Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field

Shreen El-Sapa; Munirah Aali Alotaibi

doi:10.1515/phys-2024-0085

Article Open Access

Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field

Shreen El-Sapa and Munirah Aali Alotaibi

Published/Copyright: September 27, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In this study, we examine the movement of two hard spheres aligned in a straight line within an incompressible couple stress fluid under the impact of the magnetic field. Both objects have distinct shapes and move along an axis connecting their centers with varying velocities. As a first step, an incompressible analytical analysis is performed on a fluid with couple stress properties around an axially symmetric particle. Using the superposition principle, a general solution is developed for couple stress fluid flows over two moving objects. In order to achieve the boundary conditions, the boundary collocation strategy is applied to the surfaces of the two spheres. A set of tables and graphs illustrates numerical estimates of the dimensionless drag forces acting on two spherical objects. In addition, a drop in Hartmann number or an increase in couple stress viscosity will increase the dimensionless drag force on each spherical particle.

Keywords: magnetic field; couple stress fluid; drag force; interaction; collocation method

1 Introduction

A wide variety of physical, geophysical, and industrial domains use the motion of conducting fluids in an electromagnetic field. Magnetohydrodynamics (MHD) effects can control fluid travel past solid objects in these real-world contexts. The classical problem of Hartmann flow has numerous essential applications in many fields, including MHD pumps and power generators, heating, polymer technology, aerodynamics, the petroleum industry, crude oil purification, and heat exchanger design. The study of transporting fluids past solid surfaces of diverse shapes has also emerged due to recent breakthroughs in rocketry and spacecraft. Bearman and Wadcock [1] discussed the interaction between the flows around two circular cylinders when they are shifted in a plane perpendicular to the free stream and brought near together. A study was carried out by Yutaka et al. [2] to investigate the fluid dynamic interaction between two spheres to gather fundamental knowledge about the flow of two-phase fluids, particularly in the dense phase. Faltas et al. [3] proposed a solution for the interaction between two spherical particles revolving in a micropolar fluid. Shehadeh and Ashmawy [4] examined the continuous linear movement of two aligned solid spheres in an incompressible couple stress fluid.

Several investigations have sought to determine the influence of an applied magnetic field on the movement of an electrically conductive fluid around a spherical or cylindrical object. MHD is an area of science that focuses on comprehending the behavior of fluids when exposed to a magnetic field’s influence. The presence of MHD in a fluid flow has the potential to manage flow separation, optimize heat transfer, and alter fluid flow velocity. Chandrasekhar [5] has made observations about how a magnetic field affects rotating cylinders’ viscous flow consistency. Davis et al. [6] demonstrated the hydrodynamic modification of a rigid elastic sphere that is submerged in a viscous fluid and moving into another sphere or a solid surface. An inviscid conducting fluid with a strong magnetic field was the subject of Stewartson’s [7] examination of a fully conducting spherical in steady motion. When the sphere moves in the same direction as the field, the streamlines outside the sphere appear as straight lines. However, when the sphere moves perpendicular to the field, the streamlines exhibit abrupt twists. Saad [8] conducted a study on the influence of a magnetic field on the flow around a porous sphere and cylinders enclosed by a cell. Srivastava [9] investigated the implications of MHD regarding the hydrodynamic porosity of a permeable membrane consisting of spherical particles. Madasu and Bucha [10] examined the MHD impact of the movement of fluid through a partially permeable spherical particle and derived a clear expression for drag. Also, they analyzed the flow of MHD past a cylindrical shill using Brinkman’s model for the problem of parallel flow [11]. El-Sapa [12] investigated the impact of slip and magnetic field on the motion of a solid sphere that is moving perpendicular to an infinite rigid wall in an unbounded viscous fluid. El-Sapa and Alsudais [13] investigated, on the assumption of Stock’s conditions, the effects of the magnetic field on two stiff spheres with varying slip conditions on their surfaces in a permeable media. El-Sapa and Faltas [14] studied the continuous and almost consistent straight motion of two spherical particles immersed in an infinite magnetomicropolar fluid. Several recent scholarly articles have concentrated on the subject of how a magnetic field affects the flow of fluid [15–19].

MHD and couple stress fluid are distinct ideas that can be integrated to investigate the flow characteristics of electrically conductive fluids with certain attributes. Through the examination of MHD in the context of couple stress fluid flow, scientists acquire a more profound comprehension of the intricacies of fluid dynamics in scenarios encompassing: fluid with microstructure that can conduct electricity, magnetic fields employed for regulating the flow of a substance, enhancement of filtration, and lubrication procedures. Nadeem and Akram [20] performed a research investigation on the movement of fluid with qualities of couple stress in a channel that is not symmetrical, while considering the impact of the magnetic field that is created. The solutions for the equations of magnetic field and momentum have been accurately calculated, under the assumption of the long wavelengths and a low, but non-zero Reynolds number. Hassan [21] investigated the examination of a hydromagnetic fluid flow that undergoes a chemical reaction, specifically focusing on couple stress fluids. The flow occurs through a channel that is filled with porous particles. Srinivasacharaya and Rao [22] provided numerical solutions for the constant flow of MHD blood via a branched artery with a slight narrowing in the main channel, including heat transfer and assuming blood to be fluid with couple stress. Ali et al. [23] conducted a study on the MHD flow and heat transfer of a couple stress fluid across an oscillating stretching band in a porous medium with a heat source/sink. Alotaibi and El-Sapa [24] investigated the translation motion of a rigid sphere enclosed by a concentric sphere filled with a MHD fluid that exhibits coupling stress. The study considered the effects of slippage and the Hatmann number. The investigation of couple stress fluids is seen crucial for comprehending various physical phenomena, particularly in the field of biomedicine.

Drag force refers to the opposition encountered by an object as it travels through a fluid, which be either a liquid or a gas. This opposing force acts in the direction opposite to an object’s motion. The amount of the drag force is influenced by various parameters, such as the velocity of the object, the dimensions and configuration of the item, the density, and the consistency of the fluid. Hoffmann et al. [25] analyzed the opposition encountered by a spherical object in micropolar fluids taking into account non-uniform boundary conditions for the microrotation vector. Ashmawy [26] derived a comprehensive equation for the drag experienced by a sphere immersed in a slow and time-varying flow of a micropolar fluid. Shu and Lee [27] developed fundamental solutions for micropolar fluids generated by a singular point force and a singular point pair. Later on, the aforementioned solutions were used to calculate the drag force applied to a solid sphere moving in a micropolar flow with a low Reynolds number. Sherief et al. [28] formulated a mathematical equation to determine the drag force exerted on a symmetrical object that is in motion at a consistent speed within a microstrech fluid.

Approximately 50 years ago, Gluchman et al. [29] created a novel method for treating multiple particles that are immersed in an infinitely slow viscous flow. This approach is a numerical method commonly known as the Frontier collocation technique. Their theoretically predicted drag results show a high level of agreement with the experimental evidence. The fundamental concept behind employing the collection approach to address the flow issue at a low Reynolds number is to compute the velocity field generated by every coordinate boundary. This is achieved through the methodical arrangement of fundamental solutions that are suitable for the particular constant orthogonal surface. The coefficients in the fundamental solutions are determined by the specified boundary conditions. The series, which represents the solution, is shortened and the boundary criteria are only applied at certain locations referred to as collocation points. Several studies have made use of this approach [30–34]. Conversely, several numerical techniques exist for examining this interplay between the objects, and also show the interactions and chemical applications, which are elucidated through previous studies [35,36,37,38,39,40,41].

The purpose of this study is to demonstrate how two rigid spheres traveling rectilinearly on a shared axis interact with each other during axial movement. The spheres are subjected to the impact of a magnetic field in an incompressible fluid with couple stress properties. The surfaces of the two spheres are exposed to boundary conditions where there is no couple tension and no sliding. An incompressible object moving steadily in an infinite series may represent the semi-analytical solution for the couple stress fluid via an axis-symmetric object. The general solution for the stable motion of a fluid with couple stress, as it flows past two spheres that are moving in a straight line along their centers, is constructed by combining the obtained solution with the superposition principle. Two spherical coordinate systems are employed, each with its origin located in the center of a solid sphere. The collocation approach is utilized to satisfy the prescribed border constraints on the spherical surfaces. The dimensionless drag force acting based on the solid spheres is calculated and sketched. Analyzing numerical results will be accomplished by graphing the results.

2 MHD couple stress fluid equations

Therefore, couple stresses primarily affect the creation of a length-dependent impact, which is nonexistent from the standard model of non-polar fluid dynamics. Couple stress fluid hypothesis was introduced by Stokes [42] in 1984, building over the well-known Navier–Stokes theories. The current investigation is based on the assumptions that [43] and [44] examine: (a) the fluid is incompressible because of its homogeneous, isotropic density, (b) since the Lorentz force is only magnetic, uneven polarization and electric charge are inconsequential, (c) the movement is creeping because magnetic Reynold’s number R = U a ν σ and Reynold’s number R = U a ν are deemed to display an appropriate level of humility. A term representing inertia is disregarded, (d) the couple stress is ignored, as is the body force operating across the flow direction, (e) there is no external electric field, and (f) this analysis is isothermal. Moreover, previous studies [43–45] provide the MHD couple stress fluid equations of motion as.

(2.1) ∇ ⋅ q → = 0 ,

(2.2) μ ∇ ∧ ∇ ∧ q → + η ∇ ∧ ∇ ∧ ∇ ∧ ∇ ∧ q → = − ∇ p + μ e 2 σ ( q → ∧ H → ) ∧ H → ,

where ∇ is the spherical partial differential operator, q → is the volume-averaged velocity, p is the pore average pressure, μ represents the viscosity of the fluid, η is the first couple stress viscosity coefficient, η ′ is the second couple stress viscosity coefficient. If the couple stress coefficient η is taken zero, then the equation of motion (2.2) reduces to the classical Navier–Stokes’s equation. The basis of classical electromagnetism is made up of the Lorentz force law and an ensemble of related partial differential equations known as the Maxwell–Heaviside equations are given as

(2.3) div E → = 0 , div H → = 0 , curl E → = − ∂ B → ∂ t , curl H → = J → + ∂ D → ∂ t ,

where B → = μ e H → , D → = ε E → , B → ∧ J → = σ B → ∧ ( E → + q → ∧ B → ) , the magnetic field vector is H → , the electric field vector E → , the fluid pressure is p , the magnetic permeation is μ e , μ is the viscosity coefficient, the electric conduction is σ . η , and η ′ are the couple stress viscosities. Additionally, J → , B → , and D → represent the current density, magnetic induction, and electric displacement vectors, accordingly. Whenever η = 0 (2.2) is provided the Newtonian fluids by Prasad and Sarkar [43].

The constitutive relations listed in the following describe couple stress fluids:

(2.4) τ i j = − p δ i j + 2 μ d i j − 1 2 ε i j k m s k , a ,

(2.5) m i j = m δ i j + 4 ( η ′ ω i , j + η ω j , i ) ,

where τ i j and m i j are the stress and couple stress tensors respectively, the alternating tensor is ε i j k , and the Kronecker delta is δ i j . Furthermore, the deformation stress tensor is d i j = 1 2 ( q i , j + q j , i ) , and the spin function is ω i = 1 2 ε i j k q k , j .

These inequalities are satisfied by the viscosity coefficients in the couple stress fluid calculations:

(2.6) 0 ≤ μ , 0 ≤ η , η ′ − η ≤ 0 , 0 ≤ 3 λ + 2 μ .

Consequently, the earlier provided mathematical equations have been assigned the subsequent non-dimensional characteristics:

(2.7) q → * = q → U , ω → * = a ω → U , p * = a p μ U , ∇ * = a ∇ , τ r r * = a τ r r μ U , α 2 = a 2 κ , ξ = a 2 μ η , H → * = H → H 0 , R H = μ e 2 a 2 H 0 2 σ μ .

After substituting Eq. (2.7) into (2.2), disregarding the asterisks for simplicity, and using the slowing flow from the earlier assumptions, we obtain [43]

(2.8) ∇ p + 1 ξ 2 ∇ ∧ ∇ ∧ ∇ ∧ ∇ ∧ q → + ∇ ∧ ∇ ∧ q → − R H 2 ( q → ∧ H → ) ∧ H → = 0 .

Using Lorentz’s force, the equation for a non-polar fluid (2.8) is an updated version of the Stokes equation, where R H represents the Hartmann number, if ξ approaches infinity. If R H = 0 , the Stokes approach to modeling the behavior of a fluid with couple stress.

The function stream ψ can be utilized for describing the velocity features:

(2.9) q r ( r , θ ) = − 1 r 2 sin θ ∂ ψ ∂ θ , q θ ( r , θ ) = 1 r sin θ ∂ ψ ∂ r .

Additionally, the vorticity vector can be defined as follows:

(2.10) ω → = 1 2 ∇ ∧ q → = E 2 ψ 2 r sin θ e → ϕ .

Then, with the help of Eq. (2.9) the continuity Eq. (2.1) is immediately achieved, which reduces to:

(2.11) 0 = − ∂ p ∂ r − 1 r 2 sin θ ∂ ∂ θ E 2 ψ + 1 ξ 2 1 r 2 sin θ ∂ ∂ θ E 2 ( E 2 ψ ) + R H 2 1 r 2 sin θ ∂ ψ ∂ θ ,

(2.12) 0 = − 1 r ∂ p ∂ θ + 1 r sin θ ∂ ∂ r E 2 ψ − 1 ξ 2 1 r sin θ ∂ ∂ r E 2 ( E 2 ψ ) − R H 2 1 r sin θ ∂ ψ ∂ r ,

where E 2 = ∂ 2 ∂ r 2 + 1 − ζ 2 r 2 ∂ 2 ∂ ξ 2 , ζ = cos θ . One can derive the subsequent six-order partial differential formula by removing pressure from Eqs (2.11) and (2.12)

(2.13) E 2 ( E 2 − k 1 2 ) ( E 2 − k 2 2 ) ψ = 0 ,

where the roots of (3.6) are obtained by

(2.14) k 1 2 + k 2 2 = ξ 2 , k 1 2 k 2 2 = ξ 2 R H 2 k i = ξ 2 ± ξ ξ 2 − 4 R H 2 2 , i = 1 , 2 .

3 MHD couple stress fluid over a moving rigid sphere

Under the assumption that fluid motion is axisymmetric, meaning that every value acts independently of ϕ , ( r , θ , ϕ ) represents the spherical polar structure. The rigid sphere of radius a is translated at a uniform velocity U through unbounded MHD couple stress fluid. Furthermore, the stream function, the velocities, and the vorticity components are established by the following:

(3.1) ψ ( r , θ ) = ∑ n = 2 ∞ A n r − n + 1 + B n r 1 2 K n − 1 2 ( k 1 r ) + C n r 1 2 K n − 1 2 ( k 2 r ) G n ( ζ ) ,

(3.2) q r = − ∑ n = 2 ∞ A n r − n − 1 + B n r − 3 2 K n − 1 2 ( k 1 r ) + C n r − 3 2 K n − 1 2 ( k 2 r ) P n − 1 ( ζ ) ,

(3.3) q θ = ∑ n = 2 ∞ ( 1 − n ) A n r − n − 1 + B n r − 3 2 ( n K n − 1 2 ( k 1 r ) − k 1 r K n + 1 2 ( k 1 r ) ) + C n r − 3 2 ( n K n − 1 2 ( k 2 r ) − k 2 r K n + 1 2 ( k 2 r ) ) G n ( ζ ) 1 − ζ 2 .

From Eq. (3.3), the vorticity component is

(3.4) ω ϕ = 1 2 ∑ n = 2 ∞ r − 1 2 B n k 1 2 K n − 1 2 ( k 1 r ) + C n k 2 2 K n − 1 2 ( k 2 r ) G n ( ζ ) 1 − ζ 2 .

The set of constants A n , B n , and C n must be computed using the constraints on the boundary that have been applied to the boundary region.

The formula can be used to calculate the force of the drag produced by an incompressible medium [34]. The flow of a couple stress fluid is characterized by infinite stretching along the axisymmetric object as it moves in the same direction.

(3.5) F z = 8 π μ lim r → ∞ ψ r sin 2 θ .

The following formula is obtained by combining (3.1) with the drag formula (3.5):

(3.6) F z = 2 μ π R H 2 U a 2 3 + A 2 .

As is well known, F ∞ = − 6 π μ U a represents the force of drag operating on a rigid sphere traveling in a uniform incompressible viscous flow of fluid [26]. When we compare our findings to the special situation of viscous fluid, this latter formula will be helpful.

4 Interaction of two rigid spheres within magneto couple stress fluid

This section analyzes the uniform-state translating motion of an incompressible couple stress fluid past two inline hard spheres. Two hard objects having radii of a j , j = 1 , 2 , and uniform velocities of U j , j = 1 , 2 , respectively, translate along a common axis that connects their centers to produce the fluid motion. As seen in Figure 1, the two spherical objects are situated outside of one another. It is believed that the fluid is at rest at a great distance from the two spheres. Dual spherical coordinate systems, ( r 1 , θ 1 , ϕ ) and ( r 2 , θ 2 , ϕ ) , having origins at the centers of spheres a 1 and a 2 , respectively, are conveniently considered. While the boundaries are axisymmetric, the angle ϕ has no bearing on any of the flow field functions. Furthermore, the following relations tie the two coordinate systems ( r 1 , θ 1 ) and ( r 2 , θ 2 ) to one another:

(4.1) r 1 2 = r 2 2 + h 2 − 2 r 2 h cos θ 2 , r 2 2 = r 1 2 + h 2 + 2 r 1 h cos θ 1 .

Figure 1

Sketch of two spheres in a magneto-couple stress fluid.

The Stokesian flow presumption will be used, assuming that the velocities are modest. In these conditions, all hydrodynamic functions are not dependent on ϕ and the flow is axially symmetric. In the event when the second item is absent, take q → j to represent the velocity vector of the couple stress fluid caused by the existence of the spherical particles a j . We select τ r θ ( j ) , q → ( j ) , ω → ( j ) , and m r ϕ ( j ) in the following manner:

(4.2) q → ( j ) ( r j , θ j ) = q r ( 1 ) ( r 1 , θ 1 ) + q r ( 2 ) ( r 2 , θ 2 ) , m r j ϕ ( j ) ( r j , θ j ) = m r 1 ϕ ( 1 ) ( r 1 , θ 1 ) + m r 2 ϕ ( 2 ) ( r 2 , θ 2 ) .

The separation of variables technique is the conventional approach to solve a sixth-order linear partial differential Eq. (2.13):

(4.3) ψ ( r , θ ) = ∑ j = 1 2 ∑ n = 2 ∞ A n ( j ) r j − n + 1 + B n ( j ) r j 1 2 K n − 1 2 ( k 1 r j ) + C n ( j ) r j 1 2 K n − 1 2 ( k 2 r j ) G n ( ζ j ) .

The second-kind modified Bessel equations of degree n are defined by K n ( . ) , accordingly. The constants A n ( j ) , C n ( j ) , B n ( j ) , E n ( j ) , D n ( j ) , F n ( j ) , j = 1 , 2 can be derived from the boundary conditions (4.1). The components of velocity are

(4.4) q r = − ∑ j = 1 2 ∑ n = 2 ∞ A n ( j ) r j − n − 1 + B n ( j ) r j − 3 2 K n − 1 2 ( k 1 r j ) + C n ( j ) r j − 3 2 K n − 1 2 ( k 2 r j ) P n − 1 ( ζ j ) .

(4.5) q θ = ∑ j = 1 2 ∑ n = 2 ∞ ( 1 − n ) A n ( j ) r j − n − 1 + B n ( j ) r j − 3 2 ( n K n − 1 2 ( k 1 r j ) − k 1 r j K n + 1 2 ( k 1 r j ) ) + C n ( j ) r j − 3 2 ( n K n − 1 2 ( k 2 r j ) − k 2 r j K n + 1 2 ( k 2 r j ) ) G n ( ζ j ) 1 − ζ j 2 .

From Eq. (3.3), the vorticity and couple stress components are

(4.6) ω ϕ = 1 2 ∑ j = 1 2 ∑ n = 2 ∞ r j − 1 2 B n ( j ) k 1 2 K n − 1 2 ( k 1 r j ) + C n ( j ) k 2 2 K n − 1 2 ( k 2 r j ) G n ( ζ j ) 1 − ζ j 2 .

(4.7) m r ϕ ( r , θ ) = ∑ j = 1 2 ∑ n = 2 ∞ r j − 3 2 B n ( j ) k 1 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 − ξ − 2 k 1 r j K n + 1 2 ( k 1 r j ) + C n ( j ) k 2 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 − ξ − 2 k 2 r j K n + 1 2 ( k 2 r j ) G n ( ζ j ) 1 − ζ j 2 .

For Eq. (2.14), we should have the six boundary conditions to describe the problem fully. The assumptions of Stokes [42] state that the physical interactions on the outer edge are equal to force dispersion mainly because there are no couple stresses at the spherical object interface. The two spheres translate with different velocities U j , j = 1 , 2 such that the kinematic and dynamic boundary conditions on the two surfaces of the solid objects r j = a j , j = 1 , 2 are

(4.8) q r ( r j , θ j ) = U j cos θ j , q θ ( r j , θ j ) = − U j sin θ j , m r j ϕ ( r j , θ j ) = 0 .

Inputting in the left-hand side of the boundary constraints (4.5)–(4.13) for r j = a j , j = 1 , 2 , we obtain the equations of velocity and couple stress provided by (4.4), (4.5), and (4.7):

(4.9) ∑ n = 2 ∞ A n ( 1 ) a 1 − n − 1 + B n ( 1 ) a 1 − 3 2 K n − 1 2 ( k 1 a 1 ) + C n ( 1 ) a 1 − 3 2 K n − 1 2 ( k 2 a 1 ) P n − 1 ( ζ 1 ) + ∑ n = 2 ∞ A n ( 2 ) r 2 − n − 1 + B n ( 2 ) r 2 − 3 2 K n − 1 2 ( k 1 r 2 ) + C n ( j ) r 2 − 3 2 K n − 1 2 ( k 2 r 2 ) r 1 = a 1 × P n − 1 ( ζ 2 ) = − ζ 1 ,

(4.10) ∑ n = 2 ∞ A n ( 1 ) r 1 − n − 1 + B n ( 1 ) r 1 − 3 2 K n − 1 2 ( k 1 r 1 ) + C n ( 1 ) r 1 − 3 2 K n − 1 2 ( k 2 r 1 ) r 2 = a 2 P n − 1 ( ζ 1 ) + ∑ n = 2 ∞ A n ( 2 ) a 2 − n − 1 + B n ( 2 ) a 2 − 3 2 K n − 1 2 ( k 1 a 2 ) + C n ( j ) a 2 − 3 2 K n − 1 2 ( k 2 a 2 ) P n − 1 ( ζ 2 ) = − U ¯ ζ 2 ,

(4.11) ∑ n = 2 ∞ ( 1 − n ) A n ( 1 ) a 1 − n − 1 + B n ( 1 ) a 1 − 3 2 n K n − 1 2 ( k 1 a 1 ) − k 1 a 1 K n + 1 2 ( k 1 a 1 ) + C n ( 1 ) a 1 − 3 2 n K n − 1 2 ( k 2 a 1 ) − k 2 a 1 K n + 1 2 ( k 2 a 1 ) G n ( ζ 1 ) 1 − ζ 1 2 , + ∑ n = 2 ∞ ( 1 − n ) A n ( 2 ) r 2 − n − 1 + B n ( 2 ) r 2 − 3 2 n K n − 1 2 ( k 1 r 2 ) − k 1 r 2 K n + 1 2 ( k 1 r 2 ) + C n ( 2 ) r 2 − 3 2 n K n − 1 2 ( k 2 r 2 ) − k 2 r 2 K n + 1 2 ( k 2 r 2 ) r 1 = a 1 G n ( ζ 2 ) 1 − ζ 2 2 = 1 1 − ζ 1 2 ,

(4.12) ∑ n = 2 ∞ ( 1 − n ) A n ( 1 ) r 1 − n − 1 + B n ( 1 ) a 1 − 3 2 n K n − 1 2 ( k 1 r 1 ) − k 1 r 1 K n + 1 2 ( k 1 r 1 ) + C n ( 1 ) a 1 − 3 2 n K n − 1 2 ( k 2 r 1 ) − k 2 r 1 K n + 1 2 ( k 2 r 1 ) r 2 = a 2 G n ( ζ 1 ) 1 − ζ 1 2 , + ∑ n = 2 ∞ ( 1 − n ) A n ( 2 ) a 2 − n − 1 + B n ( 2 ) a 2 − 3 2 n K n − 1 2 ( k 1 a 2 ) − k 1 a 2 K n + 1 2 ( k 1 a 2 ) + C n ( 2 ) a 2 − 3 2 n K n − 1 2 ( k 2 a 2 ) − k 2 a 2 K n + 1 2 ( k 2 a 2 ) G n ( ζ 2 ) 1 − ζ 2 2 = U ¯ 1 − ζ 2 2 ,

(4.13) ∑ n = 2 ∞ a 1 − 3 2 B n ( 1 ) k 1 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 1 a 1 ) − ξ − 2 k 1 a 1 K n + 1 2 ( k 1 a 1 ) + C n ( 1 ) k 2 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 1 a 1 ) − ξ − 2 k 1 a 1 K n + 1 2 ( k 2 a 1 ) G n ( ζ 1 ) 1 − ζ 1 2 + ∑ n = 2 ∞ r 2 − 3 2 B n ( 2 ) k 1 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 2 r 2 ) − ξ − 2 k 1 r 2 K n + 1 2 ( k 1 r 2 ) + C n ( 2 ) k 2 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 2 r 2 ) − ξ − 2 k 2 r 2 K n + 1 2 ( k 2 r 2 ) r 1 = a 1 G n ( ζ 2 ) 1 − ζ 2 2 = 0 ,

(4.14) ∑ n = 2 ∞ a 1 − 3 2 B n ( 1 ) k 1 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 1 r 1 ) − ξ − 2 k 1 r 1 K n + 1 2 ( k 1 r 1 ) + C n ( 1 ) k 2 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 1 r 1 ) − ξ − 2 k 2 r 1 K n + 1 2 ( k 2 r 1 ) r 2 = a 2 G n ( ζ 1 ) 1 − ζ 1 2 + ∑ n = 2 ∞ a 2 − 3 2 B n ( 2 ) k 1 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 2 a 2 ) − ξ − 2 k 1 a 2 K n + 1 2 ( k 1 a 2 ) + C n ( 2 ) k 2 2 ( ( n − 1 ) ξ − 2 − ξ ′ − 2 ) K n − 1 2 ( k 2 a 2 ) − ξ − 2 k 2 a 2 K n + 1 2 ( k 2 a 2 ) G n ( ζ 2 ) 1 − ζ 2 2 = 0 .

This problem might be solved using the boundary collocation strategy established by Gluckman et al. [29]. But in order to do this, the complete infinite system of equations involving the unknown constants would need to be solved, which is not conceivable. Using the boundary colocation approach developed by Gluckman et al. [29], this challenge may be overcome. The steps involved in this technique are as follows: first, each infinite series must be truncated after a certain number N ≈ 60 of terms in order for the number of unknown constants to become finite; next, a sufficient set of points on each spherical particle must be chosen as colocation points; at these points, boundary conditions must be applied in order to yield the same number of linear equations, 6 N as those containing the unknown constants; and finally, the system of equations that results is solved to yield the unknown constants, allowing us to calculate the flow field. The hydrodynamic drag force F z ( j ) with respect to F ∞ ( j ) = − 6 μ π U j a j acting on the particle a j may be obtained by using formula (3.6):

(4.15) F z ( j ) = 2 μ π α 2 U j a j 2 3 + A 2 ( j ) .

5 Numerical results

Non-Newtonian fluid flow has long been a topic of interest; many practicals have been conducted to expand the applications and advance contemporary technology. Scholars examined many forms of fluids and contributed to global modernization. One of the greatest subjects in fluid dynamics is couple stress fluids. Chemical engineering, manufacturing, polymer analysis, and biomedical engineering all make use of non-Newtonian fluids. This study shows the distributions of dimensionless drag force F z ⁄ F ∞ versus the separation distance δ ( = h ⁄ ( a 1 + a 2 ) ) for unique values of couple stress parameters, η , η ′ , the size ratio a 2 ⁄ a 1 , the velocity ratio U 2 ⁄ U 1 , the Hartmann number R H , as shown in Figures 2, 3, 4, 5, 6, 7 and Table 1. In addition, the streamlines in terms of the stream functions are displayed in Figures 8, 9, 10, 11, 12. The dimensionless Hartmann number measures the relative significance of magnetic induction drag and viscous drag. Furthermore, the electric and magnetic fields have many applications. Given that blood is a fluid that conducts electricity and the Lorentz force opposes the motion of conducting fluids according to Lenz’s law, the principles of MHD therapy can be used to slow the flow of blood through the human artery system, which can help treat certain cardiovascular disorders.

Figure 2 represents the separation distance for different first couple stress parameters, versus the normalized drag force. It is observed that the drag force, increases gradually with both the increase of separation distance and the first couple stress parameters, which physically coincide with the fact that states when the sphere translates alone in the absence of the second force, it has a significant force at the two spheres moving in the same direction with the same velocities and equal in size with certain values of R H = 1.0 , η ¯ ′ = 0.01 .
In Figure 3, the normalized drag force is plotted against the separation distance for different velocity ratio parameters. It has been observed that drag force increases gradually as separation distance increases and decreases as the velocity ratio increases. Physically, drag opposes object motion just like friction does. In this case, two solid spheres have the same size with equal speeds at R H = 1.0 , η ¯ ′ = 0.01 , and η ¯ = 0.1 .
Figure 4 illustrates the normalized drag force corresponding to the separation distance for different size ratio parameters. It has been observed that when the sphere a 2 = 10 a 1 the drag force tends to unity and also it increases with both the separation distance and the size ratio. Physically, drag opposes object motion just like friction does. In this case, the two solid spheres move in the same direction with the same speeds with the other certain values being R H = 1.0 , η ¯ ′ = 0.01 , and η ¯ = 0.1 .
Figures 5 and 6 display the normalized drag force corresponding to the separation distance for different Hartmann numbers. For high values f R H , the normalized drag force inclines with the increase of R H at η ¯ = 0.001 but for small values of R H at η ¯ = 0.1 , the normalized drag force increases with the increase of R H with the other certain values being U 2 ⁄ U 1 = 1.0 , η ¯ ′ = 0.01 , and a 2 ⁄ a 1 = 1.0 .
Figure 7 shows the normalized drag force corresponding to the separation distance for different second couple stress fluid parameters. It seems that the improvement of the normalized drag force improves with the increase of η ¯ ′ with the other certain values being U 2 ⁄ U 1 = 1.0 , η ¯ = 0.1 , a 2 ⁄ a 1 = 1.0 , R H = 1.0 .
Figures 8–12 depict the streamlines of the constant stream function with the other constant values of the parameters U 2 ⁄ U 1 , η ¯ , η ¯ ′ , a 2 ⁄ a 1 , R H , and δ .

$Figure 2 Drag force distribution versus the separation parameter δ \delta for different first couple stress parameter η ¯ \bar{\eta } with the constant values of U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 , η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , and R H = 1.0 {R}_{H}=1.0 .$

Figure 2

Drag force distribution versus the separation parameter δ for different first couple stress parameter η ¯ with the constant values of U 2 U 1 = 1.0 , a 2 a 1 = 1.0 , η ¯ ′ = 0.01 , and R H = 1.0 .

$Figure 3 Drag force distribution versus the separation parameter δ \delta for different velocities ratios U 2 U 1 \frac{{U}_{2}}{{U}_{1}} with the constant values of a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 , η ¯ = 0.1 \bar{\eta }=0.1 , η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , and R H = 0.1 {R}_{H}=0.1 .$

Figure 3

Drag force distribution versus the separation parameter δ for different velocities ratios U 2 U 1 with the constant values of a 2 a 1 = 1.0 , η ¯ = 0.1 , η ¯ ′ = 0.01 , and R H = 0.1 .

$Figure 4 Drag force distribution versus the separation parameter δ \delta for different size ratios a 2 a 1 \frac{{a}_{2}}{{a}_{1}} with the constant values of U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , η ¯ = 0.1 \bar{\eta }=0.1 , η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , and R H = 1.0 {R}_{H}=1.0 .$

Figure 4

Drag force distribution versus the separation parameter δ for different size ratios a 2 a 1 with the constant values of U 2 U 1 = 1.0 , η ¯ = 0.1 , η ¯ ′ = 0.01 , and R H = 1.0 .

$Figure 5 Drag force distribution versus the separation parameter δ \delta for different Hartmann number RH with the constant values of U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , η ¯ = 0.001 \bar{\eta }=0.001 , η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 .$

Figure 5

Drag force distribution versus the separation parameter δ for different Hartmann number RH with the constant values of U 2 U 1 = 1.0 , η ¯ = 0.001 , η ¯ ′ = 0.01 , a 2 a 1 = 1.0 .

$Figure 6 Drag force distribution versus the separation parameter δ \delta for different Hartmann numbers with the constant values of U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , η ¯ = 0.1 \bar{\eta }=0.1 , η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , and a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 .$

Figure 6

Drag force distribution versus the separation parameter δ for different Hartmann numbers with the constant values of U 2 U 1 = 1.0 , η ¯ = 0.1 , η ¯ ′ = 0.01 , and a 2 a 1 = 1.0 .

$Figure 7 Drag force distribution versus the separation parameter δ \delta for different η ¯ ′ \bar{\eta }^{\prime} with the constant values of U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , η ¯ = 0.1 \bar{\eta }=0.1 , a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 , and R H = 1.0 {R}_{H}=1.0 .$

Figure 7

Drag force distribution versus the separation parameter δ for different η ¯ ′ with the constant values of U 2 U 1 = 1.0 , η ¯ = 0.1 , a 2 a 1 = 1.0 , and R H = 1.0 .

Table 1

Dimensionless drag force on the rigid sphere a 1 for different relevant factors with η ′ ¯ = 0.01 and the two equal spheres

		F z ⁄ F 0
		R H = 0.0 ( Ref [ 4 ] )			R H = 1.0
U 2 U 1	δ	η ¯ = 0.001	η ¯ = 0.025	η ¯ = 0.1	η ¯ = 0.001	η ¯ = 0.025	η ¯ = 0.1
	1.05	0.942388	0.961502	1.007459	0.423299	0.481470	0.705180
	2.0	0.989002	1.009977	1.056119	0.580938	0.666808	1.023915
	3.0	1.000166	1.022736	1.072481	0.588492	0.675521	1.041810
1.0	4.0	1.003849	1.027253	1.078907	0.588908	0.675904	1.042120
	5.0	1.005496	1.029437	1.082338	0.588919	0.675875	1.041809
	6.0	1.006375	1.030693	1.084480	0.588925	0.675866	1.041677
	7.0	1.006902	1.031500	1.085948	0.588935	0.675872	1.041634
	8.0	1.007244	1.032059	1.087020	0.588946	0.675881	1.041624
	9.0	1.007481	1.032468	1.087837	0.588954	0.675889	1.041625
	10.0	1.007653	1.032780	1.088482	0.588961	0.675896	1.041629
	1.05	1.130650	1.185593	1.315125	0.745426	0.859470	1.393453
	2.0	1.034587	1.070941	1.153892	0.597014	0.684985	1.059540
	3.0	1.019071	1.051395	1.124588	0.589474	0.676317	1.041456
− 1.0	4.0	1.014251	1.044880	1.113957	0.589064	0.675947	1.041205
	5.0	1.012174	1.041874	1.108691	0.589054	0.675977	1.041523
	6.0	1.011095	1.040207	1.105593	0.589048	0.675986	1.041657
	7.0	1.010460	1.039167	1.103566	0.589037	0.675980	1.041700
	8.0	1.010054	1.038463	1.102141	0.589027	0.675971	1.041709
	9.0	1.009777	1.037960	1.101085	0.589018	0.675962	1.041708
	10.0	1.009578	1.037583	1.100272	0.589012	0.675955	1.041704

$Figure 8 Streamlines for certain values of η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , η ¯ = 0.001 \bar{\eta }=0.001 , a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 , R H = 1.0 {R}_{H}=1.0 , and δ = 1.05 \delta =1.05 .$

Figure 8

Streamlines for certain values of η ¯ ′ = 0.01 , U 2 U 1 = 1.0 , η ¯ = 0.001 , a 2 a 1 = 1.0 , R H = 1.0 , and δ = 1.05 .

$Figure 9 Streamlines for certain values of η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , U 2 U 1 = 1.0 \frac{{U}_{2}}{{U}_{1}}=1.0 , η ¯ = 0.001 \bar{\eta }=0.001 , a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 , R H = 1.0 {R}_{H}=1.0 , and δ = 3.0 \delta =3.0 .$

Figure 9

Streamlines for certain values of η ¯ ′ = 0.01 , U 2 U 1 = 1.0 , η ¯ = 0.001 , a 2 a 1 = 1.0 , R H = 1.0 , and δ = 3.0 .

$Figure 10 Streamlines for certain values of η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , U 2 U 1 = 2.0 \frac{{U}_{2}}{{U}_{1}}=2.0 , η ¯ = 0.001 \bar{\eta }=0.001 , a 2 a 1 = 1.0 \frac{{a}_{2}}{{a}_{1}}=1.0 , R H = 2.0 {R}_{H}=2.0 , and δ = 3.0 \delta =3.0 .$

Figure 10

Streamlines for certain values of η ¯ ′ = 0.01 , U 2 U 1 = 2.0 , η ¯ = 0.001 , a 2 a 1 = 1.0 , R H = 2.0 , and δ = 3.0 .

$Figure 11 Streamlines for certain values of η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , U 2 U 1 = 2.0 \frac{{U}_{2}}{{U}_{1}}=2.0 , η ¯ = 0.01 \bar{\eta }=0.01 , a 2 a 1 = 2.0 \frac{{a}_{2}}{{a}_{1}}=2.0 , R H = 2.0 {R}_{H}=2.0 , and δ = 3.0 \delta =3.0 .$

Figure 11

Streamlines for certain values of η ¯ ′ = 0.01 , U 2 U 1 = 2.0 , η ¯ = 0.01 , a 2 a 1 = 2.0 , R H = 2.0 , and δ = 3.0 .

$Figure 12 Streamlines for certain values of η ¯ ′ = 0.01 \bar{\eta }^{\prime} =0.01 , U 2 U 1 = 4.0 \frac{{U}_{2}}{{U}_{1}}=4.0 , η ¯ = 0.1 \bar{\eta }=0.1 , a 2 a 1 = 4.0 \frac{{a}_{2}}{{a}_{1}}=4.0 , R H = 1.0 {R}_{H}=1.0 , and δ = 3.0 \delta =3.0 .$

Figure 12

Streamlines for certain values of η ¯ ′ = 0.01 , U 2 U 1 = 4.0 , η ¯ = 0.1 , a 2 a 1 = 4.0 , R H = 1.0 , and δ = 3.0 .

6 Conclusions

This article examines the migration of two rigid objects moving in a straight line through an incompressible fluid with couple stress properties. The migration is analyzed in terms of its progression and uniformity, specifically under the influence of a magnetic field. The concept of superposition is used to generate an infinite series, which serves as the general solution for the continuous translational movements of a couple stress fluid traveling through an axisymmetric structure. Using this method to build two spherical coordinate frames with origins at the centers of the two spherical particles solves the problem. Using the boundary collocation approach to limit the spherical bounds, we may find the unknown constants in the truncated series. We provide and display the numerical results of the normalized drag applied to an object. These findings demonstrated how quickly the numerical values converge to at least six decimal places. The normalized force of drag is significantly influenced by the physical parameters δ , U 2 ⁄ U 1 , a 2 ⁄ a 1 , η ¯ , η ′ ¯ , and R H . The key observations and the numerical findings of the current study can be summarized as follows:

The impact of R H with η ¯ on the normalized force of drag on each of the sphere is more significant.
It is detected that the drag force generally rises as the couple stress parameters rise and falls as the Hartmann number rises. This is because permitting the microelements to generate less overall proportionate motion between the fluid and the particle minimizes the drag force.
At significantly higher-frequency parameters, the velocity is noted to approach zero at the center of the microannulus.
To further comprehend the behavior of fluid flow when the magnetic field strength is raised, streamlines are displayed.
Finally, by setting the viscosity coefficient to zero and R H = 0 , the specific case of binary translating spheres in a viscous fluid is generated.

This study holds numerous applications in fluid dynamics and couple stress fluids are one of the trendiest subjects. Non-Newtonian fluids have applications in polymer analysis, manufacturing, chemical engineering, and biomedical engineering.

Funding information: This research was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2024R522), Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.
Author contributions: All authors were involved in the overall planning and design of the study. Shreen El-Sapa was responsible for gathering materials, collecting data, and analyzing the results. Munirah Aali Alotaibi wrote the initial draft of the manuscript, which was then reviewed and improved upon by all authors. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2024-07-07

Revised: 2024-08-10

Accepted: 2024-08-19

Published Online: 2024-09-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0085

Keywords for this article

magnetic field; couple stress fluid; drag force; interaction; collocation method

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