Oscillatory interactions of two spheres in an unbounded couple stress fluid

Abstract

This study investigates the rectilinear oscillations of two coaxially aligned spherical particles in an unbounded couple stress fluid at low Reynolds numbers, addressing a fundamental problem in microfluidics and biomechanics where microstructure effects dominate. The importance lies in applications such as drug delivery and material processing, where understanding particle-fluid interactions is critical. The unsteady Stokes equations were solved using a superposition of fundamental solutions in spherical coordinates, centered on each particle, with no-slip boundary conditions enforced via a collocation method. Key results include the quantification of in-phase and out-of-phase drag force coefficients, revealing that increasing the couple stress parameter ($\bar{\eta }$) enhances drag forces by up to 50% for $\bar{\eta } = 0.9$ compared to Newtonian cases ($\bar{\eta } = 0$). Numerical simulations demonstrated robust convergence across dimensionless parameters (e.g., separation distance $\delta$, frequency $\alpha$), with tabulated data showing agreement within % of established solutions for steady-state and single-sphere oscillations. Novelty arises from extending prior work on viscous fluids to couple stress fluids, uncovering how microstructural effects amplify drag and alter oscillation dynamics. For instance, at $\alpha = 60$, drag forces increased by 30% for closely spaced spheres ($\delta = 1.05$), highlighting the interplay between frequency and microstructure. This work advances predictive models for complex fluids and provides design insights for microfluidic systems.

Introduction

The study of fluid mechanics, particularly the behavior of fluids with microstructure, has been significantly advanced by foundational works such as V. K. Stokes’ "Theories of Fluids with Microstructure: an Introduction" (1984). Stokes provides a comprehensive overview of the theoretical frameworks that govern the dynamics of complex fluids, which include micropolar and couple stress fluids. These types of fluids exhibit unique characteristics that differ from Newtonian fluids, especially at low Reynolds numbers where viscous forces dominate¹. One notable contribution in this field is the work by Ganatos et al.², which addresses the creeping motion of a sphere between parallel walls. The authors demonstrate a significant error margin in earlier approximations, revealing that accurate modeling is crucial for predicting fluid behavior in constrained environments². Such findings are essential for applications ranging from biomedical engineering to materials science³.

Furthermore, Ramkissoon’s investigation into drag within couple stress fluids reveals additional complexities associated with non-Newtonian flow behaviors. Coupled with Lawrence and Weinbaum’s analysis on unsteady forces acting on bodies under low Reynolds number conditions, these studies underscore the intricate interplay between fluid dynamics and body motion at microscopic scales^3,4. Collectively, these contributions enhance our understanding of microstructural influences on fluid mechanics and offer avenues for future research in related domains². Couple stress fluids represent a class of non-Newtonian fluids characterized by their ability to exhibit couple stresses, which arise from the rotation of fluid elements. This property differentiates them from classical Newtonian fluids, where only shear stress is considered. For instance, Farooq⁵ explores the generalized Couette flow of couple stress fluids between parallel plates and highlights how these fluids behave under various thermal conditions. The study employs sophisticated mathematical techniques like the Optimal Homotopy Asymptotic Method (OHAM) and New Iterative Method (NIM) to derive analytical solutions for velocity profiles and temperature distributions. In contrast, Khan⁶ investigates the interaction between couple stress fluids and microcantilever sensors in a squeezed channel, emphasizing the complex interplay between heat transfer and mass transfer in such systems.

Both studies underscore the significance of couple stresses in fluid dynamics but approach the analysis from different perspectives. Farooq’s research focuses primarily on steady-state conditions within controlled geometries, while Khan’s work delves into dynamic interactions involving sensor surfaces that are subject to external variables such as permeability and squeezing effects. Additionally, Khan’s findings indicate that increased couple stress can lead to reduced fluid velocity under specific conditions, whereas Farooq’s model provides insights into how thermal effects can alter flow characteristics without direct mention of sensor interactions. The methodologies employed also differ significantly; while Farooq utilizes advanced asymptotic methods for solution derivation, Khan adopts a numerical approach through the shooting method after transforming governing equations into ordinary differential forms. This highlights a methodological divergence that reflects broader trends in fluid dynamics research where both analytical and numerical frameworks coexist to address complex phenomena inherent in couple stress fluids⁷. The study of oscillations in couple stress fluids, particularly involving two spheres, presents a compelling area of research within fluid mechanics. Couple stress fluids are characterized by their unique ability to account for the microstructural effects that arise from the interactions between particles and fluid at small scales. This is particularly relevant when considering the dynamics of two oscillating spheres, where the interplay of forces can significantly influence their motion⁸. The behavior of such systems is not only critical for theoretical understanding but also has practical implications in various engineering applications, including drug delivery and material processing⁹.

Recent studies have highlighted the complex nature of flow fields generated by couple stress fluids around rigid bodies. For instance, Aparna et al.¹⁰ analytically examined the flow field around a single permeable sphere undergoing steady oscillations, identifying how parameters such as permeability and frequency affect drag forces. Although this research focuses on a singular sphere, it lays foundational insights applicable to multiple-sphere systems by elucidating how couple stresses contribute to enhanced flow resistance compared to traditional viscous fluids¹¹. In contrast, Vani et al.⁸ specifically addressed the oscillatory behavior between two concentric spheres in a couple stress fluid environment. Their analysis reveals that both spheres can operate under different angular speeds while maintaining synchronized frequencies. This duality introduces intricate interactions that must be accounted for when modeling real-world scenarios involving multiple solid bodies within a fluid medium⁸. Understanding these dynamics is crucial for predicting system behaviors under varying operational conditions¹⁰.

In the realm of fluid dynamics, these studies collectively enhance the understanding of sphere interactions. They emphasize the diverse aspects of fluid dynamics that are critical for various engineering applications, ranging from hydrophobic conditions to magnetic influences. Algatheem et al. investigated the dynamics of two hydrophobic spheres oscillating within an infinite Brinkman-Stokes fluid, highlighting the influence of permeability and slip conditions⁸. This work is complemented by Albalawi et al. , who further examined the effect of permeability on sphere interactions within a Stokes-Brinkmann medium¹². Conversely, El-Sapa and Albalawi focused on the impact of magnetic fields on oscillating spheres, employing a boundary collocation method to analyze unsteady drag coefficients¹³. Expanding on magnetic influences, Algatheem and El-Sapa studied magnetic field and slippage effects in magnetorheological ferrofluids¹⁴. Moreover, El-Sapa and Alhejaili analyzed the hydrodynamic interaction of coaxial spheres in a viscous fluid with slip¹⁵, and El-Sapa examined magneto-spherical particle oscillations near a planar wall¹⁶. Faltas and El-Sapa also studied rectilinear oscillations of spheres in an unbounded viscous fluid¹⁷.

This study advances the foundational steady-state analysis of Chen and Keh^18,19 to the realm of unsteady Stokes flow, while also extending the work of Faltas and El-Sapa¹⁷—which examined oscillations of two spheres in an unbounded viscous fluid without slippage—by introducing a couple stress fluid model with microstructural effects through viscosity parameters $(\eta , \eta ')$. It represents the first comprehensive solution for oscillatory interactions between two spherical particles in couple stress fluids, overcoming prior limitations in studies focused on single spheres^10,17 or Newtonian fluids^17,18. Specifically, the investigation analyzes the rectilinear harmonic oscillations of two coaxially aligned spherical particles, driven by distinct displacement amplitudes and radii, within the low Reynolds number regime. Building on Stokes’ theory¹ and Lawrence-Weinbaum’s approach⁴, we develop a novel superposition-collocation method, inspired by Ganatos et al.², to solve the unsteady hydrodynamic interactions. The hydrodynamic velocity field is determined through a superposition of fundamental solutions in spherical coordinate systems centered on each particle, with rigorously enforced no-slip boundary conditions. Numerical simulations quantify the in-phase and out-of-phase force coefficients, demonstrating robust convergence across dimensionless parameters (e.g., radius ratio, separation distance, frequency $\alpha$, and viscosity coefficients). The study reveals new phenomena in drag forces, validated against classical limits^5,17, and demonstrates practical applications in drug delivery and material processing. By systematically exploring frequency parameters under constraints like $\bar{\eta }=0$, this work extends Chen-Keh’s steady-state analysis to dynamic systems and provides insights beyond Faltas and El-Sapa’s viscous fluid results.

Foundational equations

For an unsteady incompressible couple stress fluid in the absence of body forces and body moments by¹. characterized by the condition of conservation of mass and momentum are:

$$\begin{aligned}&\nabla \cdot {\bf v} = 0, \end{aligned}$$

(1)

$$\begin{aligned}&\rho \frac{\partial {\bf v}}{\partial t} = -\nabla p - \mu \nabla \times \nabla \times {\bf v}- \eta \nabla \times \nabla \times \nabla \times \nabla \times {\bf v}. \end{aligned}$$

(2)

where ${\bf v}$ denotes the velocity and p represents the fluid pressure. Neglecting convective terms, based on Stokes assumption that flow is very slow and Reynolds number is very small $(Re<< 1)$. The material constant $\mu$ denotes the classical viscosity parameter of the fluid, $\eta$ is the new viscosity coefficient, which characterizes the couple stress effect, and $\rho$ represents the fluid density. The last term in Eq. (2), represents the effect of couple stresses, and here $\mu , \eta ,$ are material constants.

The constitutive equations for a couple stress fluid will be assumed to have the form:

$$\begin{aligned} \tau _{ij}&= -p \delta _{ij} + \lambda (v_{i,j}+v_{j,i})-\tfrac{1}{2} \epsilon _{ijk} m_{sk,s}, \end{aligned}$$

(3)

$$\begin{aligned} m_{ij}&=\tfrac{1}{3} m \delta _{ij}+4 \eta \omega _{i,j}+4\eta ' \omega _{j,i}, \end{aligned}$$

(4)

where $\eta '$ is a material constant, also, $\mu> 0,~ (3\lambda + 2\mu )> 0,~ \eta > \eta '$. The vorticity is defined as:

$$\begin{aligned} \omega = \frac{1}{2} \epsilon _{ijk} v_{kj}. \end{aligned}$$

(5)

The scalar quantity m represents one-third of the trace of the couple stress tensor, $d_{ij}$, and $\epsilon _{ijk}$ are respectively the Kronecker delta and the alternating tensor defined by

$$\begin{aligned} \delta _{ij} = {\left\{ \begin{array}{ll} 0 & \text {if } i \ne j \\ 1 & \text {if } i = j' \end{array}\right. } \end{aligned}$$

(6)

$$\begin{aligned} \epsilon _{ijk} = {\left\{ \begin{array}{ll} 1 & \text {if } (i,j,k) \text { is a cyclic permutation of } (1,2,3) \\ -1 & \text {if } (i,j,k) \text { is a non cyclic permutation of } (1,2,3) \\ 0 & \text {if at least two of the indices are equal} \end{array}\right. } \end{aligned}$$

(7)

Mathematical modeling

In this study, we consider the two-dimensional axisymmetric translational motion in spherical coordinates where the velocity ${\bf {v}}=\langle v_r, v_\theta , 0 \rangle$ can be written as:

$$\begin{aligned}& {\bf v} = \nabla \wedge \frac{\psi {\bf e}_\phi }{r\sin \theta }= \begin{pmatrix} \frac{1}{r^2 \sin \theta } \frac{\partial \psi }{\partial \theta } \\ \frac{-1}{r \sin \theta } \frac{\partial \psi }{\partial r} \end{pmatrix} , \end{aligned}$$

(8)

Substituting from Eq. (8) into (2), we have:

$$\begin{aligned}&\rho \frac{\partial v_r}{\partial t} = -\frac{\partial p}{\partial r} + \frac{\mu }{r^2 \sin \theta } \frac{\partial }{\partial \theta } \left( L_{-1} \psi \right) - \frac{\eta }{r^2 \sin \theta } \frac{\partial }{\partial \theta } \left( L_{-1}^2 \psi \right) , \end{aligned}$$

(9)

$$\begin{aligned}&\rho \frac{\partial v_\theta }{\partial t} = -\frac{1}{r} \frac{\partial p}{\partial \theta } -\frac{\mu }{r \sin \theta } \frac{\partial }{\partial r} \left( L_{-1} \psi \right) + \frac{\mu }{r \sin \theta } \frac{\partial }{\partial r} \left( L_{-1}^2 \psi \right) . \end{aligned}$$

(10)

where $L_{-1}=\frac{\partial ^2}{\partial r^2} + \frac{\sin \theta }{r^2} \frac{\partial }{\partial \theta } \left( \frac{1}{\sin \theta } \frac{\partial }{\partial \theta } \right) , \quad \zeta = \cos \theta$ is the Stokesian operator.

Now, we examine an oscillatory flow characterized by the expression $U e^{i \sigma t} {\bf e}_{\phi }$ of an incompressible couple stress fluid, with the oscillation direction aligned along ${\bf e}_{\phi }$. A spherical membrane of radius $a$, featuring a solid surface, is introduced into the flow and fixed at the origin. We consider a spherical coordinate system with the origin positioned at the center of the sphere and the $z$-axis aligned with the flow direction. The velocity field and pressure corresponding to this oscillatory flow are expressed as:

$$\begin{aligned}&\psi = \Psi (r,\theta )~ e^{i \sigma t} \end{aligned}$$

(11)

$$\begin{aligned}&\varvec{\omega }=\tfrac{1}{2}\nabla \wedge {\bf v}=\Big (\frac{E^2\psi ~{\bf e}_{\phi } }{2r\sin \theta }\Big ) e^{i \sigma t},~p=p_{0}e^{i \sigma t} \end{aligned}$$

(12)

To express the governing equations in a non-dimensional form, we introduce non-dimensional variables as:

$$\begin{aligned} r = a \bar{r},~ p = \tfrac{\mu U}{a} \bar{p},~ {\bf v} = U \bar{{\bf v}},~ \bar{\eta }=\tfrac{\eta }{a^2\mu },~ \lambda ^2=\tfrac{i \sigma a^2}{\nu }=i\alpha ^2,~ \nu =\tfrac{\mu }{\rho },~t=\tfrac{\bar{t}}{\sigma }. \end{aligned}$$

(13)

Substituting Eq. (13) into Eqs. (9)–(12) and dropping bar, yielding:

$$\begin{aligned} \frac{\partial p}{\partial r}&= -\frac{\bar{\eta }}{r^2 \sin \theta } \frac{\partial }{\partial \theta } \left( L_{-1} - \xi L_{-1} + \xi ^2 \right) \psi , \end{aligned}$$

(14)

$$\begin{aligned} \frac{1}{r} \frac{\partial p}{\partial \theta }&= \frac{\bar{\eta }}{r \sin \theta } \frac{\partial }{\partial r} \left( L^2_{-1} - \xi L_{-1} + \xi \lambda ^2 \right) \psi . \end{aligned}$$

(15)

Eliminating the pressure from Eqs. (14) and (15), we obtain a partial differential equation of the fourth order:

$$\begin{aligned} L_{-1} (L_{-1} - \lambda _1^2)(L_{-1} - \lambda _2^2) \psi = 0, \end{aligned}$$

(16)

where $\alpha$ is the frequency parameter, $\lambda ^2=i S_t R_e$, $R_e=\frac{U a}{\nu }$ is the Reynolds number, and $S_t= \frac{\sigma a}{U}$ is the Strouhal number. The roots are obtained as:

$$\begin{aligned} \left. \begin{aligned}&\lambda _1^2 \lambda _2^2=\xi \alpha ^2,~\lambda _1^2+\lambda _2^2=\xi ,\xi =1/\bar{\eta },~~\\&\lambda _{i}=\sqrt{\frac{\xi \pm \sqrt{\xi ^2-4~\xi \lambda ^2}}{2}},\quad ~i=1,2~~\quad \quad \\ \end{aligned} \right\} \end{aligned}$$

(17)

Additionally, when $\xi \rightarrow \infty$ or $\eta =0$ then Eqs. (14) and (15) represent the modified Stokes equation for non-polar fluid. It is noticed that:

$$\begin{aligned} R_e<< 1, \; \; \; S_t>>1 \; \; \; or \; \; \left( a>> \frac{U}{\sigma }\right) . \end{aligned}$$

(18)

This implies that the amplitude of the oscillation is small compared to the characteristic length of the solid body. The stress and couple stress components are calculated by:

$$\begin{aligned} \tau _{r\theta }= & \frac{1}{r}\frac{\partial v_r}{\partial \theta }-\frac{v_{\theta }}{r}+\frac{\partial v_\theta }{\partial r}-\frac{2}{\xi }\Big [\frac{\partial ^2\omega _\phi }{\partial r^2}+\frac{2}{r}\frac{\partial \omega _\phi }{\partial r}-\frac{2}{r^2}\omega _\phi \Big ], \end{aligned}$$

(19)

$$\begin{aligned} m_{r\phi }= & \frac{4}{\xi }\frac{\partial \omega _\phi }{\partial r}-\frac{4}{\xi '}\frac{\omega _\phi }{r}. \end{aligned}$$

(20)

The general solution of (16) which is regular as $r \rightarrow \infty$ is given by:

$$\begin{aligned} \psi (r, \theta ) = \sum _{n=2}^{\infty } \left[ A_n r^{-n+1} + B_n r^{\frac{1}{2}} K_{n-\frac{1}{2}}(\lambda _1 r)+ C_n r^{\frac{1}{2}} K_{n-\frac{1}{2}}(\lambda _2 r) \right] \Im _n(\zeta ), \end{aligned}$$

(21)

where $\Im _n(.)$ is the Gegenbauer function of the first kind of order n and degree $(-\frac{1}{2})$ and $K_n(.)$ is the modified Bessel function of the second kind of order n. Note that in the solution (16), we have omitted the irregular terms along the axis $\zeta = \pm 1$. The components of velocity $(v_r, v_\theta )$ are given by:

$$\begin{aligned} v_r(r, \theta )&= - \sum _{n=2}^{\infty } \left[ A_n r^{-n-1} + B_n r^{-\frac{3}{2}} K_{n-\frac{1}{2}}(\lambda _1 r)+C_n r^{-\frac{3}{2}} K_{n-\frac{1}{2}}(\lambda _2 r) \right] P_{n-1}(\zeta ), \end{aligned}$$

(22)

$$\begin{aligned} v_\theta (r, \theta )&= \sum _{n=2}^{\infty } \Big [ (1-n) A_n r^{-n-1} + B_n r^{-\frac{3}{2}} \left( n K_{n-\frac{1}{2}}(\lambda _1 r) - \lambda _1 r K_{n+\frac{1}{2}}(\lambda _1 r) \right) \nonumber \\&+C_n r^{-\frac{3}{2}} \left( n K_{n-\frac{1}{2}}(\lambda _2 r) - \lambda _2 r K_{n+\frac{1}{2}}(\lambda _2 r) \right) \Big ] \frac{\Im _{n}(\zeta )}{\sqrt{1-\zeta ^2}}. \end{aligned}$$

(23)

Moreover, the vorticity and couple stress are calculated as:

$$\begin{aligned} \omega _{\phi }(r, \theta )&=\frac{1}{2}\sum _{n=2}^{\infty } r^{\tfrac{-1}{2}}\Big [B_{n} \lambda _{1}^{2} K_{n-\tfrac{1}{2}}(\lambda _{1}r)+C_{n} \lambda _{2}^{2} K_{n-\tfrac{1}{2}}(\lambda _{2}r)\Big ] \frac{\Im _{n}(\zeta )}{\sqrt{1-\zeta ^2}}. \end{aligned}$$

(24)

$$\begin{aligned} m_{r\phi }(r, \theta )&=2\sum _{j=1}^{2}r^{\tfrac{-3}{2}} \Big [B_{n}\lambda _{1}^2 \Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{1}r) -\bar{\eta } \lambda _{1} r K_{n+\tfrac{1}{2}}(\lambda _{1}r)\Big )\nonumber \\&+ C_{n}\lambda _{2}^2\Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{2}r) -\bar{\eta } \lambda _{2} r K_{n+\tfrac{1}{2}}(\lambda _{2}r)\Big ) \Big ]\frac{ \Im _{n}(\zeta )}{\sqrt{1-\zeta ^2}}. \end{aligned}$$

(25)

where $P_n(.)$ is the Legendre polynomial of degree n.

Oscillatory motion of a solid sphere through CSF

We now investigate the translatory oscillation of a couple stress fluid surrounding a rigid sphere of radius a. This motion is induced by the rectilinear oscillation of the sphere about a vertical diameter, characterized by a time-dependent angular velocity ${\bf e}^{i \sigma t}$, where $\sigma$ denotes a constant with the dimension of angular velocity and $i = \sqrt{-1}$. We adopt a spherical polar coordinate system $(r, \theta , \phi )$, positioning the center of the sphere as the origin, with the z-axis aligned along the axis of revolution, represented by the basis vectors $({\bf e}_r, {\bf e}_\theta , {\bf e}_\phi )$. Given the axisymmetric nature of the flow, all physical quantities remain invariant with respect to $\phi$.

The regular solution of the sixth-order linear partial differential equation (15) is obtained by using the method of separation of variables is given:

$$\begin{aligned} \frac{2\psi }{\sin ^2\theta }=A_2~ r^{-1}+B_2~ r^{\tfrac{1}{2}}K_{\tfrac{3}{2}}(\lambda _1 r)+C_2~ r^{\tfrac{1}{2}}K_{\tfrac{3}{2}}(\lambda _2 r), \end{aligned}$$

(26)

where the function $K_{\tfrac{3}{2}}(.)$ is the modified Bessel functions of the second kind of order $\tfrac{3}{2}$ where $A_2,B_2,C_2$ are constants can be determined from the boundary conditions. The velocity components and vorticity are:

$$\begin{aligned}&\frac{v_{r}}{\cos \theta }=A_2 r^{-3}+B_2 r^{\tfrac{-3}{2}}K_{\tfrac{3}{2}}(\lambda _1 r)+C_2 r^{\tfrac{-3}{2}}K_{\tfrac{3}{2}}(\lambda _2 r), \end{aligned}$$

(27)

$$\begin{aligned}&\frac{2 v_{\theta }}{\sin \theta }=-A_2 r^{-3}-B_2 r^{\tfrac{-3}{2}}\big (K_{\tfrac{3}{2}}(\lambda _1 r)+\lambda _1 r K_{\tfrac{1}{2}}(\lambda _1 r)\big )-C_2 r^{\tfrac{-3}{2}}\big (K_{\tfrac{3}{2}}(\lambda _2 r)+\lambda _2 r K_{\tfrac{1}{2}}(\lambda _2 r)\big ), \end{aligned}$$

(28)

$$\begin{aligned}&\frac{4\omega _{\phi }}{\sin \theta }=B_2 \lambda _{1}^2 r^{\tfrac{-1}{2}}K_{\tfrac{3}{2}}(\lambda _1 r)+C_2 \lambda _{2}^2 r^{\tfrac{-1}{2}}K_{\tfrac{3}{2}}(\lambda _2 r), \end{aligned}$$

(29)

$$\begin{aligned}&\frac{m_{r\phi }}{\sin \theta }=-r^{\tfrac{-3}{2}} B_2\lambda _{1}^2 \Big (( 2\bar{\eta }+\bar{\eta }') K_{\tfrac{3}{2}}(\lambda _{1}r) -\bar{\eta } \lambda _{1} r K_{\tfrac{1}{2}}(\lambda _{1}r)\Big )\nonumber \\&\quad \quad -r^{\tfrac{-3}{2}} C_2\lambda _{2}^2\Big ((2\bar{\eta }+\bar{\eta }') K_{\tfrac{3}{2}}(\lambda _{2}r)-\bar{\eta } \lambda _{2} r K_{\tfrac{1}{2}}(\lambda _{2}r)\Big ). \end{aligned}$$

(30)

On the surface of the aerosol particle, $r=a$, we require three boundary conditions to find the unknowns A, B, and C. The boundary conditions at $r=a$ are:

$$\begin{aligned} v_{r}=U \cos \theta ,\quad v_{\theta }=-U \sin \theta ,\quad m_{r\phi }=0, \end{aligned}$$

(31)

We get the following system that can be solved to obtain the constants $A_2,B_2,C_2$ as:

$$\begin{aligned}&A_2 a^{-3}+B_2 a^{\tfrac{-3}{2}}K_{\tfrac{3}{2}}(\lambda _1 a)+C_2 a^{\tfrac{-3}{2}}K_{\tfrac{3}{2}}(\lambda _2 a)=U, \end{aligned}$$

(32)

$$\begin{aligned}&A_2 a^{-3}+B_2 a^{\tfrac{-3}{2}}\big (K_{\tfrac{3}{2}}(\lambda _1 a)+\lambda _1 a K_{\tfrac{1}{2}}(\lambda _1 a)\big )+C_2 a^{\tfrac{-3}{2}}\big (K_{\tfrac{3}{2}}(\lambda _2 a)+\lambda _2 a K_{\tfrac{1}{2}}(\lambda _2 a)\big )=2U, \end{aligned}$$

(33)

$$\begin{aligned}&B_2\lambda _{1}^2 \Big (( 2\bar{\eta }+\bar{\eta }') K_{\tfrac{3}{2}}(\lambda _{1}a) -\bar{\eta } \lambda _{1} a K_{\tfrac{1}{2}}(\lambda _{1}a)\Big )+C_2\lambda _{2}^2\Big ((2\bar{\eta }+\bar{\eta }') K_{\tfrac{3}{2}}(\lambda _{2}a)-\bar{\eta } \lambda _{2} a K_{\tfrac{1}{2}}(\lambda _{2}a)\Big )=0. \end{aligned}$$

(34)

where

$$\begin{aligned} A_2&= U a^{3/2} \left[ a^{3/2} - \Delta ^{-1} \bar{\eta } a \left( \lambda _1^2 K_{1/2}(\lambda _1 a) K_{3/2}(\lambda _2 a) - \lambda _2^2 K_{1/2}(\lambda _2 a) K_{3/2}(\lambda _1 a) \right) \right. \\&+ \Delta ^{-1} (2\bar{\eta } + \eta ) (\lambda _1^2 - \lambda _2^2) K_{3/2}(\lambda _1 a) K_{3/2}(\lambda _2 a) \\ B_2&= \Delta ^{-1} \lambda _2^2 \left( (2\bar{\eta } + \eta ) K_{3/2}(\lambda _2 a) - \bar{\eta } \lambda _2 a K_{1/2}(\lambda _2 a) \right) U \\ C_2&= -\Delta ^{-1} \lambda _1^2 \left( (2\bar{\eta } + \eta ) K_{3/2}(\lambda _1 a) - \bar{\eta } \lambda _1 a K_{1/2}(\lambda _1 a) \right) U \\ \Delta&= a^{-1/2} \lambda _1 \lambda _2 \left[ (2\bar{\eta } + \eta ) \left( \lambda _2 K_{3/2}(\lambda _2 a) K_{1/2}(\lambda _1 a) - \lambda _1 K_{3/2}(\lambda _1 a) K_{1/2}(\lambda _2 a) \right) \right. \\&- \bar{\eta } (\lambda _2^2 - \lambda _1^2) a K_{1/2}(\lambda _2 a) K_{1/2}(\lambda _1 a) \end{aligned}$$

Lawrence and Weinbaum⁴, developed an expression for the force on the oscillating body for axisymmetric motion:

$$\begin{aligned} F = \pi \mu \displaystyle \oint _C \left[ \varpi ^2 \frac{\partial }{\partial n} \left( \frac{E^2 \psi }{\varpi ^2} \right) - \varpi \lambda ^2 \frac{\partial \psi }{\partial n} \right] ds, \end{aligned}$$

(35)

where $\varpi = r \sin \theta$ is the distance from the axis of symmetry and $\vec {n}$ is the outward normal to the surface in which s is the coordinate along the generating arc.

$$\begin{aligned} \frac{F}{F_\infty } = K + i K' \end{aligned}$$

(36)

where K, $K'$ are the force coefficients. Physically the force coefficients K, $K'$ represent, respectively, the in-phase and out-of-phase force oscillations. The modified drag force experienced by an incompressible couple stress fluid flow extending to infinity on an axisymmetric particle translating in it, along its direction of motion can be evaluated using the formula :

$$\begin{aligned} F = \lambda ^2 (\mathcal {V} + 2\pi A_2), \end{aligned}$$

(37)

where V is the volume of the particle a. It is known that, the drag force acting on a rigid sphere moving in a steady incompressible viscous fluid flow is given by³:

$$\begin{aligned} F_0 = -6\pi \mu a U. \end{aligned}$$

(38)

Axisymmetric interaction between two solid spheres through CSF

In this mathematical framework, we investigate a system comprising two oscillatory solid spherical particles, characterized by radii $a_1$ and $a_2$ (with the condition $a_2 \ge a_1$), each suspended within a viscous medium. The particles $a_j$ (where $j = 1, 2$) oscillate at a common angular frequency $\omega$, exhibiting distinct amplitudes $\frac{U_1}{\sigma }$ and $\frac{U_2}{\sigma }$ along the linear axis connecting their centers. These spheres are positioned externally relative to one another, with their centers separated by a distance h. The flow field is quiescent at infinity. For analytical convenience, we adopt two spherical coordinate systems, denoted as $(r_1, \theta _1, \varphi )$ and $(r_2, \theta _2, \varphi )$, which are anchored at the centers of particles $a_1$ and $a_2$, respectively. The geometric configuration of this scenario is depicted in Fig. 2. The relationship between the coordinate pairs $(r_1, \theta _1)$ and $(r_2, \theta _2)$ is articulated as follows:

$$\begin{aligned} r_1^2 = r_2^2 + h^2 - 2r_2 h \cos \theta _2, \end{aligned}$$

(39)

or by

$$\begin{aligned} r_2^2 = r_1^2 + h^2 + 2r_1 h \cos \theta _1. \end{aligned}$$

(40)

Let $\vec {q}^{(j)}$, ($j = 1, 2$) be the velocity vector of the fluid due to the presence of the spherical particle $a_j$ in the absence of the other particle. Let

$$\begin{aligned} \vec {q}^{(j)} = v_r^{(j)}(r_j, \theta _j) \vec {e}_{r_j} + v_\theta ^{(j)}(r_j, \theta _j) \vec {e}_{\theta _j}. \end{aligned}$$

(41)

The velocity components of the particle $a_j$ in the direction of the unit vectors $(\vec {e}_{r_j}, \vec {e}_{\theta _j}, \vec {e}_{\varphi })$ are:

$$\begin{aligned} V_r^{(j)}&= U_j \cos \theta _j, V_\theta ^{(j)} = -U_j \sin \theta _j, V_\varphi ^{(j)} = 0. \end{aligned}$$

(42)

The kinematical conditions describing the impenetrability of the surfaces of the spherical particles are:

$$\begin{aligned} v_r|_{r_j = a_j} = U_j \cos \theta _j, \qquad j = 1, 2 \end{aligned}$$

(43)

The dynamical (no-slip) conditions take the following form:

$$\begin{aligned} v_\theta |_{r_j = a_j} = -U_j \sin \theta _j, \qquad j = 1, 2 \end{aligned}$$

(44)

The no-spin slip condition is:

$$\begin{aligned} m_{r\phi }|_{r_j = a_j} =0, \qquad j = 1, 2. \end{aligned}$$

(45)

Fig. 1

Schematic representation of two oscillating spheres in CSF.

Moreover, at large distances from the spherical particles, the velocity components tend to zero, that is:

$$\begin{aligned} v_r \rightarrow 0, \quad v_\theta \rightarrow 0 \quad \text {as} \quad r \rightarrow \infty , \end{aligned}$$

(46)

The linearity of the governing equation of motion and boundary conditions permits us to use the principle of superposition, therefore the velocity components $v_r$, $v_\theta$ of the flow are:

$$\begin{aligned}&v_r(r_1, \theta _1; r_2, \theta _2) = v_r^{(1)}(r_1, \theta _1) + v_r^{(2)}(r_2, \theta _2), \end{aligned}$$

(47)

$$\begin{aligned}&v_\theta (r_1, \theta _1; r_2, \theta _2) = v_\theta ^{(1)}(r_1, \theta _1) + v_\theta ^{(2)}(r_2, \theta _2), \end{aligned}$$

(48)

$$\begin{aligned}&m_{r\phi }(r_1, \theta _1; r_2, \theta _2) = m_{r\phi }^{(1)}(r_1, \theta _1) + m_{r\phi }^{(2)}(r_2, \theta _2), \end{aligned}$$

(49)

The components $v_r$ and $v_\theta$ can be built up as follows:

$$\begin{aligned} v_{r}=&-\sum _{j=1}^{2} \sum _{n=2}^{\infty }\Big [A_{n}^{(j)} r_{j}^{-n-1}+B_{n}^{(j)} r_{j}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{1}r_{j})+ C_{n}^{(j)} r_{j}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{2}r_{j}) \Big ] P_{n-1}(\zeta _{j}). \end{aligned}$$

(50)

$$\begin{aligned} v_{\theta }=&\sum _{j=1}^{2} \sum _{n=2}^{\infty }\Big [(1-n) A_{n}^{(j)} r_{j}^{-n-1}+B_{n}^{(j)} r_{j}^{\tfrac{-3}{2}} \Big (n K_{n-\tfrac{1}{2}}(\lambda _{1}r_{j})-\lambda _{1} r_{j} K_{n+\tfrac{1}{2}}(\lambda _{1}r_{j})\Big )\nonumber \\&+ C_{n}^{(j)} r_{j}^{\tfrac{-3}{2}} \Big (n K_{n-\tfrac{1}{2}}(\lambda _{2}r_{j})-\lambda _{2} r_{j} K_{n+\tfrac{1}{2}}(\lambda _{2}r_{j})\Big ) \Big ] \frac{\Im _{n}(\zeta _{j})}{\sqrt{1-\zeta ^2_{j}}}. \end{aligned}$$

(51)

The vorticity and couple stress components are calculated as follow:

$$\begin{aligned} \omega _{\phi }&=\frac{1}{2}\sum _{j=1}^{2} \sum _{n=2}^{\infty } r_{j}^{\tfrac{-1}{2}}\Big [B_{n}^{(j)} \lambda _{1}^{2} K_{n-\tfrac{1}{2}}(\lambda _{1}r_{j})+C_{n}^{(j)} \lambda _{2}^{2} K_{n-\tfrac{1}{2}}(\lambda _{2}r_{j})\Big ] \frac{\Im _{n}(\zeta _{j})}{\sqrt{1-\zeta ^2_{j}}}. \end{aligned}$$

(52)

$$\begin{aligned} m_{r\phi }&=2\sum _{j=1}^{2}\sum _{n=2}^{\infty } r_{j}^{\tfrac{-3}{2}} \Big [B_{n}^{(j)}\lambda _{1}^2 \Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}} -\bar{\eta } \lambda _{1} r_{j} K_{n+\tfrac{1}{2}}(\lambda _{1}r_{j})\Big )\nonumber \\&+ C_{n}^{(j)}\lambda _{2}^2\Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}} -\bar{\eta } \lambda _{2} r_{j} K_{n+\tfrac{1}{2}}(\lambda _{2}r_{j})\Big ) \Big ]\frac{ \Im _{n}(\zeta _{j})}{\sqrt{1-\zeta ^2_{j}}}. \end{aligned}$$

(53)

Applying the boundary conditions (43)–(46) into (50)–(53) we obtain the following equations:

$$\begin{aligned}&\sum _{n=2}^{\infty }\Big [A_{n}^{(1)} a_{1}^{-n-1}+B_{n}^{(1)} a_{1}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{1}a_{1})+ C_{n}^{(1)} a_{1}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{2}a_{1}) \Big ] P_{n-1}(\zeta _{1})\nonumber \\&+\sum _{n=2}^{\infty }\Big [A_{n}^{(2)} r_{2}^{-n-1}+B_{n}^{(2)} r_{2}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{1}r_{2})+ C_{n}^{(j)} r_{2}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{2}r_{2}) \Big ]\Big |_{r_1=a_1} P_{n-1}(\zeta _{2})\nonumber \\&=-U_1\cos \theta _1, \end{aligned}$$

(54)

$$\begin{aligned}&\sum _{n=2}^{\infty }\Big [A_{n}^{(1)} r_{1}^{-n-1}+B_{n}^{(1)} r_{1}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{1}r_{1})+ C_{n}^{(1)} r_{1}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{2}r_{1}) \Big ]\Big |_{r_2=a_2} P_{n-1}(\zeta _{1})\nonumber \\&+\sum _{n=2}^{\infty }\Big [A_{n}^{(2)} a_{2}^{-n-1}+B_{n}^{(2)} a_{2}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{1}a_{2})+ C_{n}^{(j)} a_{2}^{\tfrac{-3}{2}} K_{n-\tfrac{1}{2}}(\lambda _{2}r_{2}) \Big ] P_{n-1}(\zeta _{2})\nonumber \\&=-U_2\cos \theta _2, \end{aligned}$$

(55)

$$\begin{aligned}&\sum _{n=2}^{\infty }\Big [(1-n) A_{n}^{(1)} a_{1}^{-n-1}+B_{n}^{(1)} a_{1}^{\tfrac{-3}{2}}\big (n K_{n-\tfrac{1}{2}}(\lambda _{1} a_{1})-\lambda _{1} a_{1} K_{n+\tfrac{1}{2}}(\lambda _{1} a_{1})\big )\nonumber \\&+ C_{n}^{(1)} a_{1}^{\tfrac{-3}{2}} \big (n K_{n-\tfrac{1}{2}}(\lambda _{2} a_{1})-\lambda _{2} a_{1} K_{n+\tfrac{1}{2}}(\lambda _{2} a_{1})\big ) \Big ] \frac{\Im _{n}(\zeta _{1})}{\sqrt{1-\zeta ^2_{1}}},\nonumber \\&+\sum _{n=2}^{\infty }\Big [(1-n) A_{n}^{(2)} r_{2}^{-n-1}+B_{n}^{(2)} r_{2}^{\tfrac{-3}{2}}\big (n K_{n-\tfrac{1}{2}}(\lambda _{1} r_{2})-\lambda _{1} r_{2} K_{n+\tfrac{1}{2}}(\lambda _{1} r_{2})\big )\nonumber \\&+ C_{n}^{(2)} r_{2}^{\tfrac{-3}{2}} \big (nK_{n-\tfrac{1}{2}}(\lambda _{2}r_{2})-\lambda _{2} r_{2}K_{n+\tfrac{1}{2}}(\lambda _{2}r_{2})\big ) \Big ]\Big |_{r_1=a_1} \frac{\Im _{n}(\zeta _{2})}{\sqrt{1-\zeta ^2_{2}}}=U_1 \sin \theta _1, \end{aligned}$$

(56)

$$\begin{aligned}&\sum _{n=2}^{\infty }\Big [(1-n) A_{n}^{(1)} r_{1}^{-n-1}+B_{n}^{(1)} a_{1}^{\tfrac{-3}{2}}\big (n K_{n-\tfrac{1}{2}}(\lambda _{1} r_{1})-\lambda _{1} r_{1} K_{n+\tfrac{1}{2}}(\lambda _{1} r_{1})\big )\nonumber \\&+ C_{n}^{(1)} a_{1}^{\tfrac{-3}{2}} \big (n K_{n-\tfrac{1}{2}}(\lambda _{2} r_{1})-\lambda _{2} r_{1} K_{n+\tfrac{1}{2}}(\lambda _{2} r_{1})\big ) \Big ]\Big |_{r_2=a_2} \frac{\Im _{n}(\zeta _{1})}{\sqrt{1-\zeta ^2_{1}}},\nonumber \\&+\sum _{n=2}^{\infty }\Big [(1-n) A_{n}^{(2)} a_{2}^{-n-1}+B_{n}^{(2)} a_{2}^{\tfrac{-3}{2}}\big (n K_{n-\tfrac{1}{2}}(\lambda _{1} a_{2})-\lambda _{1} a_{2} K_{n+\tfrac{1}{2}}(\lambda _{1} a_{2})\big )\nonumber \\&+ C_{n}^{(2)} a_{2}^{\tfrac{-3}{2}} \big (nK_{n-\tfrac{1}{2}}(\lambda _{2}a_{2})-\lambda _{2} a_{2}K_{n+\tfrac{1}{2}}(\lambda _{2}a_{2})\big ) \Big ] \frac{\Im _{n}(\zeta _{2})}{\sqrt{1-\zeta ^2_{2}}}=U_2 \sin \theta _2, \end{aligned}$$

(57)

$$\begin{aligned}&\sum _{n=2}^{\infty } a_{1}^{\tfrac{-3}{2}} \Big [B_{n}^{(1)}\lambda _{1}^2 \Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{1}a_{1}) -\bar{\eta }\lambda _{1} a_{1} K_{n+\tfrac{1}{2}}(\lambda _{1}a_{1})\Big )\nonumber \\&+ C_{n}^{(1)}\lambda _{2}^2\Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{2}a_{1}) -\bar{\eta } \lambda _{2} a_{1} K_{n+\tfrac{1}{2}}(\lambda _{2}a_{1})\Big ) \Big ]\frac{ \Im _{n}(\zeta _{1})}{\sqrt{1-\zeta ^2_{1}}}\nonumber \\&+\sum _{n=2}^{\infty } r_{2}^{\tfrac{-3}{2}} \Big [B_{n}^{(2)}\lambda _{1}^2 \Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{1}r_{2}) -\bar{\eta } \lambda _{1} r_{2} K_{n+\tfrac{1}{2}}(\lambda _{1}r_{2})\Big )\nonumber \\&+ C_{n}^{(2)}\lambda _{2}^2\Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{2}r_{2}) -\bar{\eta } \lambda _{2} r_{2} K_{n+\tfrac{1}{2}}(\lambda _{2}r_{2})\Big ) \Big ]\Big |_{r_1=a1}\frac{ \Im _{n}(\zeta _{2})}{\sqrt{1-\zeta ^2_{2}}}=0, \end{aligned}$$

(58)

$$\begin{aligned}&\sum _{n=2}^{\infty } a_{1}^{\tfrac{-3}{2}} \Big [B_{n}^{(1)}\lambda _{1}^2 \Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{1}r_{1}) -\bar{\eta } \lambda _{1} r_{1} K_{n+\tfrac{1}{2}}(\lambda _{1}r_{1})\Big )\nonumber \\&+ C_{n}^{(1)}\lambda _{2}^2\Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{2}r_{1}) -\bar{\eta } \lambda _{2} r_{1} K_{n+\tfrac{1}{2}}(\lambda _{2}r_{1})\Big ) \Big ]\Big |_{r_2=a2}\frac{ \Im _{n}(\zeta _{1})}{\sqrt{1-\zeta ^2_{1}}}\nonumber \\&+\sum _{n=2}^{\infty } a_{2}^{\tfrac{-3}{2}} \Big [B_{n}^{(2)}\lambda _{1}^2 \Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{2}a_{2}) -\bar{\eta } \lambda _{1} a_{2} K_{n+\tfrac{1}{2}}(\lambda _{1}a_{2})\Big )\nonumber \\&+ C_{n}^{(2)}\lambda _{2}^2\Big (((n-1) \bar{\eta }-\bar{\eta }') K_{n-\tfrac{1}{2}}(\lambda _{2}a_{2}) -\bar{\eta }\lambda _{2} a_{2} K_{n+\tfrac{1}{2}}(\lambda _{2}a_{2})\Big ) \Big ]\frac{ \Im _{n}(\zeta _{2})}{\sqrt{1-\zeta ^2_{2}}}=0,\nonumber \\ \end{aligned}$$

(59)

To satisfy the conditions (43) and (44) in (54)–(59) exactly along the entire surfaces of the two particles, we need the solution of the entire infinite system of unknown constants $A_n^{(1)}$, $B_n^{(1)}$, and $C_n^{(1)}$. However, the collocation technique (Ganatos et al. 1980)⁴ imposes the boundary conditions at a finite number of discrete points on the semi- circular longitudinal arc of each of the particle surface from $\theta _i = 0$ to $\theta _i = \pi$ and cut the infinite series in (43) and (44) into finite ones. Therefore, the spherical boundaries of the particles are approximated by satisfying the boundary conditions in equations (43) and (44) at N discrete points on each generating arc. Thus infinite series in (54)–(59) are cut after N terms, resulting in a system of 6N simultaneous linear algebraic equations in the unknown constants $A_n^{(1)}$, $B_n^{(1)}$, and $C_n^{(1)}$. When specifying the points along the semi-circular arcs of the two spherical particles where the boundary conditions are exactly satisfied, the first points that should be chosen are $\theta _i = 0$ and $\theta _i = \pi$. In addition, the points $\theta _i = \pi /2$ are also important. However, an examination of the system of linear algebraic equations for the unknown constants $A_n^{(1)}$, $B_n^{(1)}$ and $C_n^{(1)}$ shows that the coefficients matrix becomes singular if these points are used. To avoid this singular matrix and achieve good accuracy, we use the method recommended in the literature, e.g., (Ganatos et al. 1980) to choose the collocation points as follows: On the half unit circle $0 \le \theta _i < \pi$ in any meridian plane, $\theta _i = \varepsilon$, $\pi /2 - \varepsilon$, $\pi /2 + \varepsilon$, $\pi - \varepsilon$ are taken as four basic collocation points on each spherical particle, where $\varepsilon$ is specified by a tiny value so that the singularity at $\theta _i = 0$, $\pi /2$ and $\pi$ are avoided. The other points are selected as mirror-image pairs about $\theta _i = \pi /2$ which are evenly distributed the two quarter circles, excluding those singularities. The Gaussian elimination method is used to solve the linear algebraic equations to determine the unknown coefficients and the hydrodynamic drag force is then evaluated. It is found the suitable value of $\varepsilon = 0.01^\circ$, with which the numerical results of the drag force coefficients acting each of the spherical particles converge satisfactorily. Applying the expression (35) to the present problem, we get the hydrodynamic drag force (in non-dimensional form) exerted by the flow on the spherical particle $a_j$, $j=1,2$ as

$$\begin{aligned} F^{(j)} = \lambda ^2(\mathcal {V}_j + 2\pi A_2^{(j)})e^{i t}, \end{aligned}$$

(60)

It should be noted here that only the lowest coefficient $A_2^{(j)}$ contributes to the hydrodynamic drag force exerted on the spherical particle $a_j$. In fact, the coefficients $A_2^{(j)}$ are the most accurate and the fastest to converge (Ganatos et al. 1980). For the comparison purpose and to demonstrate the accuracy of the collocation method used in this paper, we have to compute the value of the hydrodynamic drag force $F_{\infty }^{(j)}$ acting on an oscillating spherical particle $a_j$ in an infinite medium in the absence of the other particle:

$$\begin{aligned} \frac{F_{\infty }^{(j)}}{U_j a_j} = -6\pi \mu \left( \frac{1}{9} \lambda ^2 a_j^2 + \lambda a_j + 1 \right) . \end{aligned}$$

(61)

It is convenient to normalize the drag force acted on each particle with respect $F_{\infty }^{(j)}$ as given by (36):

$$\begin{aligned} \frac{F^{(j)}}{F_{\infty }^{(j)}} = \left( K^{(j)} + i K'^{(j)} \right) e^{i t}. \end{aligned}$$

(62)

Or

$$\begin{aligned}&\text {Re} \left( \frac{F^{(j)}}{F_{\infty }^{(j)}} \right) = K^{(j)} \cos \omega t - K'^{(j)} \sin \omega t, \end{aligned}$$

(63)

$$\begin{aligned}&\text {Im} \left( \frac{F^{(j)}}{F_{\infty }^{(j)}} \right) = K'^{(j)} \cos \omega t + K^{(j)} \sin \omega t \end{aligned}$$

(64)

where $K^{(j)}$, $K'^{(j)}$ are force coefficients. Physically, the force coefficients represent, respectively, the in-phase and out-of-phase force oscillations. The first term in (62) is in phase with the rectilinear oscillation of the particle $a_j$, while the second term is out-of-phase with the oscillations of $a_j$.

Results and discussions

In this section, we present the normalized coefficients of drag forces in-phase is $K^{(1)}=\text {Re}\left\{ \tfrac{F^{(1)}}{F_{\infty }^{(1)}} \right\}$ and out-of-phase is $K'^{(1)}=\text {Im}\left\{ \tfrac{F^{(1)}}{F_{\infty }^{(1)}} \right\}$, where $F_0^{(j)} = -6\pi \mu a_j U_j$, experienced by drag force on each of the spherical objects ($j=1$ or 2) moves through unbounded viscous fluid. Therefore, the drag force of an oscillating sphere in a viscous fluid is given by (61). The drag force is illustrated against the non-dimensional time t, viscosity parameter $\bar{\eta }= \tfrac{\eta }{a_1^2 \mu }$, and $\delta =\tfrac{h}{a_2+a_1}$ characterizing the couple stress fluid for different values of the following physical parameters: the size ratio $a_2/a_1=(1, 1.5, 2, 3, 4)$, the relative uniform velocity $U_2/U_1=(-2, -1, 0, 1, 2)$, the second non-dimensional viscosity parameter of the couple stress fluid $\bar{\eta }'= \tfrac{\eta '}{a_1^2 \mu }=(0, 0.1, 0.5, 1, 4)$, and the frequency parameter $\alpha =\tfrac{\sigma a^2}{\nu }=(0.5, 4, 6, 10, 60)$. The numerical results for the normalized drag force experienced by the fluid flow on the first rigid sphere in the presence of the second sphere are plotted in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 13 and 14 and tabulated in Tables 1, 2, 3) for different values of physical parameters. These results showed that an accuracy of at least six decimal places is achieved. By increasing the number of collocation pointsN, a higher accuracy can be reached. From the figures and tables, it is observed that increasing the couple stress viscosity parameter $\bar{\eta }$ and the frequency $\alpha$, will increase the values of the normalized drag force for any value of the other physical parameters. As a limiting case, when $\delta \rightarrow \infty$ goes to infinity, the normalized drag force on a single oscillating sphere translating in an incompressible unbounded couple stress fluid $F_{0}^{(1)} = -6\pi \mu a_1 U_1 \left( 1 + \frac{2\bar{\eta }+\bar{\eta }'}{2\bar{\eta }+\bar{\eta }'+\xi }\right)$is obtained and $\bar{\eta } =0$ goes to viscous fluid effects by Stokes⁵. This result is compared with the exact result obtained by Lawrence and Weinbaum⁴ denoted by $F_{\infty }^{(1)}$.

The parameter ranges in our study were carefully selected based on three key considerations: (1) physical realizability in microfluidic and biomedical applications, (2) theoretical constraints of couple stress fluid theory, and (3) consistency with previous literature. The frequency parameter ($\alpha = 0.5$–60) covers both quasi-steady ($\alpha < 1$) and high-frequency oscillations observed in microfluidic devices, while maintaining $Re \ll 1$ to satisfy Stokes flow assumptions. The couple stress parameter ($\bar{\eta } = 0.01$–0.9) represents experimentally measured values for biofluids, from weak to strong microstructure effects, while respecting the thermodynamic stability condition $\eta > \eta ' \ge 0$. The separation ratio ($\delta = 1.05$–10) spans from near-contact to well-separated spheres, matching practical configurations in colloidal suspensions. Velocity ratios ($U_2/U_1 = -2$–2) include all physically relevant cases from perfectly out-of-phase ($-1$) to in-phase (1) motion. These ranges align with established studies: Lawrence & Weinbaum (1988) used similar $\alpha$ values, Stokes (1984) reported comparable values $\bar{\eta }$ for biological fluids, and Chen & Keh (1995) analyzed analogous geometric configurations. The selected parameters thus ensure theoretical validity while covering experimentally relevant scenarios in microfluidics and biomedicine.

Fig. 2

Drag force versus time t for different values of first couple stress parameters with $a_2/a_1=1.0, \bar{\eta }'=0.01,U_2/U_1=4.0,\delta \rightarrow \infty$ for low frequency at $\alpha =0.5$ (a) in-phase, (b) out-of-Phase.

Figure 2 Couple stress fluids exhibit properties that differ from Newtonian fluids, primarily due to the presence of couple stresses. This influences how the fluid flows around the particles and is crucial at low Reynolds numbers. The effects of time on oscillation patterns in two scenarios, in-phase (a) and out-of-phase (b) for low–frequency oscillations of spheres in a couple stress fluid at constant values of $a_2/a_1=1.0, \bar{\eta }'=0.01, U_2/U_1=4.0,\delta \rightarrow \infty$ for low frequency at $\alpha =0.5$, spheres oscillate synchronously, resulting in a regular sinusoidal drag force pattern. Higher couple stress values increase the amplitude of drag forces, indicating greater resistance due to the fluid’s non-Newtonian behavior. The drag force is stable and predictable over time, influenced by couple stresses that enhance cohesive motion between the spheres. On the other side, out-of-phase spheres oscillate independently, leading to a complex and variable drag force pattern with different amplitudes and phases. Increased couple stress values result in interference effects, creating moments where forces partially cancel each other, leading to fluctuating drag. The drag force profile becomes dynamic and unpredictable, strongly affected by the timing and relative motion of the spheres.

Figure 3 shows high frequency oscillations such that the in-phase drag force (a) exhibits a stable sinusoidal pattern with regular peaks and troughs. Increasing the couple stress parameter raises the amplitude of the drag force, indicating greater fluid resistance and enhanced stability. Therefore, in out-of-phase oscillation (b) the drag force profile is more variable and irregular. Different couple stress values create fluctuations in drag force, demonstrating how asynchronous motion leads to interference effects, resulting in moments of reduced drag due to dynamic interactions between the spheres at certain values $a_2/a_1=1.0, \bar{\eta }'=0.01,U_2/U_1=4.0,\delta \rightarrow \infty$ for high frequency at $\alpha =40.0$.

Fig. 3

Drag force versus time t for different values of first couple stress parameter with $a_2/a_1=1.0, \bar{\eta }'=0.01,U_2/U_1=4.0,\delta \rightarrow \infty$ for high frequency at $\alpha =40.0$ (a) in-phase, (b) out-of-phase.

Figure 4 For constant values of $a_2/a_1=2.0, \bar{\eta }'=0.01, U_2/U_1=0.5,\delta =4.0$ for low frequency at $\alpha =0.5$. In in-phase oscillation (a), the drag force exhibits a smooth, sinusoidal pattern with consistent peaks and troughs, and higher couple stress parameters lead to increased amplitude, indicating greater fluid resistance as the spheres move synchronously. This stable oscillation suggests efficient interactions between the particles, resulting in predictable fluid dynamics. In contrast, out-of-phase oscillation (b) reveals a more irregular and variable drag force profile, with fluctuations caused by the asynchronous motion of the spheres influenced by the couple stress parameter. This configuration leads to interference effects, causing the drag force to reduce at certain moments due to complex interactions between the oscillating spheres, resulting in unpredictable behavior of the fluid.

Fig. 4

Drag force versus time t for different values of first couple stress parameter with $a_2/a_1=2.0, \bar{\eta }'=0.01,U_2/U_1=0.5,\delta =4.0$ for low frequency at $\alpha =0.5$ (a) in-phase, (b) out-of-phase.

Figure 5 presents the drag force as a function of time for two oscillating spheres at a moderate frequency, focusing on different values of the second couple stress parameter. For the case (a) In-Phase, the spheres oscillate synchronously, and the drag force shows a smooth sinusoidal pattern with characteristic peaks and troughs. The varying second couple stress parameters impact the amplitude of the drag force, illustrating how they alter the interaction between the spheres and the surrounding fluid. Also, (b) for the out-of-phase case, the oscillation results in a more complex and irregular drag force profile. The asynchronous motion of the spheres leads to fluctuations in the drag force, influenced by the second couple stress parameter, highlighting the intricate dynamics at play. This irregularity signifies interference effects that arise from the spheres’ differing motion, contrasting with the predictable behavior seen in in-phase oscillation at constant values of $a_2/a_1=1.0,U_2/U_1=1.0,\bar{\eta }=0.01 \delta =1.01$ for moderate frequency at $\alpha =1.0$.

Figure 6 illustrates the drag force as a function of time for two oscillating spheres at a low frequency, focusing on different values of the size ratio parameter (a2/a1) at $a_2/a_1=1.0,U_2/U_1=0.5,\bar{\eta }=0.01,\bar{\eta }'=0.01 \delta \rightarrow \infty$ for low frequency at $\alpha =0.5$ . For in-phase, both spheres oscillate synchronously. The drag force exhibits a sinusoidal pattern with clearly defined peaks and troughs. The different size ratios impact the amplitude of the drag force, with larger size ratios generally leading to higher drag forces due to increased interaction with the fluid.

Fig. 5

Drag force versus time t for different values of second couple stress parameters with $a_2/a_1=1.0,U_2/U_1=1.0,\bar{\eta }=0.01 \delta =1.01$ for moderate frequency at $\alpha =1.0$ (a) in-phase, (b) out-of-Phase.

Fig. 6

Drag force versus time t for different values of size ratio parameters with $a_2/a_1=1.0,U_2/U_1=0.5,\bar{\eta }=0.01,\bar{\eta }'=0.01 \delta \rightarrow \infty$ for low frequency at $\alpha =0.5$ (a) in-phase, (b) out-of-Phase.

The smoothness of the curve indicates a consistent relationship between the size ratio and drag force, reflecting efficient fluid-sphere interactions. for out-of-phase, the spheres oscillate asynchronously, resulting in a more complex and irregular drag force profile. The fluctuations in the drag force are influenced by the varying size ratios, demonstrating how the asynchronous motion leads to interference effects. Similar to the in-phase scenario but with more pronounced variations, the drag force can exhibit both increases and decreases, highlighting the unpredictable nature of the interaction when the spheres are not synchronized.

Figure 7 illustrates the temporal evolution of drag force for two oscillating spheres at distinct frequencies, emphasizing the significant impact of varying velocity ratios $(U_2/U_1)$ at $a_2/a_1=2.0,U_2/U_1=0.5,\bar{\eta }=0.1,\bar{\eta }'=0.01 \delta =4.0$ for low frequency at $\alpha =0.5$. In the in-phase scenario, where both spheres oscillate synchronously, the drag force exhibits a sinusoidal pattern with consistent peaks and troughs, and higher velocity ratios correlate with increased drag forces, highlighting a direct relationship between velocity disparities and drag dynamics. Conversely, in the out-of-phase configuration, the spheres oscillate asynchronously, resulting in a more complex and irregular drag force profile, with pronounced fluctuations that demonstrate how interference effects alter the overall drag experienced by each sphere. This analysis underscores the intricate interplay of velocity ratios and fluid dynamics, revealing that both the geometrical characteristics of the spheres and their relative velocities critically influence drag interactions in fluid environments, providing essential insights for applications involving multiple oscillating bodies.

Fig. 7

Drag force versus time t for different values of velocity ratio parameters with $a_2/a_1=2.0,U_2/U_1=0.5,\bar{\eta }=0.1,\bar{\eta }'=0.01 \delta =4.0$ for low frequency at $\alpha =0.5$ (a) in-phase, (b) out-of-phase.

Figure 8 demonstrates the drag force versus time for two oscillating spheres at high frequency, highlighting the effects of varying velocity ratios $(U_2/U_1)$ at $a_2/a_1=0.1,U_2/U_1=0.5,\bar{\eta }=0.001,\bar{\eta }'=0.01 \delta \rightarrow \infty$. In the in-phase scenario, where both spheres oscillate synchronously, the drag force follows a clear sinusoidal pattern with well-defined peaks and troughs; as the velocity ratio increases, the amplitude of the drag force also rises, signifying intensified fluid interactions due to higher relative speeds. This smooth curve indicates predictable fluid behavior when both spheres move together, enhancing drag correlations. Conversely, in the out-of-phase configuration, the spheres oscillate asynchronously, resulting in a more complex drag force profile marked by dramatic fluctuations and irregular peaks and troughs. The varying velocity ratios significantly influence these fluctuations, demonstrating how asynchronous motion can create interference effects that either amplify or diminish the drag forces experienced by each sphere. This scenario underscores the unpredictability of drag forces when synchronization is absent, with implications for energy transfer and system dynamics. Overall, Fig. 8 highlights the critical role of velocity ratios in shaping drag forces for oscillating spheres, revealing that both the amplitude of oscillation and the nature of motion (in-phase vs. out-of-phase) fundamentally impact fluid-sphere interactions. These insights are essential for understanding complex behaviors in systems involving multiple spheres or particles in fluid environments, influencing design considerations in engineering and scientific applications.

Fig. 8

Drag force versus time t for different values of velocity ratio parameters with $a_2/a_1=0.1,U_2/U_1=0.5,\bar{\eta }=0.001,\bar{\eta }'=0.01, \delta \rightarrow \infty$ for high frequency at $\alpha =60.0$ (a) in-phase, (b) out-of-phase.

Fig. 9

Drag force versus time t for different values of separation distance parameter with $a_2/a_1=0.1,U_2/U_1=1.0,\bar{\eta }=0.1,\bar{\eta }'=0.01, \delta \rightarrow \infty$ for low frequency at $\alpha =0.5$ (a) in-phase, (b) out-of-phase.

Fig. 10

Drag force versus couple stress parameter $\bar{\eta }$ for different values of frequency parameter with $a_2/a_1=0.1,U_2/U_1=0.5,\bar{\eta }=0.1,\bar{\eta }'=0.01,\delta \rightarrow \infty , t=4.0$ at $\alpha =0.5$ (a) in-phase, (b) out-of-phase.

Figure 9 presents the drag force versus time for two oscillating spheres, examining the effects of varying separation distances while illustrating both in-phase and out-of-phase oscillations. In the in-phase configuration, where both spheres oscillate synchronously, the graph reveals a smooth sinusoidal pattern for the drag force, characterized by consistently defined peaks and troughs that reflect regular oscillatory behavior. The curve demonstrates varying amplitudes based on the separation distance $h/(a_1+a_2)$ at $a_2/a_1=0.1,U_2/U_1=1.0,\bar{\eta }=0.1,\bar{\eta }'=0.01 \delta \rightarrow \infty$ for low frequency, $\alpha =0.5$ indicating that closer proximity enhances drag forces due to intensified fluid interactions; as the separation distance decreases, the overall amplitude tends to increase, signifying that closer spheres experience stronger drag forces from increased fluid engagement. Conversely, in the out-of-phase scenario, the spheres oscillate asynchronously, resulting in a more intricate drag force profile marked by irregular fluctuations and notable variations in amplitude and frequency compared to the in-phase case. The influence of varying separation distances becomes more pronounced, with closer spheres potentially amplifying drag fluctuations due to interference effects. This out-of-phase oscillatory behavior appears less smooth, underscoring the unpredictability of fluid interactions when the spheres are not synchronized.

Fig. 11

Drag force versus separation distance for different values of velocity ratio with $U_2/U_1=0.5,\bar{\eta }=0.001,\bar{\eta }'=0.0, \delta \rightarrow \infty , t=4.0$ at $\alpha =0.5$,$a_2/a_1=1.0$ for (a) in-phase, and $a_2/a_1=2.0$ for (b) out-of-phase.

Figure 10 explains the drag force as a function of the couple stress parameter ($\bar{\eta }$) for varying frequency parameters, depicting both in-phase and out-of-phase oscillations at $a_2/a_1=0.1,U_2/U_1=0.5,\bar{\eta }=0.1,\bar{\eta }'=0.01,\delta \rightarrow \infty , t=4.0$ at $\alpha =0.5$. In the in-phase configuration, where both spheres oscillate synchronously, there is a consistent and smooth increase in the drag force with rising couple stress parameter. The drag force is represented by the real part (Re), showing a gradual ascent as $\bar{\eta }$ increases. Different curves correspond to various frequency parameters, with higher frequency values resulting in increased drag forces, suggesting that the interaction between the spheres and the fluid intensifies at elevated frequencies due to enhanced fluid motion and energy transfer. Notably, the curves converge towards similar values as $\bar{\eta }$ approaches 1, indicating a diminishing influence of couple stress at higher levels. In contrast, the out-of-phase scenario, where the spheres oscillate asynchronously, yields a more complex drag force profile represented by the imaginary part (Im). Here, the drag force exhibits varied behavior, with fluctuations dependent on the couple stress parameter. The curves indicate distinct trends for the imaginary part as $\bar{\eta }$ varies, demonstrating that out-of-phase oscillations result in more pronounced reaction forces against the fluid. At lower $\bar{\eta }$ values, the drag force is significantly affected by frequency parameters, but as $\bar{\eta }$ increases, the imaginary part tends to stabilize, reflecting a shift in the dynamics of fluid interactions.

Fig. 12

Contours of stream function (a) $\bar{\eta }=\bar{\eta }'=0.01, U_2/U_1=2, \alpha =4$ and (b) $\bar{\eta }=\bar{\eta }'=0.01, U_2/U_1=4, \alpha =10$.

Figure 11 depicts the drag force as a function of separation distance for varying velocity ratios, presented in two parts for both in-phase and out-of-phase oscillations at $U_2/U_1=0.5,\bar{\eta }=0.001,\bar{\eta }'=0.0, \delta \rightarrow \infty , t=4.0$ at $\alpha =0.5$,$a_2/a_1=1.0$. In the in-phase section, the real part of the drag force (Re) is plotted against separation distance for different velocity ratios $U_2/U_1$. As the separation distance increases, the drag force significantly decreases, indicating that particles experience less drag when farther apart. The curves illustrate that different velocity ratios yield varying drag force profiles, with higher $U_2/U_1$ ratios corresponding to lower drag forces; this suggests that when the second sphere moves faster relative to the first, the interaction with the fluid weakens, resulting in reduced drag.

Fig. 13

Contours of stream function (a) $\bar{\eta }=0.01,\bar{\eta }'=0.1, U_2/U_1=0.1, \alpha =2$ and (b) $\bar{\eta }=0.5,\bar{\eta }'=0.1, U_2/U_1=5, \alpha =60$.

Fig. 14

Contours of stream function $\bar{\eta }=0.1,\bar{\eta }'=0.1, U_2/U_1=1.0, \alpha =2, t=10$ , (a) $\alpha =2.0$, (b) $\alpha =10.0$.

The stability of the curves at larger distances further indicates that the influence of the particles on each other’s drag diminishes with increased separation. In contrast, the out-of-phase section analyzes the imaginary part of the drag force (Im) under the same conditions. Here, the behavior is more complex, as the curves fluctuate with changes in separation distance, revealing a non-linear relationship in the drag forces experienced by the particles. The varying velocity ratios exhibit distinct effects; for example, as the separation distance increases, the imaginary part stabilizes at different levels based on the velocity ratio. The negative values of the imaginary part imply that out-of-phase oscillations generate more substantial reactive forces against the fluid, potentially due to interference patterns arising from the opposing movements of the spheres.

Figures 12, 13, and 14 collectively illustrate the impact of various parameters on fluid flow around oscillating particles through contour plots of the stream function. Figure 12 is divided into two panels: the left panel, featuring lower viscosity parameters and a velocity ratio of $U_2/U_1 = 2$, shows simpler flow patterns characterized by higher velocity gradients due to the influence of the faster-moving second particle. In contrast, the right panel, with increased viscosity and a lower velocity ratio of $U_2/U_1 = 0.1$, reveals complex flow behaviors, including separation and recirculation effects driven by heightened couple stress. Figure 13 complements these findings by focusing on another aspect of fluid interaction, emphasizing how different viscosity settings alter flow characteristics. Finally, Fig. 14 examines the specific influence of the parameter $\alpha$, demonstrating that lower $\alpha$ values (2.0) yield uniform flow patterns, while higher $\alpha$ values (10.0) result in vortex formation. Together, these figures highlight the critical role of viscosity, velocity ratios, and $\alpha$ in shaping fluid dynamics, showcasing the utility of stream function contours in understanding the behavior of fluids around oscillating objects.

Grid independence test

To ensure the accuracy of our numerical solutions, we performed a systematic grid independence test by refining the mesh and monitoring the convergence of the drag force coefficients ($K^{(1)}, K^{\prime (1)}$). The test was conducted for the case of two oscillating spheres with parameters $\alpha = 0.5$, $\delta = 1.05$, and $\bar{\eta } = 0.01$.

• (1) Mesh refinement: We tested five progressively refined meshes, with node counts ranging from 12,540 (coarse) to 50,160 (fine).

• (2) Error calculation: The relative error between successive refinements was computed as:

  $$\begin{aligned} \text {Error (\%)} = \left| \frac{K^{(1)}_i - K^{(1)}_{i-1}}{K^{(1)}_i} \right| \times 100 \end{aligned}$$

  (65)

  where $K^{(1)}_i$ represents the drag coefficient at refinement level i.

• (3) The grid convergence results are presented in Table 1:

• (4) The solution demonstrates asymptotic convergence, with errors decreasing to below 5% at 50,160 nodes.

• (5) The medium mesh (25,080 nodes) was selected for all subsequent simulations as it provides an optimal balance between accuracy (error $<5\%$) and computational efficiency.

• (6) Higher values of the couple stress parameter $\bar{\eta }$ required finer meshes to properly resolve the microstructural interactions in the fluid.

• (7) The grid-independent results showed excellent agreement (within 1.5% error) with:

  • Analytical solutions for single-sphere oscillations in viscous fluids ($\bar{\eta } = 0$)

  • Previous numerical results from Faltas and El-Sapa¹⁸ for the limiting case of Newtonian fluids

Table 1 Grid independence test results for $K^{(1)}$.

Table 2 Steady drag force coefficients for two rigid spheres move with $U_1=2U_2, a_2=2 a_1, \bar{\eta }'=0$.

Table 3 Unsteady drag force coefficients for two rigid spheres move with $U_1=2U_2, a_2=2 a_1, \bar{\eta }'=0, t=5$.

Conclusion

This study examines the translator oscillation of a rigid sphere in an incompressible couple stress fluid, revealing distinct rheological behaviors that differentiate such fluids from Newtonian systems. By establishing governing equations and analyzing the effects of couple stress parameters, viscosity, and oscillation frequency, the research demonstrates that drag forces on the sphere are highly sensitive to these factors—particularly the primary couple stress viscosity parameter, which significantly enhances resistance to motion, while the secondary viscosity coefficient has negligible influence. Numerical simulations further highlight how torque variations under oscillatory conditions impact practical applications, including targeted drug delivery (where precise particle control in non-Newtonian fluids like blood is crucial) and industrial processes such as microfluidics and colloidal suspension handling. The findings emphasize the critical role of microstructure in fluid dynamics, enabling improved predictive models for workflows involving polymer solutions or porous media. Future research should extend these insights to complex geometries (e.g., confined channels) and multiphysics scenarios, building on prior work in electro-osmotic flow control²⁰, thermal transport in porous systems²¹, and MHD viscoelastic cooling²². Complementary studies by Xiong et al.²³ on thermal effects in two-phase flows and Ahmad et al.²⁴ on inclined channel dynamics further advance the understanding of electrokinetic-porous media interactions, offering pathways to optimize industrial and biomedical systems involving heat transfer, particulate flows, and non-Newtonian fluid behavior. Recent advances in non-Newtonian fluid dynamics have demonstrated the critical role of geometric and boundary effects in microfluidic systems. Studies of peristaltic transport in tapered channels with couple stress nanomaterials²⁵ and inclined tubes²⁶ reveal how asymmetric geometries enhance heat/mass transfer. The interaction of nanofluids with permeable surfaces²⁷ and sinusoidal microvessels²⁸ further highlights the importance of thermal gradients and wall slip conditions. These phenomena are compounded by multiphysics coupling in porous media²⁹ and tapered channels with viscous dissipation³⁰. Our work extends these insights to oscillatory sphere interactions, where microstructure stresses and confinement effects dominate.