right)}={\mathrm{T}}_{\mathrm{X}}\left(\mathsf{\omega }\right)=\mathrm{Re}\left[{\mathrm{T}}_{\mathrm{X}}\left(\mathsf{\omega }\right)\right]+\mathrm{i}\mathrm{Im}\left[{\mathrm{T}}_{\mathrm{X}}\left(\mathsf{\omega }\right)\right]; \mathrm{X}=\{\mathrm{F},\mathrm{S}\},\mathrm{O}\left(\mathsf{\omega }\right)=\{\mathrm{Q}\left(\mathsf{\omega }\right),\mathsf{\sigma }\left(\mathsf{\omega }\right)\}$$

where the output vector O is given by the flow rate Q and wall stress σ, E is the oscillating electric field, and Re [Tx(ω)] is the real part and Im [Tx(ω)] the imaginary part of the quantity in question [3,4]. The transfer functions will have in-phase (real) and out-of-phase (imaginary) components with respect to the input electric field. In principle, the result of the control-type transfer function (Figure 3II) is sufficient to determine F (Figure 3III). Given the complexity of the material information (viscoelasticity, membrane elasticity, etc.), an organizing design principle is needed, and here we use the analogue mechanical network model (Figure 3I). The mechanical network guides the selection and arrangement (in series, in parallel) of the elastic and viscous mechanical elements. Feeding this design information (Figure 3I) in conjunction with the control analysis (Figure 3II) yields the sought-after relations between viscoelastic flows Q and wall stress transfer σ (Figure 3III).

Based on the facts and observations presented above, the specific objectives of this paper are:

(1)

To deduce a new general asymmetric electro rheological model for a flexoelectric membrane attached to a circular capillary tube that contains Maxwell’s viscoelastic fluids and is subjected to a fluctuating small amplitude electric field of arbitrary frequency;

(2)

To compute the flow and stress complex transfer functions vs. frequency as a function of the applied frequency and the material properties of the thermodynamic system;

(3)

To use the modeling results to characterize the complex loops of the flow and stress transfer functions;

(4)

To identify the numerical values of the dimensionless groups that lead to electromechanical force-flow conversion of potential interest to the functioning biological systems.

The outline of this paper is as follows (Figure 4).

2. Mechanical Network Model

In this section, we first present the physical system for Maxwell bulk fluids that are in contact with the flexoelectric membrane and then use the mechanical spring-dashpot network model to derive the total stress tensor σ of the device as it creates a flow rate Q in the fluid phases and forces through the wall shear stress [3,4,5]. We note that the flexoelectric membrane is attached to a cylindrical tube, and the bulk fluid flow imposes a wall shear stress on the tube walls.

2.1. Device Material Structure and Geometry

The device considered in this paper is shown in Figure 5, since gravity plays no role, due to the overall orientation of the device. A flexoelectric membrane (red sector, M) is tethered to a cylindrical capillary system {r = a, z = L}. At the left and right sectors of the membrane, there are two Maxwell viscoelastic liquids [3,4,5,28].

The material parameters that define the device’s viscoelasticity are: (a) viscosities {η_b, η_t}, (b) relaxation times {λ_b, λ_t}, and (c) elastic flexoelectric membrane {G}, respectively [3,4,5,28]. The flexoelectric effect of the membrane caused by the imposed sinusoidal E_z(t) field and that there is not an externally imposed pressure gradient in the axial direction [3,4,5].

2.2. Mechanical Analogue

The mechanical analogue of the flow and force generator device (see Figure 5) is shown in Figure 6. The viscoelastic fluid elements are in parallel to the elastic flexoelectric membrane of modulus G. The selection of an in-parallel network is a key ingredient in the device model and defines the couplings between fluid flows and elastic membrane and can be easily generalized in future work to other fluids and to other elastic elements besides the membrane G.

The viscoelastic liquid phases are characterized by Maxwell linear operators. In the viscoelastic phase (Figure 5, right), the total deformation is given by γ = (γv)_R + (γe)_L, and the shear stress in the viscous and elastic contribution satisfies σv = σe = σ_R [28]. Then, taking the time derivative of the deformation dγ/dt = d(γv)_R /dt + d(γe)_L/dt and assuming that the viscous and elastic contribution in the system can be modeled as Newtonian fluid and Hooke solid, respectively, the following expression is obtained [28]:

$$2\mathbf{D}=\stackrel{·}{\mathsf{\gamma }}=\frac{1}{{\mathsf{\eta }}_{\mathrm{R}}}{\mathsf{\sigma }}_{\mathrm{R}}+\frac{1}{{\mathrm{G}}_{\mathrm{R}}}\frac{\mathsf{\partial }{\mathsf{\sigma }}_{\mathrm{R}}}{\mathsf{\partial }\mathrm{t}}$$

(1)

Using the inverse operator formalism with Dt = ∂/∂t, we get [3,4]:

$${\mathsf{\sigma }}_{\mathrm{R}}={\mathsf{\eta }}_{\mathrm{R}}\frac{1}{1+{\mathsf{\lambda }}_{\mathrm{R}}\mathrm{D}\mathrm{t}}2D$$

(2)

where λ_R = η_R/G_R, G_R, and η_R are the relaxation time, elastic modulus, and viscosity of the Maxwell constitutive equation corresponding to the viscoelastic phase (Figure 5, right). Likewise, the left-side phase in Figure 5 is described by [3,4,28]:

$$2D=\stackrel{·}{\mathsf{\gamma }}=\frac{1}{{\mathsf{\eta }}_{\mathrm{L}}}{\mathsf{\sigma }}_{\mathrm{L}}+\frac{1}{{\mathrm{G}}_{\mathrm{L}}}\frac{\partial {\mathsf{\sigma }}_{\mathrm{L}}}{\partial \mathrm{t}}$$

(3)

and

$${\mathsf{\sigma }}_{\mathrm{L}}={\mathsf{\eta }}_{\mathrm{L}}\frac{1}{1+{\mathsf{\lambda }}_{\mathrm{L}}\mathrm{D}\mathrm{t}}2D$$

(4)

where λ_L, G_L, and η_L are the relaxation time, elastic module, and viscosity of the left phase in Figure 5. The elasticity of the flexoelectric membrane is given by a Hooke solid equation [28]:

$${\mathsf{\sigma }}_{\mathrm{M}}=\mathrm{G}\mathsf{\gamma }$$

(5)

where G can be expressed in the following form [3,4]:

$$\mathrm{G}=\frac{2{\mathsf{\gamma }}_0+\left(2{\mathrm{k}}_0+{\mathrm{k}}_1\right)\Im }{4\mathrm{L}}$$

(6)

and γ is the second-order strain tensor [3,4,5,11]. In Equation (6), γ₀ is the interfacial surface tension at zero electric field, k₀ and k₁ are the membrane bending rigidity and torsion elastic moduli, and L is the characteristic axial length of the device [5,11] (See Appendix C for details of the shape membrane equation). The total stress σ of mechanical network is given by the sum of the three stress contributions [3,4]:

$$\mathsf{\sigma }={\mathsf{\sigma }}_{\mathrm{L}}+{\mathsf{\sigma }}_{\mathrm{M}}+{\mathsf{\sigma }}_{\mathrm{R}}$$

(7)

In Equation (7), all the deformations in the elements are equal: γv = γ_R = γ_M and:

$$\mathsf{\sigma }={\mathsf{\eta }}_{\mathrm{L}}\left(\frac{1}{1+{\mathsf{\lambda }}_{\mathrm{L}}\mathrm{D}\mathrm{t}}\right)2D+\mathrm{G}\mathsf{\gamma }+{\mathsf{\eta }}_{\mathrm{R}}\left(\frac{1}{1+{\mathsf{\lambda }}_{\mathrm{R}}\mathrm{D}\mathrm{t}}\right)2D$$

(8)

and taking the time derivative

$$\frac{\partial \mathsf{\sigma }}{\partial \mathrm{t}}={\mathsf{\eta }}_{\mathrm{L}}\left(\frac{1}{1+{\mathsf{\lambda }}_{\mathrm{L}}\mathrm{D}\mathrm{t}}\right)2\frac{\partial }{\partial \mathrm{t}}D+\mathrm{G}2D+{\mathsf{\eta }}_{\mathrm{R}}\left(\frac{1}{1+{\mathsf{\lambda }}_{\mathrm{R}}\mathrm{D}\mathrm{t}}\right)2\frac{\partial }{\partial \mathrm{t}}D$$

(9)

Using the reciprocal of the inverse operator, Equation (9) gives a new asymmetric cubic stress-quadratic deformation rate evolution model:

$$\frac{\partial \mathsf{\sigma }}{\partial \mathrm{t}}+{\Sigma }_{\mathsf{\lambda }}\frac{{\partial }^2\mathsf{\sigma }}{\partial {\mathrm{t}}^2}+{\Pi }_{\mathsf{\lambda }}\frac{{\partial }^3\mathsf{\sigma }}{\partial {\mathrm{t}}^3}=2\mathrm{G}\left(D+\left({\Sigma }_{\mathsf{\lambda }}+\frac{{\Sigma }_{\mathsf{\eta }}}{\mathrm{G}}\right)\frac{\partial D}{\partial \mathrm{t}}+{\Pi }_{\mathsf{\lambda }}\left(1+\frac{{\Sigma }_{\mathrm{G}}}{\mathrm{G}}\right)\frac{{\partial }^{2}D}{\partial {\mathrm{t}}^2}\right)$$

(10)

The fluidity operator version for Equation (10) is:

$$2D={\mathrm{O}}_{\Phi }\left({\mathrm{D}}_{\mathrm{t}}\right)\mathsf{\sigma };{\mathrm{O}}_{\Phi }\left({\mathrm{D}}_{\mathrm{t}}\right)=\frac{1}{{\mathrm{O}}_{\mathsf{\eta }}\left({\mathrm{D}}_{\mathrm{t}}\right)};{\mathrm{D}}_{\mathrm{t}}=\frac{\partial }{\partial \mathrm{t}}$$

(11)

where O_Φ(Dt) = 1/Oη(Dt) is the fluidity operator (inverse of the complex viscosity Oη) of the Maxwell fluids and the solid flexoelectric membrane [3,4]. Note that the system describes a discontinuity due to the solid flexoelectric membrane, so we assume that the width of the flexoelectric membrane is small, and the asymmetric fluidity operator:

$${\mathrm{O}}_{\Phi }\left({\mathrm{D}}_{\mathrm{t}}\right)=\frac{1}{{\mathrm{O}}_{\mathsf{\eta }}\left({\mathrm{D}}_{\mathrm{t}}\right)}=\frac{{\mathrm{D}}_{\mathrm{t}}}{\mathrm{G}\left({\mathrm{D}}_{\mathrm{t}}\right)}=\frac{{\mathrm{D}}_{\mathrm{t}}}{\mathrm{G}}\frac{1+{\Sigma }_{\mathrm{\lambda }}{\mathrm{D}}_{\mathrm{t}}+{\Pi }_{\mathrm{\lambda }}{\mathrm{D}}_{\mathrm{t}}^2}{1+\left({\Sigma }_{\mathrm{\lambda }}+\frac{{\Sigma }_{\mathsf{\eta }}}{\mathrm{G}}\right){\mathrm{D}}_{\mathrm{t}}+{\Pi }_{\mathrm{\lambda }}\left(1+\frac{{\Sigma }_{\mathrm{G}}}{\mathrm{G}}\right){\mathrm{D}}_{\mathrm{t}}^2}$$

(12)

where the following material properties are defined [3,4]:

$${\Sigma }_{\mathsf{\lambda }}{=\mathsf{\lambda }}_{\mathrm{R}}{+\mathsf{\lambda }}_{\mathrm{L}}{; \Pi }_{\mathsf{\lambda }}{=\mathsf{\lambda }}_{\mathrm{R}}{\mathsf{\lambda }}_{\mathrm{L}}{; \Sigma }_{\mathrm{G}}{=\mathrm{G}}_{\mathrm{R}}{+\mathrm{G}}_{\mathrm{L}}{; \Sigma }_{\mathsf{\eta }}{=\mathsf{\eta }}_{\mathrm{R}}{+\mathsf{\eta }}_{\mathrm{L}}$$

(13)

Here, Σ_G = G_R + G_L is the total bulk elasticity, Σ_λ = λ_R + λ_L the total viscoelasticity, Π = λ_Rλ_L the memory product, and Ση = η_R + η_L the total bulk viscosity [3,4]. In Equation (12), the fluidity operator O_Φ(D_t) provides the link between stress σ and deformation rate D. Equation (10) is a variant of a Maxwell model, which describes a thermodynamic system associated with two viscoelastic liquid phases and a solid flexoelectric membrane and describes creep and stress relaxation mechanisms [3,4]. Equation (12) depends on the time differential operator Dt~1/t, which scales with the inverse of time. Note that a key point in the new developed rheological model is the asymmetry in the time derivatives of the shear stress and shear strain tensors.

The first one is of the third order, whereas the second one is of order two. This controls the mathematical shape of the resonance spectrum. Under periodic forcing of frequency ω, Dt = ω and Equation (12) admits two asymptotic limits: (i) the flexo-viscous terminal (ω → 0) mode and (ii) flexo-bulk elastic large (ω → ∞) frequency mode:

$$\begin{array}{l}\left(\mathrm{i}\right) \mathrm{Flexo}-\mathrm{viscous} \mathrm{mode}:\mathrm{Im}\underset{\mathsf{\omega }\to 0}{\lim }{\mathrm{O}}_{\mathsf{\eta }}\simeq \frac{\mathrm{G}}{\mathsf{\omega }},\mathrm{Re}\underset{\mathsf{\omega }\to 0}{\lim }{\mathrm{O}}_{{}_{\mathsf{\eta }}} \simeq {\mathsf{\eta }}_{\mathrm{R}}+{\mathsf{\eta }}_{\mathrm{L}};\\ \left(\mathrm{ii}\right) \mathrm{Flexo}-\mathrm{bulk}-\mathrm{elastic} \mathrm{mode}:\mathrm{Im}\underset{\mathsf{\omega }\to \infty }{\lim }{\mathrm{O}}_{\mathsf{\eta }}\simeq \frac{\mathrm{G}+{ \mathrm{G}}_{\mathrm{R}}{+\mathrm{G}}_{\mathrm{L}}}{\mathsf{\omega }},\mathrm{Re}\underset{\mathsf{\omega }\to 0}{\lim }{\mathrm{O}}_{{}_{\mathsf{\eta }}}\simeq 0\end{array}$$

(14)

where Im denotes imaginary and Re denotes real. Under small frequency, Equation (14) and Figure 6 indicate that bulk elasticity plays no role because the slow deformation rate activates only the bulk viscous dashpots; the frequency scaling in this regime is 1/ω. Under large frequency, Equation (14) and Figure 6 indicate that bulk viscosities of the two dashpots plays no role because the fast deformation rate activates only the bulk elastic springs while the dashpots are frozen. Then, we conclude that full viscoelasticity is only observed under intermediate frequencies, as discussed in full detail below. In partial summary, the network model offers an efficient route to derive the total stress tensor σ and the complex fluidity and viscosity operators {O_Φ(ω), O_η(ω)} of the device [3,4]. Viscoelastic flow Q and wall stress σ_w can be generated at intermediate frequencies ω. These results will be used to find the norm of the complex flow and stress transfer functions in the device. These new results provide symmetric and asymmetric operators and their proper mechanical configuration.

3. Transfer Functions from the Fourier Domain

In this section, we derive the transfer function using the following sequential steps: (i) using the linear momentum balance equation, we first find the axial velocity V; (ii) integrating the axial velocity, we find the volumetric flow rate Q; and, finally, (iii) identifying the device’s inputs and outputs in the Fourier domain, we find the flow FTF and stress transfer functions STF.

3.1. Continuity and Transport Momentum Equations

The mass balance equation (without chemical reaction):

$$\frac{\partial {\mathsf{\rho }}_{\mathrm{x}}}{\partial \mathrm{t}}+\nabla \cdot {\mathsf{\rho }}_{\mathrm{x}}V=0; \mathrm{x}=\{\mathrm{L},\mathrm{R}\}$$

(15)

The general Cauchy linear momentum balance equation is:

$$\frac{\partial }{\partial \mathrm{t}}{\mathsf{\rho }}_{\mathrm{x}}V=-\nabla \cdot \mathrm{T}; \mathrm{x}=\left\{\mathrm{L},\mathrm{R}\right\}$$

(16)

where T includes the scalar pressure p and the unit tensor I, ρ_x is the density of the liquid phases, V ⊗ V is the dyadic product tensor of the velocity vector field V and

$$T={\mathrm{p}}_{\mathrm{X}}I+{\mathsf{\rho }}_{\mathrm{X}}V\otimes V-\mathsf{\sigma }=\frac{{\mathrm{c}}_{\mathrm{f}}\Im }{\mathrm{L}}{\Phi }_{\mathrm{E}}I+{\mathsf{\rho }}_{\mathrm{X}}V\otimes V-\mathsf{\sigma }$$

(17)

In Equation (17), c_f is the flexoelectric constant ([=] Newtons/Coulomb), ℑ is the shape factor for the hemispheric cup ℑ = 8/a², L is the characteristic axial length, and Φ_E is the electrical potential field [3,4,5,11].

3.2. Complex Axial Velocity and Boundary Conditions

Assuming incompressibility in the liquid viscoelastic phases, isothermal process, the gravitational forces are neglected, the velocity vector field V has only one component different from zero, i.e., V × e_z = (0, 0, V_z(r, t)), and the process depends on time [3,5]. The fluid is sheared by the effect of the applied electric field, and once the Equations (16) and (17) are combined, the following expression is obtained for the velocity field:

$$V\left(\mathrm{r},\mathrm{t}\right)=\frac{{\mathrm{c}}_{\mathrm{f}}\left(\Im /\mathrm{L}\right){\mathrm{O}}_{\Phi }\left(\mathrm{D}\mathrm{t}\right)}{{\nabla }^2+{\mathrm{i}}^2{\mathrm{O}}_{\Phi }\left(\mathrm{D}\mathrm{t}\right)\Sigma \mathsf{\rho }\mathrm{D}\mathrm{t}}\nabla {\Phi }_{\mathrm{E}}\left(\mathrm{t}\right)$$

(18)

In Equation (18), O_Φ(Dt) is the linear operator defined in Equations (12) and (13), Σρ = ρ_R + ρ_L and Dt = ∂/∂t is the time partial derivative of the rheological equation of state [3,4]. The linear partial differential model given by Equation (18) satisfies the following boundary conditions: (i) non-slip mechanisms at the wall, and (ii) the symmetry of the velocity field [3,4]:

$${\mathrm{V}}_{\mathrm{z}}\left(\mathrm{r}=\mathrm{a},\mathrm{t}\right)={\left.V\left(\mathrm{r},\mathrm{t}\right)\cdot {\mathrm{e}}_{\mathrm{z}}\right|}_{\mathrm{r}=\mathrm{a}}=0$$

(19)

$${\left.\frac{\partial {\mathrm{V}}_{\mathrm{z}}}{\partial \mathrm{r}}\right|}_{\mathrm{r}=0}={\left.\left[\frac{\partial }{\partial \mathrm{r}}\left(V\left(\mathrm{r},\mathrm{t}\right)\cdot {\mathrm{e}}_{\mathrm{z}}\right)\right]\right|}_{\mathrm{r}=0}=0$$

(20)

3.3. Formulation in the Fourier Domain and Complex Wave Vector β(ω)

Multiplying Equation (18) for the axial unit vector e_z and applying the standard Fourier transform

$$\mathrm{F}\left(\mathsf{\omega }\right)=\frac{1}{\sqrt{2\mathsf{\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }\mathrm{f}\left(\mathrm{t}\right){\mathrm{e}}^{-\mathrm{i}\mathsf{\omega }\mathrm{t}}\mathrm{d}\mathrm{t}}$$

(21)

the axial velocity V_z (r,ω) in the Fourier domain is obtained:

$${\mathrm{V}}_{\mathrm{z}}\left(\mathrm{r},\mathsf{\omega }\right)=\frac{{\mathrm{c}}_{\mathrm{f}}\left(\Im /\mathrm{L}\right){\mathrm{O}}_{\Phi }\left(\mathrm{i}\mathsf{\omega }\right)}{\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\mathrm{r}\frac{\partial }{\partial \mathrm{r}}+{\mathsf{\beta }}^2\left(\mathsf{\omega }\right)}\nabla \mathrm{z}{\Phi }_{\mathrm{E}}\left(\mathsf{\omega }\right)$$

(22)

In Equation (22), the key reciprocal length scale parameter β (ω) appearing in the denominator of Equation (22) is given by:

$$\mathsf{\beta }\left(\mathsf{\omega }\right)=\mathrm{Re}\left[\mathsf{\beta }\left(\mathsf{\omega }\right)\right]+\mathrm{I}\mathrm{Im}\left[\mathsf{\beta }\left(\mathsf{\omega }\right)\right]={\mathrm{i}}^{3/2}\sqrt{\Sigma \mathsf{\rho }\cdot \mathsf{\omega }\cdot {\mathrm{O}}_{\Phi }\left(\mathrm{i}\mathsf{\omega }\right)}$$

(23)

The parameter β(ω) is defined as the complex wave vector associated with the axial velocity V_z (ω), which is the inverse of a characteristic length [3,4]:

3.4. Scales, Dimensionless Numbers, and Device Mode Classification Based on Kinematics and Rheology

In this section, we discuss (i) the length and time scales and dimensionless numbers of this device and (ii) use them to present a device mode classification in terms of flexo-viscous-bulk elastic contributions.

(a)

Scales and dimensionless numbers

Equations (12) and (23) provide a basis to characterize the device modes according to frequency and dimensionless numbers; quantities with overbars are dimensionless (See Appendix A). In order to simplify the analysis of the physical system, we propose scaling laws to simplify the physical interpretation and to find the dimensionless groups needed to identify the macroscopical forces in the system. The details of the dimensionless variables and groups is explained in Appendix B. In what follows, all the variables are in dimensionless form.

The wave vector $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$ in Equation (23) is a crucial parameter and has units of the inverse of a characteristic length and depends on imposed frequency, the dimensionless number E (defined below), and the rheology and flow through the fluidity operator ${\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)$.

$$\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)=\mathrm{Re}\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]+\mathrm{I}\mathrm{Im}\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]={\mathrm{i}}^{3/2}{\mathrm{E}}^{-1/2}\sqrt{\overline{\mathsf{\omega }}\cdot {\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}$$

(24)

Equation (24) shows that the complex wave vector $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$ depends on the product of the frequency $\overline{\mathsf{\omega }}$; the fluidity operator ${\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)$ can be expressed as follows:

$${\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)=\frac{1}{{\overline{\mathrm{O}}}_{\mathsf{\eta }}\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}=\mathrm{i}\overline{\mathsf{\omega }}\frac{1-\overline{\mathrm{m}}}{\overline{\mathrm{m}}}\frac{1+\mathrm{i}\overline{\mathsf{\omega }}+{\overline{\Pi }}_{\mathsf{\lambda }}{\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}^2}{1+\left(1+\frac{1-\overline{\mathrm{m}}}{\overline{\mathrm{m}}}\cdot {\overline{\Sigma }}_{\mathsf{\eta }}\right)\mathrm{i}\overline{\mathsf{\omega }}+\frac{{\overline{\Pi }}_{\mathrm{\lambda }}}{\overline{\mathrm{m}}}{\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}^2}$$

(25)

In the fluidity operator Equation (25), there are three fundamental dimensionless groups: (i) total bulk viscosity (${\overline{\Sigma }}_{\mathsf{\eta }}$), (ii) memory (${\overline{\Pi }}_{\mathrm{\lambda }}$), and (iii) elastic membrane ratio ($\overline{\mathrm{m}}$). In Equation (24), the complex wave vector $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$ can be expressed in terms of the complex compliance ${\overline{\mathrm{J}}}^{*}\left(\overline{\mathsf{\omega }}\right)$, since we are interested in analyzing the start-up mechanisms in the system [28] (inertial mechanisms).

$$\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)={\mathrm{i}}^2{\mathrm{E}}^{-1/2}\cdot \overline{\mathsf{\omega }}\cdot \sqrt{{\overline{\mathrm{J}}}^{*}\left(\overline{\mathsf{\omega }}\right)}$$

(26)

In Equation (26), the complex compliance J*(ω) is given by:

$${\overline{\mathrm{J}}}^{*}\left(\overline{\mathsf{\omega }}\right)=\frac{1}{{\overline{\mathrm{G}}}^{*}\left(\overline{\mathsf{\omega }}\right)}=\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{\mathrm{i}\overline{\mathsf{\omega }}}$$

(27)

The dimensionless elastic number E (Equation (26)) is the ratio of two fundamental numbers: (i) the Weissenberg (We) and (ii) the Reynolds number (Re) [28]. The elastic number E is given by [29,30,31]:

$$\mathrm{E}=\frac{\mathrm{W}\mathrm{e}}{\mathrm{Re}}=\frac{{\Sigma }_{\mathrm{\lambda }}}{\Sigma \mathsf{\rho }{\mathrm{a}}^2/{\Sigma }_{\mathsf{\eta }}}$$

(28)

Note that there is no imposed macroscopic velocity in the system, and for this reason it is better to explain all the mechanisms in terms of the elastic number E, instead of other dimensionless numbers such as (i) Deborah, (ii) Mach, (iii) Newtonian Womersley number, and (iv) vibratile Reynolds [32,33]. The wavelength vector b (Equation (26)) deals with the pulsating Womersley number related to the competition between the viscous and oscillatory mechanisms [32,33]:

$${\mathrm{W}}_0^2=\left({\mathrm{a}}^2\frac{\Sigma \mathsf{\rho }}{\Sigma \mathsf{\eta }}\right)\mathsf{\omega }={\mathrm{t}}_{\mathrm{v}}\mathsf{\omega }$$

(29)

where t_v is visco-inertial time scale. In terms of the elastic number E, the non-Newtonian Womersley number (Wo) is given by [34,35,36,37]:

$${\mathrm{W}}_0=\sqrt{\frac{{\mathrm{t}}_{\mathrm{v}}}{{\mathrm{t}}_{\mathrm{v}\mathrm{e}}}\left({\mathrm{t}}_{\mathrm{v}\mathrm{e}}\mathsf{\omega }\right)}=\sqrt{\frac{1}{{\mathrm{t}}_{\mathrm{v}\mathrm{e}}/{\mathrm{t}}_{\mathrm{v}}}\left({\mathrm{t}}_{\mathrm{v}\mathrm{e}}\mathsf{\omega }\right)}={\mathrm{E}}^{-1/2}\sqrt{\overline{\mathsf{\omega }}}$$

(30)

where time scale t_ve can be read off [34,35,36,37]. Note that Wo and E numbers are related through the inverse relation in Equation (28). The Womersley number describes the competition between the flow frequency and the dissipation associated with the viscous (Newtonian) or viscoelastic (non-Newtonian) effects, and it increases with frequency [34,35,36,37,38].

(b)

Device mode classification based on flow kinematics

Equation (23) indicates the crucial importance of the complex velocity wave vector $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$; units of beta: 1/length. Hence, we expect that this reciprocal length scale provides a direct link to the classification of the modes presents in the device as the frequency increases. The computed norm of the complex wave vector as a function of dimensionless frequency is shown in Figure 7. It clearly shows the three-mode response: flexo-viscous with linear frequency scaling, intermediate flexo-viscoelastic with square-root frequency scaling, and flexo-bulk elastic linear scaling [29,30]. The role of the elastic number is a vertical shift [31,32]. This primitive quantity $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$ associated with the velocity field shows the scaling of its norm with the Womersley (Wo) [33,34,35,36,37,38] and elastic (E) number [31,32] is:

$$\begin{array}{l}(\mathrm{i}) \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{s}:\mathsf{\beta }\propto \mathrm{E}\overline{\mathsf{\omega }}={\mathrm{E}}^2{\mathrm{W}}_0{}^2={\left(\mathrm{W}\mathrm{e}/\mathrm{Re}\right)}^2{\mathrm{W}}_0{}^2\\ (\mathrm{i}\mathrm{i}) \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}:\mathsf{\beta }\propto \mathrm{E}\sqrt{\overline{\mathsf{\omega }}}=\sqrt{\mathrm{E}}{\mathrm{W}}_0=\sqrt{\mathrm{W}\mathrm{e}/\mathrm{Re}}{\mathrm{W}}_0\\ (\mathrm{i}\mathrm{i}\mathrm{i}) \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}-\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}:\mathsf{\beta }\propto \mathrm{E}\overline{\mathsf{\omega }}={\left(\mathrm{W}\mathrm{e}/\mathrm{Re}\right)}^2{\mathrm{W}}_0{}^2\end{array}$$

(31)

(c)

Device mode classification based on rheology

A complementary device mode classification can be conducted based on the complex viscosity function ${\overline{\mathrm{O}}}_{\mathsf{\eta }}$ (see Equation (11)). In Figure 7, complex viscosity is plotted vs. frequency for our visco-elastic rheological model. Herrera-Valencia and Rey (2018) showed that these dimensionless numbers describe a 3D-prismatic space [3]. They established that the system optimum conditions are a floppy membrane ($\overline{\mathrm{m}}=1\times {10}^{-4}$), total bulk viscosity (large dissipation, ${\overline{\Sigma }}_{\mathsf{\eta }}=1$), viscoelastic contrast between the bulk fluids (${\overline{\Pi }}_{\mathsf{\lambda }}=1\times {10}^{-4}$), and balanced flexoelectric and bulk elastic mechanisms (Qr = 1), defined in Equation (34) below [3].

Figure 8 shows the three modes envisioned by analyzing the dashpot-spring network mentioned above:

$$\begin{array}{l}\left(\mathrm{i}\right) \mathrm{Flexo}-\mathrm{viscous} \mathrm{mode}:\mathrm{Im}\underset{\mathsf{\omega }\to 0}{\lim }{\overline{\mathrm{O}}}_{\mathsf{\eta }}\simeq \frac{\overline{\mathrm{G}}}{\overline{\mathsf{\omega }}}=\frac{\overline{\mathrm{G}}}{{\mathrm{W}}_0{}^2\mathrm{E}},\mathrm{Re}\underset{\mathsf{\omega }\to 0}{\lim }{\overline{\mathrm{O}}}_{{}_{\mathsf{\eta }}} \simeq {\overline{\Sigma }}_{\mathsf{\eta }}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\\ \left(\mathrm{ii}\right) \mathrm{Flexo}-\mathrm{viscou}-\mathrm{elastic} \mathrm{mode}:\left|{\overline{\mathrm{O}}}_{\mathsf{\eta }}\right|=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\\ \left(\mathrm{iii}\right) \mathrm{Flexo}-\mathrm{bulk}-\mathrm{elastic} \mathrm{mode}:\underset{\mathsf{\omega }\to \infty }{\lim }\left|{\mathrm{O}}_{\mathsf{\eta }}\right|\simeq \frac{\overline{\mathrm{G}}+{\overline{\Sigma }}_{\overline{\mathrm{G}}}}{\overline{\mathsf{\omega }}}=\frac{\overline{\mathrm{G}}+{\overline{\Sigma }}_{\overline{\mathrm{G}}}}{{\mathrm{W}}_0{}^2\mathrm{E}}\end{array}$$

(32)

Hence, the rheological characterization clearly distinguishes the three modes; more importantly, it shows the band of frequencies where we can expect resonances and hence more flow and more shear wall stress.

Below, we will discuss how the complex viscosity-frequency responds to changes in material properties, in particular at intermediate frequencies.

3.5. Complex Axial Velocity

The solution of Equation (18) is given in terms of a combination of zero order Bessel functions:

$$\overline{\mathrm{V}}\mathrm{z}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)={\overline{\mathrm{C}}}_1{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\overline{\mathrm{r}}\right]+{\overline{\mathrm{C}}}_2{\mathrm{Y}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\overline{\mathrm{r}}\right]+\mathrm{Q}\mathrm{r}\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{{\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)}^2}\overline{\nabla }\mathrm{z}{\overline{\Phi }}_{\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)$$

(33)

where Qr is the ratio between the flexoelectric and bulk elastic mechanisms, and it is given by:

$$\mathrm{Q}\mathrm{r}=\frac{\mathrm{a}{\mathrm{c}}_{\mathrm{f}}\Im {\mathrm{E}}_0/\mathrm{L}}{{\Sigma }_{\mathrm{G}}}=\frac{\mathrm{F}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}-\mathrm{M}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{s}}{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}-\mathrm{E}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{M}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{s}}$$

(34)

In Equation (33), {J₀, Y₀} are the classical special Bessel functions [3]. From the boundary conditions (Equations (19) and (20)), the constants ${\overline{\mathrm{C}}}_1\left(\overline{\mathsf{\omega }}\right)$ and ${\overline{\mathrm{C}}}_2\left(\overline{\mathsf{\omega }}\right)$ are calculated as follows:

$${\overline{\mathrm{C}}}_1\left(\overline{\mathsf{\omega }}\right)=\mathrm{Q}\mathrm{r}\frac{1}{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]}\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{{\overline{\mathsf{\beta }}}^2\left(\overline{\mathsf{\omega }}\right)}\left({\mathrm{i}}^2\overline{\nabla }\mathrm{z}{\overline{\Phi }}_{\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)\right)$$

(35)

$${\mathrm{C}}_2\left(\overline{\mathsf{\omega }}\right)=0$$

(36)

The constant ${\overline{\mathrm{C}}}_2\left(\overline{\mathsf{\omega }}\right)$ must be set up to zero because, physically, the velocity cannot be infinity at $\overline{\mathrm{r}}=0$. Then, the general profile for the axial velocity $\overline{\mathrm{V}}\mathrm{z}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)$ is:

$${\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)=\mathrm{Re}\left[{\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)\right]+\mathrm{i}\mathrm{Im}\left[{\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)\right]=\mathrm{Q}\mathrm{r}\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{{\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)}^2}\left(\frac{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]-{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\overline{\mathrm{r}}\right]}{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]}\right)\left(\overline{\nabla }\mathrm{z}{\overline{\Phi }}_{\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)\right)$$

(37)

Equation (37) is the sum of an r-dependent function and an r-independent driving force term that always vanishes at the bounding walls. The effectiveness of the driving term is regulated by the flexoelectricity number Qr, and it is zero for a purely elastic membrane. The norm of the complex velocity is:

$${\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)=\sqrt{{\left(\mathrm{Re}\left[{\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)\right]\right)}^2+{\left(\mathrm{Im}\left[{\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)\right]\right)}^2}$$

(38)

The velocity at the center of the cylindrical geometry is:

$$\overline{\mathrm{V}}\max \left(\overline{\mathsf{\omega }}\right)={\overline{\mathrm{V}}}_{\mathrm{z}}\left(\overline{\mathrm{r}}=0,\overline{\mathsf{\omega }}\right)=\mathrm{Q}\mathrm{r}\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{{\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)}^2}\left(\frac{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]-1}{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]}\right)\left(\overline{\nabla }\mathrm{z}{\overline{\Phi }}_{\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)\right)$$

(39)

3.6. Complex Volumetric Flow Rate and Transfer Functions

The right and left volumetric flow rates ${\overline{\mathrm{Q}}}_{\mathrm{R}}\left(\overline{\mathsf{\omega }}\right)$ and ${\overline{\mathrm{Q}}}_{\mathrm{L}}\left(\overline{\mathsf{\omega }}\right)$ (see Figure 5) are induced by the speed of the flexoelectric membrane as it curves into the contacting viscoelastic liquid phases:

$$\overline{\mathrm{Q}}\left(\overline{\mathsf{\omega }}\right)=\left|{\overline{\mathrm{Q}}}_{\mathrm{R}}\left(\overline{\mathsf{\omega }}\right)\right|=\left|{\overline{\mathrm{Q}}}_{\mathrm{L}}\left(\overline{\mathsf{\omega }}\right)\right|=2{\displaystyle \underset{0}{\overset{1}{\int }}\overline{\mathrm{V}}\mathrm{z}\left(\overline{\mathrm{r}},\overline{\mathsf{\omega }}\right)\overline{\mathrm{r}}\mathrm{d}\overline{\mathrm{r}}}$$

(40)

Replacing the axial velocity profile (Equation (37)) into the volumetric flow rate (Equation (40))

$$\overline{\mathrm{Q}}\left(\overline{\mathsf{\omega }}\right)=\mathrm{Q}\mathrm{r}\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{{\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)}^2}\left(1-2\frac{{\mathrm{J}}_1\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]/\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)}{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]}\right)\left(\overline{\nabla }\mathrm{z}{\overline{\Phi }}_{\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)\right)$$

(41)

Equation (41) can be rewritten in the following compact form:

$$\underset{\mathrm{F}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}}{\underbrace{\overline{\mathrm{Q}}\left(\overline{\mathsf{\omega }}\right)}}=\underset{\mathrm{F}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{transfer} \mathrm{function}}{\underbrace{{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)}}\cdot \underset{\mathrm{F}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{s}\mathrm{t}\mathrm{R}\mathrm{e}\mathrm{s}\mathrm{s}}{\underbrace{{\overline{\mathsf{\sigma }}}_{\mathrm{F}\mathrm{E}}}}$$

(42)

In Equation (42), ${\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)$, ${\overline{\mathsf{\sigma }}}_{\mathrm{F}\mathrm{E}}$ are the complex flow transfer function and the wall flexoelectric stress force in the system (See Appendix C). The wall flexoelectric stress force ${\overline{\mathsf{\sigma }}}_{\mathrm{F}\mathrm{E}}$ is given in terms of the negative of the electrical potential:

$${\overline{\mathsf{\sigma }}}_{\mathrm{F}\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)=\frac{1}{2}{\overline{\mathrm{E}}}_{\mathrm{z}}\left(\overline{\mathsf{\omega }}\right)=\frac{1}{2}\left(-\overline{\nabla }\mathrm{z}{\overline{\Phi }}_{\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)\right)$$

(43)

3.7. Flow Transfer Function (${\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)$)

Starting from Equations (41) and (42), the dimensionless complex flow transfer function, ${\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)$ can be expressed as follows:

$${\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)=\underset{\mathrm{F}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}-\mathrm{E}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}}{\underbrace{\mathrm{Q}\mathrm{r}}}\cdot \underset{\mathrm{R}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{y}}{\underbrace{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}}\cdot \underset{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{\overline{\mathrm{F}}\left(\overline{\mathsf{\beta }}\right)}}$$

(44)

where the nonlinear dispersion contribution $\overline{\mathrm{F}}\left(\overline{\mathsf{\beta }}\right)$ is given by:

$$\overline{\mathrm{F}}\left(\overline{\mathsf{\beta }}\right)=\frac{8{\mathrm{i}}^2}{{\overline{\mathsf{\beta }}}^2\left(\overline{\mathsf{\omega }}\right)}\left(1-2\frac{{\mathrm{J}}_1\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]/\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)}{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]}\right)$$

(45)

It is important to note that the dispersion mechanisms depend only on the wave vector $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$ parameter, through the elastic E number, the rheology (fluidity operator), and the frequency. Mathematically, the norm of the flow transfer function (Equation (44)) is given by:

$$\left|{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }},\overline{\mathrm{m}},{\overline{\Pi }}_{\mathrm{\lambda }},{\overline{\Sigma }}_{\mathsf{\eta }},\mathrm{E}\right)\right|=\sqrt{{\left(\mathrm{Re}[{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }},\overline{\mathrm{m}},{\overline{\Pi }}_{\mathrm{\lambda }},{\overline{\Sigma }}_{\mathsf{\eta }},\mathrm{E}\right)]\right)}^2+{\left(\mathrm{Im}[{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }},\overline{\mathrm{m}},{\overline{\Pi }}_{\mathrm{\lambda }},{\overline{\Sigma }}_{\mathsf{\eta }},\mathrm{E}\right)]\right)}^2}$$

(46)

3.8. Stress Transfer Function (Ts)

In this section, the dimensionless stress transfer function ${\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\omega }}\right)$ is derived. The wall flexoelectric stress (mentioned above) is:

$${\overline{\mathsf{\sigma }}}_{\mathrm{F}\mathrm{E}}\left(\overline{\mathsf{\omega }}\right)=\frac{1}{2}{\overline{\mathrm{E}}}_{\mathrm{z}}\left(\overline{\mathsf{\omega }}\right)$$

(47)

The wall stress σ(ω) can be estimated in a straightforward calculation using Equations (11) and (12):

$$\overline{\mathsf{\sigma }}\left(\overline{\mathsf{\omega }}\right)={\left.\left(\frac{1}{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}\right)\frac{\partial \overline{\mathrm{V}}\mathrm{z}\left(\overline{\mathsf{\omega }}\right)}{\partial \overline{\mathrm{r}}}\right|}_{\overline{\mathrm{r}}=1}=\left(\frac{1}{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}\right){\overline{\mathrm{C}}}_1\left(\overline{\mathsf{\omega }}\right)\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right){\mathrm{J}}_1\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]$$

(48)

Once the constant ${\overline{\mathrm{C}}}_1\left(\overline{\mathsf{\omega }}\right)$ (Equation (31)) is inserted into Equation (41), the dimensionless stress transfer function $\overline{\mathrm{T}}\mathrm{s}\left(\overline{\mathsf{\omega }}\right)$ (STF) is obtained:

$$\underset{\mathrm{Inertial}-\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}{\underbrace{\overline{\mathsf{\sigma }}\left(\overline{\mathsf{\omega }}\right)}}=\underset{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{e}\mathrm{r} \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{{\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)}}\cdot \underset{\mathrm{F}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}{\underbrace{{\overline{\mathsf{\sigma }}}_{\mathrm{FE}}\left(\overline{\mathsf{\omega }}\right)}}$$

(49)

The dimensionless stress transfer function $\overline{\mathrm{T}}\mathrm{s}\left(\overline{\mathsf{\omega }}\right)$

$${\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)=2\mathrm{Q}\mathrm{r}\frac{{\mathrm{J}}_1\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]/\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)}{{\mathrm{J}}_0\left[\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right]}$$

(50)

The norm of the complex stress transfer function is given by:

$$\left|{\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\omega }},\overline{\mathrm{m}},{\overline{\Pi }}_{\mathrm{\lambda }},{\overline{\Sigma }}_{\mathsf{\eta }},\mathrm{E}\right)\right|=\sqrt{{\left(\mathrm{Re}[{\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\omega }},\overline{\mathrm{m}},{\overline{\Pi }}_{\mathrm{\lambda }},{\overline{\Sigma }}_{\mathsf{\eta }},\mathrm{E}\right)]\right)}^2+{\left(\mathrm{Im}[{\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\omega }},\overline{\mathrm{m}},{\overline{\Pi }}_{\mathrm{\lambda }},{\overline{\Sigma }}_{\mathsf{\eta }},\mathrm{E}\right)]\right)}^2}$$

(51)

When increasing the frequency, beta $\left(\overline{\mathsf{\beta }}\left(\mathsf{\omega }\right)={\mathrm{i}}^{3/2}{\mathrm{E}}^{-1/2}\sqrt{\overline{\mathsf{\omega }}\cdot {\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}\right)$ samples the zeroes of the oscillatory Bessel function, indicating oscillations in the flow and force transfer functions. The main equations characterize this device and the three-mode characterization using kinematics and rheology. The first important equation is the complex flow transfer function (FTF, Equations (44)–(46)), which gives us the relation between the input (Ez) and the output (Q). The second important equation is the stress transfer function (STF, Equations (49)–(51)). This equation deals with the electrical wall stress (input) and the inertial-electrical wall stress (output). These results are essential to compute the resonance curves and the stress-flow loops of the electro-mechanical device (Real and Imaginary flow-stress loops are explained in detail in Appendix D).

4. Device Functionality: Flow and Force Generation

The purpose of the device studied in this paper is to create volumetric flow and to generate wall forces, and we find that an implicit relation between the two characteristic transfer functions (flow ${\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)$ and force $\overline{\mathrm{T}}\mathrm{s}\left(\overline{\mathsf{\omega }}\right)$) best characterizes the device functionality. Thus, the connection between the FTF and STF $\left\{{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right),\overline{\mathrm{T}}\mathrm{s}\left(\overline{\mathsf{\omega }}\right)\right\}$, as a function of the flexo-elastic-bulk viscoelastic mechanisms, needs to be understood and characterized. Without ambiguity, we do not discuss here real and imaginary components as that is done in the next numerical section (See Appendix E for 3D parametric classification of the mean dimensionless macroscopical groups). The relation between the two complex transfer functions is repeated here for clarity and continuity; it is given by:

$${\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\omega }}\right)=\mathrm{F}\left[{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)\right]$$

(52)

Using Equations (44), (45) and (50), the device functionality is given by:

$${\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)=\mathrm{Q}\mathrm{r}+\frac{{\overline{\mathsf{\beta }}}^2\left(\overline{\mathsf{\omega }}\right)}{8{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\mathrm{i}\overline{\mathsf{\omega }}\right)$$

(53)

Using the definition of wave vector $\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)$ (Equation (24)), the following equivalent form is obtained:

$${\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)=\mathrm{Q}\mathrm{r}+\frac{{\mathrm{i}}^2\cdot {\mathrm{E}}^{-1}\cdot \left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{8}\cdot {\overline{\mathrm{T}}}_{\mathrm{F}}\left(\mathrm{i}\overline{\mathsf{\omega }}\right)$$

(54)

At small and large frequencies, Equation (54) reflects the expected solid modes:

$$\begin{array}{l}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}:\left\{\begin{array}{l}\underset{\overline{\mathsf{\omega }}\to 0}{\lim }{\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)=\mathrm{Q}\mathrm{r}\\ \underset{\overline{\mathsf{\omega }}\to 0}{\lim }{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\beta }}\left(\overline{\mathsf{\omega }}\right)\right)=0\end{array}\right.\\ \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}-\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}-\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c} \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}:\left\{\begin{array}{l}\underset{\overline{\mathsf{\omega }}\to \infty }{\lim }{\overline{\mathrm{T}}}_{\mathrm{S}}\left(\overline{\mathsf{\omega }}\right)=0\\ \underset{\overline{\mathsf{\omega }}\to \infty }{\lim }{\overline{\mathrm{T}}}_{\mathrm{F}}\left(\overline{\mathsf{\omega }}\right)\to -8\mathrm{Q}\mathrm{r}\frac{{\overline{\mathrm{O}}}_{\Phi }\left(\mathrm{i}\overline{\mathsf{\omega }}\right)}{{\overline{\mathsf{\beta }}}^2\left(\overline{\mathsf{\omega }}\right)}\end{array}\right.\end{array}$$

(55)

5. Computational Results

Next, we present a systematic study as follows: (a) use Equation (45) to find the flow transfer function FTF, and (b) use Equation (50) to determine the stress transfer function STF. The main dimensionless numbers are: (i) total bulk viscosity, (ii) memory, (iii) elastic ratio, and (iv) elastic number E. The final output of this analysis provides (a) flow and stress transfer functions, (b) device functionality (Equation (53)), (c) material properties, and (d) electric field frequencies (ω) that simultaneously optimize (i.e., achieve high values) flow and force generation.

5.1. Elastic Number E

In Figure 9a,b, the norm of FTF and STF vs. frequency is analyzed for the decreasing E number. Except for Figure 10a, in all the simulations the value of the dimensionless groups, the representative dimensionless material parameters employed in the simulations, are: (i) $\overline{\mathrm{m}}=1\times {10}^{-4}$, (ii) ${{\displaystyle \overline{\Sigma }}}_{\mathsf{\eta }}=1$ (iii) ${\overline{\Pi }}_{\mathsf{\lambda }}=1\times {10}^{-4}$ and Qr = 1.

It is important to note that both transfer functions display primary and secondary resonances. The figures shows: (a) at low frequencies, the norm of the FTF and STF are constant and independent of the frequency, (b) the dominant frequency occurs at the first primary peak and is much larger than the other secondary resonance peaks, (c) the flow transfer function has a train of secondary peaks whose amplitude gradually decrease, (d) for the dimensionless numbers chosen, the value of the resonance frequency is close to $\overline{\mathsf{\omega }}\cong 20$, and (e) the resonance of the flow transfer function is larger than for the stress transfer function system. We note that the norm of FTF shows similar resonance behavior as other bio-physic systems (occlusion in human blood arteries) [4,5,6,7,8].

5.2. Membrane Elasticity

In Figure 10a, the norm of the FTF as a function of frequency is analyzed in terms of the elastic ratio m: $\overline{\mathrm{m}}=\left\{\left(\mathrm{a}\right):1\times {10}^{-4},\left(\mathrm{b}\right):1\times {10}^{-1};\left(\mathrm{c}\right):5\times {10}^{-1}\right\}$. The behavior is analogous to that shown in Figure 9. The net effect of increasing the elastic ratio is to decrease the resonance peak in both transfer functions; to avoid redundancy, we only show the FTF.

5.3. Velocity Profile

In Figure 10b, the norm of the complex velocity vs. radial coordinate is shown for representative material conditions. The dimensionless numbers used in the simulations are the same as Figure 8 and Figure 9a. Panel A shows the effect of the frequency over the norm of the velocity profile. It is clear that for some particular frequencies, the system displays oscillatory behavior instead of the classical velocity profile. Panels B–D show the effect of the elastic number E, which is related to the viscoelastic and inertial mechanisms. Notice that when the inertia mechanisms are small, it displays a symmetric velocity profile, whereas the opposite effect is clearly seen when the inertia mechanisms are dominant over the viscous and viscoelastic forces. Finally, the oscillatory behavior is a coupled mechanism between frequency, viscous, and viscoelastic behavior through a critical E number.

Again, in these velocity profiles we see that increasing E = We/Re dampens oscillations, consistent with previous observations. The oscillations shown in Figure 10b come from the contribution of the resonance and anti-resonance of the transfer function. The numbers of them, deals are controlled for all the mechanisms in the system (bulk-viscous, viscoelastic, bulk-elastic, flexoelectric, etc.).

5.4. Flow-Stress Loop Diagrams

Figure 11 shows the norms of stress transfer STF vs. flow transfer FTF functions for Elastic E numbers: {(i) 100, (ii) 4, (iii) 1.0}. The numerical value of dimensionless numbers used in the simulation are the same of Figure 9a. Panel A: At high values in Elastic’s number and small frequency, the FTC and STF have the same numerical value (black circle). When the frequency increases, both transfer functions show decreasing behavior, up to an asymptotic value of zero. Panel B: In contrast, when the viscoelasticity increases through the E number, i.e., E = 4, the system has the same values at small frequencies. However, when it increases, the stress transfer function increases as a function of the flow transfer function (transition from the black to yellow point). At a critical frequency (stress saturation point: yellow color), the stress and flow decrease as a function of the frequency, and for a threshold frequency (green color), it seems to be a transition from the real and complex roots of the coupled STF-FTF device, and it starts to oscillate as a consequence of the resonance behavior of the system (See Appendix E). Panel C shows the STF—FTF response with an Elastic number E = 1.0. The black circle deals with the behavior at small frequencies, i.e., Lim $\overline{\mathsf{\omega }}$→0, the black point is localized in the point (STF, FTF) = (1,1). When the dimensionless frequency increases, the STF and FTF show monotonically increasing behavior until a maximum value, followed by a stress saturation returning point (yellow color). This turning point coincides with the dominant resonance peaks (green color of the FTF and STF), which is associated with the mean Maxwell relaxation time. In the range of small and moderate frequencies, i.e., ${\displaystyle \overline{\mathsf{\omega }}}$ ∈ [1,10], there is a critical frequency (green point) where the system starts to oscillate and the coupled transfer functions show a multivalued function in the dynamical response. From a mathematical point of view, a plausible explanation of these oscillations deals with the ratio of the Bessel modified function of the first and second kind; it is possible that the system undergoes transitions from real to imaginary roots (See Appendix F for the mathematical details of the complex roots). The key physical message of Figure 11 is that the viscoelasticity, bulk-viscous, bulk-elastic, and flexoelectric behavior control the shape and orientation of the force–flow loops in the system.

5.5. Force–Flow Loops: Resonances and Transitions

Figure 12 summarizes key findings regarding resonances and monotonic-oscillatory transitions. Panels A, B show the flow, stress function as a function of frequency. Panel C shows the flow–force loop (See Appendix D for real and imaginary loops). The material conditions are as in Figure 10 and Figure 11. Here, the value of the elastic number is E = 1.

Figure 12 shows the following important points:

(a)

The material conditions needed to observe multiple resonance peaks in the FTF spectrum are: (i) an asymmetry in the viscoelastic liquid phases, i.e., one of them is totally viscoelastic and the other weakly viscoelastic, i.e., ${\overline{\Pi }}_{\mathsf{\lambda }}=\mathsf{\epsilon }$, ε << 1. This means that one of the viscoelastic relaxation times is close to the unit ${\overline{\mathsf{\lambda }}}_{\mathrm{L}}=1-\mathsf{\epsilon }$, and the other one is very small, ε =1 × 10⁻⁴. (ii) The system has large viscosity, i.e., $\overline{\Sigma }\mathsf{\eta }=1$. (iii) The membrane elasticity m = ε = 1 × 10⁻⁴. (iv) E is greater than zero, i.e., viscoelastic mechanisms (Weissenberg number We) (viscoelastic forces) dominate over the inertial and viscous forces (Reynolds number Re), E = We/Re > 0.

(b)

The maximum resonance peak is linked to the inertia, viscoelastic, and flexo-elastic processes, and its magnitude depends on the elastic number E = We/Re.

(c)

The multiple peaks are associated with a possible change in the roots of the ratio of the Bessel functions from real to complex values. The nature of these oscillations depends on coupled mechanisms associated with the inertia, viscous, elastic, and flexoelectric through the dimensionless numbers.

(d)

The global peak is found at a critical frequency whose value depends on the main Maxwell relaxation time, and this time is related with the frequency through ${\overline{\mathsf{\omega }}}_{\mathrm{peak}}=1/1-\mathsf{\epsilon }$. The secondary peak corresponds to the less dominant relaxation time, i.e., ${\overline{\mathsf{\omega }}}_{\mathrm{peak}}=1/\mathsf{\epsilon }$.

(e)

The adequate model to predict resonance behavior is a new asymmetric rheological model, which is a variant of Burgers’ constitutive equation (Equation (10)). Other models (Jeffreys, Burgers, fractional derivative models) contribute to modify the resonance behavior through the viscosity operator defined in Equations (11) and (12) [3,4].

(f)

Multiple resonance peaks are found in other biological and complex systems [39,40]. For example, in biorheology, the pulsating pressure-driven force increases the volumetric flow, and its effect is dominated by dynamic permeability [41,42,43,44,45]. In veins with central and peripherical occlusions [46], the frequency can be tuned to obtain the maximum pulsating volumetric flow rate [46], as well as in oil recovery operations with surfactants such as worm-like micellar solutions [13,47,48,49,50].

(g)

The train of peaks can contribute to increase the FTF and STF, which are a starting point to study OHCs of mammalians and ear diseases [3,4,24,41].

6. Conclusions

The merit of this research is to propose a new asymmetric electro-rheological model (Equation (10)) to understand the dynamical behavior of flexoelectric membranes immersed in liquid viscoelastic media. The key issue is to understand the interactions between the driven electrical force (input), the volumetric flow rate, and the wall stress (outputs). This methodology is general and can be extended to include a higher viscoelastic linear and fractional model through the new asymmetric fluidity operator. The new asymmetric fluidity operator (Equation (12)) led us to obtain resonance behavior for the flow and stress transfer linear functions through macroscopical forces such as (i) inertia, (ii) bulk-viscous, (iii) bulk-elastic, (iv) flexoelectric, and (v) membrane elasticity in the physical system. This new approach has a particular parallel mechanical configuration (Maxwell/Hook/Maxwell) and its respective rheological equations of state (the Burgers-like model). The resonance behavior is totally controlled by the asymmetry of the viscoelastic phases, through fundamental dimensionless numbers (Appendix B). The new approach in this research is the elastic number (Equation (28)), which led us to display a power spectrum with one dominant resonant peak and a train of tunable secondary peaks. Lastly, special attention was placed on formulating and implementing an engineering device methodology that integrates design/control/prediction [3,4,5,10,11,12].