Interaction of Two Rigid Spheres Oscillating in an Infinite Liquid under the Control of a Magnetic Field

. The transient creeping motion of two rigid spheres oscillating in a boundless viscous ﬂ uid beneath the impact of the magnetic ﬁ eld is investigated. There is no slippage associated with a Stokes ﬂ ow on the two rigid spherical surfaces with di ﬀ erent sizes and radii. The solutions can be obtained using the boundary collocation scheme at low Reynolds numbers. The unsteady real and imaginary drag coe ﬃ cients are estimated on the hard spherical particles. These coe ﬃ cients are computed in tables for various parameters and illustrated graphically. The frequency parameter and Hartmann number also play a signi ﬁ cant role in this study where the drag coe ﬃ cients decrease or increase by 100 percent after the value of κ = 5 : 0 . Using available literature data, we tested the accuracy and reliability of our results.


Introduction
There has been a lot of research into how the dynamics of fluid techniques interact with oscillating particles. As a result of these studies, wave packs experienced by offshore forms can be estimated and ocean vehicle motion can be predicted from a functional view. The relations between two rigid spheres oscillating in a viscid flow along their linking axis of symmetry are introduced in [1]. By using direct numerical simulations [2], the periodic couplings of two hard spheres that move uniformly side by side in a fluid stream are studied. On the other hand, an investigation of hard spheres swimming into an oscillating fluid flow is investigated in [3]. Therefore, [4] introduces interactions around oscillating particles in a steady streaming flow. Further, the study of the impact of adjoining boundaries upon rotationally oscillating spheres in viscous fluids is presented experimentally and theoretically in [5]. In a pendulous flow, two solid spheres immersed in a viscid fluid move perpendicularly to the direction of the flow, and a little separation distance between them is presented by [6]. Therefore, Faltas and El-Sapa [7] proposed a solution to the problem of a couple of spherical objects swimming in a viscid liquid at a low Reynolds number by employing the collocation technique.
In cancer therapy, magnetohydrodynamics (MHD) plays a significant role in magnetic drug targeting. There is a model demonstrating a setup for examining the effects of an external magnetic field containing a magnetic carrier substance that interacts with blood flow. In medication, MHD liquid streams and fluctuations in different mathematical shapes applicable to human body parts are fascinating and essential exploration regions. Plumpton and Ferraro [8] researched the influence of the homogenous magnetic field upon the torsional fluctuations of a perpetual sphere. Also, Stewartson [9] introduced symmetrically oscillating objects at the infinite conductivity limit. In 1965, [10] investigated the impacts of incompressible magnetic fields, Coriolis powers, and their communication on physical drag. In 2015, Barakat [11] presented an issue of the MHD soundness of an oscillating liquid within sight of a longitudinal magnetic field. This work settled utilizing a PC calculation that decided the steady and unsteady zones under the changed values of the acting magnetic field. Besides, Wentzell [12] correspondingly contemplated the torsional motions of a profoundly directing low viscosity of a drop enclosed in a uniform magnetic field. Under the impact of the magnetic field, there are many applications of MHD such as the solitary waves propagation, and stability conditions of the cold plasma are analyzed using the reductive perturbation method by Adel et al. [13,14] and also in nuclear reactors and nanofluids in [15,16].
Furthermore, the numerical explanations of the equations coupled between magnetic and velocity fields for a completely created MHD move via an unplugged direct channel of the rectangular area. Hence, Cai et al. [17] assumed the combination strategy collocation with a plan of a storm under the impact of a magnetic field. A careful examination of MHD applications and mathematical demonstrations in organic frameworks was directed by Rashidi et al. [18]. Also, Dhivya et al. [19] used a Wrench Nicholson plot and limited contrast plans for solving an issue of a characteristic convective progression of an isochoric and synthetically responsive viscid liquid about an oscillating vertical chamber encased in a permeable structure. A computational strategy for a thick incompressible magnetohydrodynamic stream in a pivoting channel under magnetic field impacts was also determined by Li et al. [20]. The semianalytical method for the study of the translational movement of two rigid spheres is considered by El-Sapa [21] and El-Sapa and Faltas [22] to compute the mobility coefficients and set the circumstances on the spheres' surfaces mathematically utilizing the limit collocation technique. A semianalytical model for analyzing rectilinear fluctuations of a solid particle engaged in an incompressible micropolar fluid specified by a rigid plane wall has been developed by Yadav et al. [23]. In a plane wall, the particle oscillates rectilinearly. Moreover, recently, El-Sapa and Alhejaili [24] explored the influence of slippage length on the movement of two inflexible spheres oscillating via a Stokes flow about their axis of symmetry through the line linking their poles and the global solutions constructed upon the superposition of the actual solutions in the two spherical coordinate techniques by a collocation procedure.
This work focuses on oscillating two rigid spheres moving inside a viscid liquid along the line joining their centers at different velocities affected by a magnetic field. Semianalytical solutions for the velocity fields are introduced. In addition, the hydrodynamic drag force coefficients of the real and the imaginary parts for different frequencies, detachment space, size ratio, Hartmann number, and speed proportions are obtained and discussed. In general, the forces decrease or increase gradually at the value of κ = 5:0. For instance, the steady state with pure oscillations and no-slip express is confirmed as convergent and accurate.

Magneto-Stokes Field Equations
Low Reynolds number assumptions are covered by the general equations of the incompressible magnetoviscid liquid [17,22]: considering the velocity q ! , density ρ, fluid pressure p, kinematic viscosity of the fluid νð= μ/ρÞ, and dynamic viscosity μ. Therefore, F ! E is the outward magnetoforce given as Consequently, c is the light speed, B             Journal of Applied Mathematics [23] where the vector of magnetization is disregarded except if the magnetic field is the areas of extreme strength, speed of the liquid is not normal to the vector of magnetic induction that maintains the stream axisymmetric and creeping movements, so where B 0 is a constant. Besides, we have equation (3) that pursues where η = c 2 /σμ 0 is the parameter of diffusivity, σ is the electroconductivity, and R H is the number of Hartmann.

The Mathematical Formulation
Further, it is assumed that the unbending molecule of radius a oscillates axisymmetrically under the circumstances q ! /U = cos ωt e ! z as r ⟶ ∞ on the limit and impedes in an infinite viscid liquid under the normal magnetic field. Thus, e The two-dimensional velocity vector is 3.2. The Dimensionless Quantities. By using U for the scale of velocity, the particle, and a for the length in Re By using equation (8) into equation (2) and using equation (5), we have In addition, we have where R m = aU/μ and S t = lω/U are the Reynolds and the Strouhal numbers, respectively, ðS t > >1, R m < <1Þ. From equation (9), the field equations with stream function terms are 1 r Eliminating the pressure from equations (11) and (12), a fourth-order partial differential equation was obtained by using the stream function: is the frequency parameter, E 2 = ð∂ 2 /∂r 2 Þ + ð1 − ξ 2 /r 2 Þð∂ 2 /∂ξ 2 Þ is the Stokesian operator, and ξ = cos θ. The consistent solution of (13) is as follows: where the Gegenbauer function is I n ð:Þ of the first kind of order n and degree −1/2. The modified Bessel function is K n ð:Þ of the second kind of order n. Thus, the components of velocity are calculated by where P n ð:Þ is the polynomial of the Legendre of degree n. In the oscillating volume particle V, a nondimensional drag force operates by [15] and modified by [22]: Additionally, equation (18) gives A drag force into equation (18) can be normalized by acting upon a solid sphere a 1 swimming within an infinite viscous fluid region in the absence of another hard sphere without slippage, which is introduced by [15]:

The Problem Solution
The two rigid spheres of radii are a j , j = 1, 1, and suppose that a 2 ða 2 > a 1 Þ. The influence by a magnetic field, axially oscillated inside an endless fluid flow of Stokes, suggests that the two hard spheres vibrate with respective amplitudes U 1 and U 2 going with the associating line of their centers which are isolated by a constant distance h. Then, the fluid flow stops at limitlessness. The system of spherical frameworks is utilized and formed over the focus of the two unbending spheres. In the meantime, the discoveries of this work are Re  Journal of Applied Mathematics characterized in Figure 1. This connection between the two coordinates ðr 1 , θ 1 Þ and ðr 2 , θ 2 Þ is given by r 2 2 = r 2 1 + h 2 + 2h r 1 cos θ 1 , θ 1 ) or r 2 1 = r 2 2 + h 2 − 2hr 2 cos θ 2 , θ 1 ). Suppose the fluid velocities of the two solid spheres are 4.1. Boundary Conditions. As a result of the components of the spheres' velocities approaching zero at an extended distance, the surfaces of solid spheres a j , j = 1, 1 exhibit the following conditions: (1) Impenetrability conditions: (2) Dynamical conditions: By applying the superposition principle, we have The function of the stream and the components of velocity are written in the following forms: Applying the boundary conditions from equations (21) and (22) into (24) and (25), we obtained the following four equations: The procedure of the Gauss elimination is assumed to solve the above equations to get the constants A ðjÞ n and B ðjÞ n , j = 1, 2. Then, by equations (18) and (19), we can obtain the following expression of the hydrodynamic nondimensional drag  Journal of Applied Mathematics force coefficients on the particle a 1 : where where K j and K j ′ are defined physically as in-phase and out-of phase forces of the oscillations, respectively.

Results and Discussions
In this paper, we describe the influence of the normal magnetic field upon two hard spheres that oscillate along their connecting lines and associate their centers. We show that the unsteady normalized drag force coefficients of the real part Re fF ð1Þ z /F ð1Þ ∞ g and the imaginary part Im fF ð1Þ z /F ð1Þ ∞ g act on the solid sphere a 1 , respectively, shown in Figures 2-9 and Tables 1-3. Therefore, the two parts of forces are determined analytically and numerically for differ-ent relevant parameters as the number of the Hartmann R H , the frequency κ, the parameter of separation h/ða 2 + a 1 Þ, the velocity's ratio U 2 /U 1 , and the size ratio a 2 /a 1 . Physically, this means that the value and direction of the magnetic field oscillate with time at any point in the unbounded region. Furthermore, this means that the combined effect of the magnetic field and the oscillation on the particles will now experience a changing force, causing them to move with the wave. In fact, the real drag force coefficient is enhanced with the increase of the frequency and the magnetic field in the case of a small period of time and conversely for a large period of time. On the other side, the imaginary drag force coefficient disintegrates in the increase of the frequency parameters, but for the magnetic effects, it differs from a high or low level due to the period of time. Figure 2 exposes an analysis of Re fF ð1Þ z /F ð1Þ ∞ g versus the frequency parameter for different values of time t = 0:1, 0:5 , 1:0, and 4:0, and the number of the Hartmann number is R H = 0:0, 2:0, 6:0, and 10:0 that the two hard spheres oscillate with equal velocities and sizes with separation h/a 2 + a 1 = 1:5. Also, clearly, the nondimensional forces increment and diminish from least to greatest values at t = 0:1 and 4:0; with the growth of R H , the forces reverse their effects, where at t = 0:1, they increase with the increase of R H and converse at t = 4:0. Furthermore, this phenomenon has the lowest significance when the frequency becomes low and highest  Journal of Applied Mathematics significance when there is high hesitancy, but for the values t = 0:5 and 1:0, it rises with the growth of Hartmann numbers, and also, it has the most increased significance for low hesitancy and gradually goes to lowest values for the increased frequency. As expected, increasing R H decreases flow amplitude. In this case, the magnetic field acts as a resistance to the flow. Figure 3 shows an analysis of Im fF ð1Þ z /F ð1Þ ∞ g against the frequency for different values of time t = 0:1, 0:5, 1:0, and 4:0 , and the Hartmann numbers R H = 0:0, 2:0, 6:0, and 10:0 move with equal velocities and equal size with h/a 2 + a 1 = 1:5. Hence, the dimensionless frictional force begins to improve and also diminishes from the greatest significance for low hesitancy to the lowest significance for the elevated hesitancy at t = 0:5 and 1:0, and hence, the drag force reverses the impact at t = 4:0 versus the frequency, while it reduces with the growth of R H at t = 0:5 and 1:0 and improves at t = 0:1 and 4:0. Figure 4 exhibits an analysis of Re fF ð1Þ z /F ð1Þ ∞ g versus the frequency for different separation distance h/a 2 + a 1 = 1:05, 1:5, 4:0, ∞ at times t = 0:1, 0:5, 1:0, and 4:0 for a 2 /a 1 = 4:0 action in the identical direction with equal speeds and R H = 10:0. In addition, the nondimensional force increases with the expansion of the divergence space between the spheres from the lowest value at a lower frequency to the greatest value at an increased frequency. In addition, at t = 0:5, the real coefficient rises with the expansion of the separation distance and reverses its impacts at ðX, YÞ = ð13:2,8:7058Þ with the same impact at t = 1:0 at this point ðX, YÞ = ð8,3:63531Þ, while at t = 4:0, it reduces and then grows with the expansion of the gap space at ðX, YÞ = ð9:2,−4:8439Þ. Figure 5 reveals the unsteady imaginary coefficient Im fF ð1Þ z /F ð1Þ ∞ g versus the frequency for different separation space h/a 2 + a 1 = 1:05, 1:5, 4:0, ∞ at times t = 0:1, 0:5, 1:0, and 4:0 for a 2 /a 1 = 4:0 with an equal size and R H = 10:0. Moreover, the force coefficient reduces with the separation distance growth at t = 0:1, 0:5, and 1:0 but reverses its direction at t = 4:0. While, for the beginning three time values, it started from high at the low frequency to low at the high frequency and finally reversing its direction. Figure 6 presents an analysis of the real part Re fF ð1Þ z /F ð1Þ ∞ g against the frequency for various velocity ratios U 2 /U 1 = −2:0, −1:0, 1:0, and 2:0 at times t = 0:1, 0:5, 1:0, and 4:0 for a 2 /a 1 = 2:0, h/a 2 + a 1 = 1:5, and R H = 10:0. Therefore, the nondimensional force diminishes with the 10 Journal of Applied Mathematics improvement of the velocity ratios at t = 0:1 and begins from a minimum at the lower frequency to a maximum at the heightened frequency. In addition, the values of t = 0:5 and 1:0 decrease with the improvement of the ratio of the speeds and flip its impacts at the point ðX, YÞ = ð 13:6,8:18006Þ while it rises and then reduces from less to most at the time t = 4:0. Figure 7 represents the normalized imaginary force Im fF ð1Þ z /F ð1Þ ∞ g versus the frequency for distinct speeds U 2 /U 1 = −2:0, −1:0, 1:0, and 2:0 and t = 0:1, 0:5, 1:0, and 4:0 with h/a 2 + a 1 = 1:5 and R H = 10:0. Due to this, the force improves as the velocity ratio grows at t = 0:1, 0:5, and 1:0 and initiates from the most at the lower frequency to less at the heightened frequency. Therefore, at t = 4:0, it reduces with the expansion of the velocity's ratio from min to max. Figure 8 displays an investigation of Re fF ð1Þ z /F ð1Þ ∞ g versus the frequency for distinct size ratio at times t = 0:1, 0:5, 1:0, and 4:0 for the two rigid spheres move in the identical path with h/a 2 + a 1 = 1:5 and R H = 10:0. Also, the normalized force declines with the improvement of size ratio at t = 0:1, and for the time values t = 0:5 and 1:0, it reverses its effects at the point of reflection. However, it goes from a low value at a low frequency that is improving with respect to the size ratio and then goes to a high value at a high frequency where it is reducing as the ratio size improves at the point of inversion, ðX, YÞ = ð9:2,−4:75159Þ at the time value t = 4:0. Figure 9 displays an analysis of the frictional force Im f F ð1Þ z /F ð1Þ ∞ g versus the frequency for different size ratios a 2 / a 1 = 1:0, 2:0, 4:0, and 10:0 for the two hard spheres moving in the same direction with equal size and h/a 2 + a 1 = 1:5 and R H = 10:0. Moreover, the nondimensional drag force improves as the size ratio rises at t = 0:1, 0:5, and 1:0, starting from the highest value to the lowest value at the high frequency, but at t = 4:0, the size ratio declined with increasing frequency, starting from the low frequency and then increasing to the heightened frequency. The two coefficients of drag forces are calculated numerically in Tables 1 and 2, but  Table 3 represents the limiting case of the steady state and the comparison between this study and the case of the absence of a magnetic field in the work of Faltas and El-Sapa [7]. Finally, all the figures have the following behavior: the drag force coefficients are proportional to the frequency, and the Hartmann number distinguishes low and high frequency where they differ according to the time at high frequency.

Conclusion
The impact of the magnetic field on the interaction of two oscillating rigid spheres moving into an endless Stokes flow launching along the axis of symmetry is contemplated. We employed a semianalytical strategy and a collocation method to get the solution and hence calculate the nondimensional friction force coefficients. In the meantime, we determined the drag coefficients for different size proportions, speed proportions, separation distance, frequency parameters, and Hartmann numbers. Then again, it shows that in the case of low frequencies and the drag force, it starts at the most extreme and subsequently diminishes. Thus, it is more critical to notice the effects of the Hartman number that shows that the real part of the drag force diminishes by expanding the Hartmann number over a long time, though the imaginary part of the drag force increments by expand-ing the Hartmann number over a long time, and in the two cases, it begins from a base for low frequency and afterward comes to a maximum for high hesitation. Accordingly, the nondimensional drag force coefficients increment as the detachment distance develops at specific points in the frequency, and it reverses its impacts. Subsequently, it diminishes for the real force and increments for the coefficient of the imaginary drag force as the speed proportion rises. Faltas and El-Sapa [7] and Chen and Keh proposed great support and accuracy in the limiting cases. The future scope of this study may be in many fields of technology and in biomedicine, and magnetic oscillating particles have been used for years. In biomedicine, they are utilized in imaging, drug delivery, and magnetic hyperthermia (MH). Under the influence of alternating high-frequency magnetic fields, MH increases local temperatures in target cells by activating magnetic nanoparticles locally. The application of this study Journal of Applied Mathematics in the practical field may be presented as an experimental investigation of a sphere performing torsional oscillations in a Stokes flow. An experimental setup was developed that allowed the movement of the sphere to be remotely controlled by a magnetic field.

Data Availability
Data are available upon request to the corresponding author.

Conflicts of Interest
The authors declare that there are no conflicts of interest.

Authors' Contributions
Shreen El-Sapa contributed in the conceptualization, methodology, software provision, and data curation and wrote the original draft. Wedad Albalawi performed supervision, visualization, and investigation.