EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1566-1592 ISSN 1307-5543 – ejpam.com Published by New York Business Global The axisymmetric migration of an aerosol particle embedded in a Brinkmann medium of a couple stress fluid with slip regime Shreen El-Sapa1,2∗, Areej A. Almoneef2 1 Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt 2 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O.Box 84428, Riyadh 11671, Saudi Arabia Abstract. This study examined the relationship between stresses and couple stresses within the context of the couple stress fluid theory. The application of the effective medium approach, based on the Brinkman equation, is also discussed under the effect of slip and spin slip on the surface of the solid sphere immersed in porous couple stress fluid. The motion is generated by moving a solid sphere at a constant speed. Using Stokesian assumptions, the equation of motion ignores nonlinear terms. The surface conditions for slip and spin slip of couple stress fluids have been applied. Tables and graphs are used to represent the normalized drag force exerted by the fluid flow on the solid sphere. According to the numerical results, the drag force increases monotonically with permeability and couple stress. As a result, the results obtained are consistent with those found in the literature for viscous fluids, and the special cases are determined. 2020 Mathematics Subject Classifications: 76-XX, 76DXX Key Words and Phrases: Couple stress fluid, Slip and Spin slip, Permeability 1. Introduction Recent investigations pertaining to the theory of couple stress fluid manner are ex- tremely valuable, and such analyses provide sufficient insight into the behavior of rheolog- ical complex fluids, such as polymeric suspensions containing long-chain molecules, liquid crystals, lubricants, and human blood. According to couple stress fluid theory, particle sizes are considered when describing a non-Newtonian fluid. A theory of microcontinuum obtained by Stokes [1] is considered to take into account the particle size effects in order to capture the significance of couple stresses. As a generalization of the traditional theory of fluids, Stokes microcontinuum allows for polar impacts, such as the existence of couple ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4549 Email address: s elsapa82@sci.dmu.edu.eg (Shreen El-Sapa), aaalmoneef@pnu.edu.sa (Areej A. Almoneef) https://www.ejpam.com 1566 © 2022 EJPAM All rights reserved. Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1567 stresses, body couples, and antisymmetric stress tensors. For example, in pipe Poiseuille flow, even after all the variables have been nondimensionalized in the usual way, the ve- locity profile is a function of the pipe radius Stokes[2]. Thus, couple stresses may be important under lubrication conditions, where thin films are usually formed. Additionally, the theory of couple stress fluid flow introduces nonlin- ear terms in the relationship between shear stresses and velocity gradients. Accordingly, the lubricant should be considered non-Newtonian and described by two constants, shear viscosity, and couple stress. Therefore, Elsharkawy and Guedouar [3] developed a reverse solution for finite journal bearings lubricated with couple stress fluids to evaluate eccen- tricity proportion and couple stress parameters for an experimentally estimated pressure diffusion. In recent years, the interaction between two rigid spheres in an incompressible fluid flow has become increasingly important [4–8] and [9]. Furthermore, oscillatory flows allow one to observe and calculate multiple elastic effects of dilute polymers in a conve- nient manner. As a result of free stream oscillations, important physical problems arise, such as [10, 11]. Further, slip flow is more practical than no-slip flow (slide surface, flow through wet- ted surface, brick wall). It plays a significant role to illustrate the macroscopic influences in the study of fluid-solid surface interactions when fluid slips at a rigid wall. It is as- sumed that there is no slip in most investigations of peristaltic flow, however, wall slip is considered in a few types of research. Moreover, Saad [12] solved multiple problems such as the unstable plane Couette flow of an incompressible couple stress fluid with the slip conditions at both of the plates are assumed to be of different parameters and couple stresses at the boundaries are not present, the application of a time-dependent pressure gradient under the assumptions of the slip conditions and the zero couple stresses at the two boundaries, and the unsteady Poiseuille flow of an incompressible couple stress fluid inside an infinite cylinder. Moreover, Saad and Ashmawy [13] investigated the advective flow of an incompressible couple stress fluid between two parallel plates with slip boundary conditions put on the two plates and vanishing couple stress conditions at the interfaces. The fluid dynamical process that takes place in many body organs, including vascu- lar beds, lungs, kidneys, and tumorous vessels, must also be explained adequately, which makes study of porosity absolutely essential. The porosity of the media also plays a cru- cial role in many biomechanical investigations in determining the fluid transport. Parallel flows of fluids with various viscosities and densities through porous media are a common feature of technical procedures. Similar parallel flows can be seen in packed bed reactors used in the chemical industry, petroleum production engineering, sedimentation, diluted polymers, suspensions, journal bearing lubrication, and many other processes. To model flow across porous media, Brinkman [14] established a momentum equation with a New- tonian viscous drag term and a Darcy drag term, which together balance the pressure gradient. When the permeability of the medium approaches 0, the Brinkman equation becomes the Darcy equation, and when the permeability approaches infinity, it becomes Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1568 the Stokes equation. The equation has been widely used to assess porous media with high porosity, such as fiberglass wool. Sharma and Thakur [15] assumed a layer of electrically conducting couple stress fluid warmed from beneath in a permeable medium in the existence of the magnetic field. Al- saedi et al. [16] analyzed the peristaltic motion of a couple stress fluids occupying the porous medium through large wavelength and low Reynolds number. Opanuga et al. [17] studied the second law analysis of hydromagnetic couple stress fluid via a channel sup- plied with non Darcian permeable medium beneath the heat of the fluid exchanges with the ambient following Newtonian law. The thermal fluctuation in a layer of couple stress nanofluid saturated permeable medium was studied by Chand et al [18]. Yadav et al [19] introduced the collaborative impact of temperature-reliant viscosity and interior heat generation on the impression of convective movement in couple stress fluids saturated in a thin porous layer by utilizing the linear stability principles. There are multi applications on porous medium and magnetic field such as [20–28]. Thus, the objective of the current study is to examine the regular translational move- ment of a solid sphere immersed in an incompressible couple stress fluid with a slip regime and slip-spin while also getting the fundamental relationships analytically. The solution of the equation of momentum has been created analytically in terms of a spherical coordinate system. The assessed boundary conditions are then satisfied on the solid sphere surface. The drag force of the fluid flow on the rigid sphere is then calculated numerically and represented graphically for various values of the physical relevant parameters and also two special cases, Case I: perfect slip and spin-slip, and Case II: no-slip and no-spin slip. 2. Mathematical formulation of the problem Fig. 1 shows the axisymmetric migration of an aerosol particle embedded in an un- bounded incompressible Brinkmann medium of couple stress fluids under the impact of slip regime. An aerosol particle is a spherical particle of radius a moves along z axis with an uniform flow U . In the permeable region, the flow is governed by Brinkman’s equation. Also, under the assumptions of low Reynolds number, the inertial terms in the fluid mo- mentum are neglected.The Stokes microcontinuum theory [1] is considered the simplest theory of fluids, which allows the polar effects such as the presence of couple stresses, body couples and nonsymmetric tensors. The modified governing equations of momentum and the continuity of an incompressible fluid with couple stress in the presence of Brinkman’s medium are given by [6] and [24]: ∇ · q⃗ = 0, (1) ∇p+ µ∇∧∇ ∧ q⃗ + η∇∧∇ ∧∇ ∧∇ ∧ q⃗ + µ K q⃗ = 0. (2) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1569 Figure 1: The physical representation of the problem. where q⃗ is the volume averaged velocity parallel to the wall,p is the pore average pressure, µ represents the viscosity of the fluid, η is the first couple stress viscosity coefficient, and η ′ is the second couple stress viscosity coefficient. If the couple stress coefficient η is taken zero, then the equation of motion (2) reduces to the classical Navier–Stokes’s equation. In addition, K is the Darcy permeability of the permeable medium and also is a scalar for isotropic porous medium. Otherwise, K a second order tensor [25] and the Brinkman equation of couple stress fluid reduces to the Darcy equation when K → 0 and to the Stokes equation of couple stress fluid when K → ∞. The material constants are [17]: µ ≥ 0, η ≥ 0 η − η ′ ≥ 0. (3) The components of velocity field are of the form: q⃗ = −∇ ∧ ( ψe⃗ϕ r sin θ ) . (4) The constitutive equations for the couple stress fluids in dyadic form are: (i) The stress tensor, T and the strain tensor,∆: T = −pI+ 2µE + 1 2 I ∧∇ . M, (5) E = 1 2 [ ∇q⃗ + (∇q⃗)T ] . (6) (ii) The couple stress tensor, M M = mI + 4η∇ω⃗ + 4η ′ (∇ω⃗)T , (7) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1570 where I is the unit dyadic, (.)T denotes for transpose, and ω is the vorticity vector which it is defined in terms of the stream function ψ by: ω⃗ = 1 2∇∧ q⃗ = 1 2 1 r sin θ E2ψ e⃗ϕ. (8) 3. Axisymmetric expressions of the stresses and couple stresses for the couple stress fluids theory The strain tensor in terms of the velocity components are obtained by the following from, Appendix (A.1): ω⃗ =1 2Err = ∂qr ∂r , Erθ = Eθr = 1 2 [1 r ∂qr ∂θ − qθ r + ∂qθ ∂r ] , Eθθ = [1 r ∂qr ∂θ + qθ r ] , Eθϕ = Eϕθ = 0, Erϕ =Eϕr = 0, Eϕϕ = [qr r + cot θqθ r + qθ r ] .  (9) The couple stresses in terms of the vorticity are calculated by from Appendix (A.2): Mrr =m, Mrθ, Mrϕ = 4η ∂ωϕ ∂r − 4η ′ ωϕ r , Mθr =0, Mθθ = m, Mθϕ = 4η 1 r ∂ωϕ ∂θ − 4η ′ 1 r cot θ ωϕ, Mϕr =− 4η 1 r ωϕ + 4η ′ ∂ωϕ ∂r , Mϕθ = −4η 1 r cot θωϕ + 4η ′ 1 r ∂ωϕ ∂θ , Mϕϕ = m,  (10) where m = 1 3 trM = 1 3 (Mrr + Mθθ + Mϕϕ) ,is the trace of the couple stress and for any axisymmetric equation you may take, m = 0. From equations (9), we get the following term: ∇.M = ẽϕ [ 1 r2 ∂ ∂r (r2Mrϕ) + 1 r sin θ ∂ ∂θ (sin θ Mθϕ) + Mϕr r + cot θ r Mϕθ ] And so, I ∧∇.M = (−ẽr ẽθ + ẽθẽr ) [∂Mrϕ ∂r + 2Mrϕ +Mϕr r + cot θ r (Mθϕ +Mϕθ) + 1 r ∂Mθϕ ∂θ ] (11) Therefore, the stresses are determined from equation (4): Trr =− p+ 2µErr, Trϕ, Tθθ = −p+ 2µEθθ, Tθϕ =0, Tθr = m, Tϕθ = 0, Tϕϕ = −p+ 2µEϕϕ, Trθ =2µErθ − 1 2 [∂Mrϕ ∂r + 2Mrϕ +Mϕr r + cot θ r (Mθϕ +Mϕθ) + 1 r ∂Mθϕ ∂θ ] , Tθr =2µEθr + 1 2 [∂Mrϕ ∂r + 2Mrϕ +Mϕr r + cot θ r (Mθϕ +Mϕθ) + 1 r ∂Mθϕ ∂θ ] .  (12) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1571 For dimensionless governing equations, we used the following nondimensional variables: ˆ⃗q = q⃗ U , p̂ = ap µU , ∇̂ = a∇, T̂rr = aTrr µU , ω̂ϕ = aωϕ U . (13) Substituting equation (12) in equation (2) and then dropping the tildes, we get: ∇p+∇∧∇ ∧ q⃗ + 1 ζ2 ∇∧∇ ∧∇ ∧∇ ∧ q⃗ +K2q⃗ = 0. (14) where k= a2 K , ζ 2 = µa2 η are the permeability parameter and the length dependent parameter of the first couple stress fluid coefficient, respectively. If ζ → ∞ equation (13) represents the modified Stokes equation for nonpolar fluid. 4. Differential equation satisfied by the stream function,ψ Further, the fluid movement is axisymmetric, so all the field functions are independent of ϕ and the spherical polar coordinate system is (r, θ, ϕ) where the origin located at the center of the sphere. Since the velocity vector q⃗ and the vorticity vector ω⃗ are assumed as: q⃗ = qr(r, θ)e⃗r + qθ(r, θ)e⃗θ, ω⃗ = ωϕ(r, θ)e⃗ϕ. (15) From equation (1), the velocity components can be represented in terms of the stream function ψ from equation (3) as the following: qr = − 1 r2θ ∂ψ ∂θ , qθ = 1 rθ ∂ψ ∂r . (16) From equations (8)-(9) and (11) we obtained the following non-dimensional stresses: Trr = −p+ 2 ∂qr ∂r , (17) Trθ = [1 r − qθ r + ∂qθ ∂r ] − 1 2ζ2 [∂2ωϕ ∂r2 + 2 r ∂ωϕ ∂r − 2ωϕ r2 ] , (18) Mrϕ = 4 ζ2 ∂ωϕ ∂r − 4 ζ ′2 ωphi r , (19) where, ζ ′2 = µa2 η′ is the length dependent parameter on the second couple stress fluid coefficient. Thus, the governing equations have the form: Mrϕ = 4 ζ2 ∂ωϕ ∂r − 4 ζ ′2 ωphi r , (20) where, ζ ′2 = a2µ η′ is the length dependent parameter on the second couple stress fluid coefficient. Thus, the governing equations have the form: ζ−2 r2 sin θ ∂ ∂θ L−1 ( L−1 − ζ2 ) ψ + κ2 r2 sin θ ∂ψ ∂θ − ∂p ∂r = 0, (21) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1572 −ζ−2 r sin θ ∂ ∂r L−1 ( L−1 − ζ2 ) ψ − κ2 r sin θ ∂ψ ∂r − 1 r ∂p ∂θ = 0. (22) One can easily eliminate the pressure from equations (21) and (22) to get the following partial differential equation: L−1 ( L−1 − α2 1 )( L−1 − α2 2 ) ψ = 0, (23) where the axisymmetric Stokesian operator: L−1 = ∂2 ∂r2 + 1−ξ2 r2 ∂2 ∂ξ2 , ξ = cos θ, α2 1α 2 2 = ζ2κ2, α2 1 + α2 2 = ζ2, α2 1 = ζ2 + ζ √ ζ2 − 4κ2 2 , α2 2 = ζ2 − ζ √ ζ2 − 4κ2 2 .  (24) 5. The solution of the Problem The regular solution of the sixth order linear partial differential equation (23) is ob- tained by using the method of separation of variables is given: ψ = 1 2 ( Ar−1 +Br 1 2K3 2 (α1r) + Cr 1 2K3 2 (α2r) ) sin2 θ, (25) where the function K3 2 (.) is the modified Bessel functions of the second kind of order 3 2 where A,B,C are constants can be determined from the boundary conditions. Substitut- ing equation (25) into equation (16), we get the velocity components: qr = ( Ar−3 +Br −3 2 K3 2 (α1r) + Cr −3 2 K3 2 (α2r) ) cos θ, (26) qθ = 1 2 ( −Ar−3−Br −3 2 [ K3 2 (α1r) + α1rK1 2 (α1r) ] −Cr −3 2 [ K3 2 (α2r) + α2rK1 2 (α2r) ]) sin θ, (27) Inserting the two expressions (26) and (27) into the relation (8) , we get: ωϕ = 1 4 ( Bα2 1r −1 2 K3 2 (α1r) + Cα2 2r −1 2 K3 2 (α2r) ) sin θ. (28) The expression for pressure is: p = −−1 2r2 κ2A cos θ. (29) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1573 6. Boundary conditions On the surface of the aerosol particle, r = a, we require three boundary conditions to find the unknowns A,B, and C. Recently, some researchers have studied the slip effect on the solid spheres (9),(26)-(28). (i) The kinematic impenetrability boundary condition is: qr = U cos θ, (30) (ii) The dynamic slip boundary condition where the tangential velocity of the couple stress fluid relative to the solid at a point on its surface is proportional to the tangential stress prevailing at that point. The latter is known as the slip boundary condition introduced by Navier in 1823: qθ = −U sin θ + 1 β1 Trθ, (31) (iii) The spin slip couple stress condition is given as: ωϕ = 1 β2 Mrϕ, (32) where β1 is the velocity slip coefficient and β2 is the spin slip coefficient. Therefore, • The slips coefficients depend on the nature of the solid and fluid surface. • In the limiting case of β1 → 0, there is a perfect slip and the solid sphere acts like a spherical gas bubble if β1 → ∞, we get the standard no-slip condition. • Also, for β2 → 0 we have the couple stress zero at the boundary which means that the mechanical interactions at the surface are equipollent to a force distribution only. On the other hand, when β2 → ∞ this is equivalent to prevent relative rotation of the fluid element at the surface of the aerosol. The expressions for stress and couple stress components from Appendix (A.4) are : Trr = ( 1 2r −4(κ2r2+12)A+ 2Br −5 2 [ 3K3 2 (α1r) + α1rK1 2 (α1r) ] +2Cr −5 2 [ 3K3 2 (α2r) + α2rK1 2 (α2r) ]) cos θ, (33) Trθ = ( 3r−4A+1 2Br −5 2 [ (α2 1r 2 + 6− α4 1ζ −2r2)K3 2 (α1r) + 2α1rK1 2 (α1r) ] +1 2Cr −5 2 [ (α2 2r 2 + 6− α4 2ζ −2r2)K3 2 (α2r) + 2α2rK1 2 (α2r) ]) sin θ, (34) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1574 Therefore, relation (20) gives: Mrϕ =− ( Bα2 1r −3 2 [ (2ζ−2 + ζ− ′2)K3 2 (α1r) + α1rζ −2K1 2 (α1r) ] +Cα2 2r −5 2 [ (2ζ−2 + ζ− ′2)K3 2 (α2r) + α2rζ −2K1 2 (α2r) ]) sin θ. (35) 7. Drag force The formula of drag force operating on the axisymmetric rigid particle along the axis of symmetry has been shown by h3: Fz = 2πa2 ∫ π 0 ( Trr cos θ − Trθ sin θ ) r=a sin θdθ (36) Inserting the expressions (33) and (34) into (36), we arrive at the following: Fz πa2 = 2 3 ( κ2a−2A+2Ba −5 2 [ α2 1a 2ζ−2(α2 1 − ζ2)K3 2 (α1r) ] +2Ca −5 2 [ α2 2a 2ζ−2(α2 2 − ζ2)K3 2 (α2r) ) (37) By applying the boundary conditions (30)-(32) , we get the unknown constant A , to obtain the following system of equations: Aa −3 2 +BK3 2 (α1a) + CK3 2 (α2a) = −Ua 3 2 , (38) A(1 + 6β−1 1 a−1) +Ba 3 2 [( 1 + β−1 1 a−1(α2 1a 2 + 6− α4 1ζ −2a2) ) K3 2 (α1a) + α1a ( 1 + 2β−1 1 a−1 ) K1 2 (α1a) ] +Ca 3 2 [( 1 + β−1 1 a−1(α2 2a 2 + 6− α4 2ζ −2a2) ) K3 2 (α2a) + α2a ( 1 + 2β−1 1 a−1 ) K1 2 (α2a) ] =2Ua3, (39) Bα2 1 [( 1 + 4β−1 2 a−1(2ζ−2 + ζ ′2) ) K3 2 (α1a) + α1ζ −2β−1 2 K1 2 (α1a) ] +Cα2 2 [( 1 + 4β−1 2 a−1(2ζ−2 + ζ ′2) ) K3 2 (α2a) + α2ζ −2β−1 2 K1 2 (α2a) ] = 0, (40) where, A = −Ua3 −Ba 3 2K3 2 (α1a)− Ca 3 2K3 2 (α2a), B = 3Ua3α2 2∆ −1(1+2β−1 1 a−−1) [( 1+4β−1 2 a−1(2ζ−2+ζ ′2) ) K3 2 (α2a)+α2ζ −2β−1 2 K1 2 (α2a) ] , Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1575 C = −3Ua3α2 1∆ −1(1+2β−1 1 a−1)[ ( 1+4β−1 2 a−1(2ζ−2+ζ ′2) ) K3 2 (α1a)+α1ζ −2β−1 2 K1 2 (α1a) ] , ∆ = β−1 1 ζ−2a 1 2 (α2 1α 4 2 − α2 2α 4 1) ( 1 + 4β−1 2 a−1(2ζ−2 + ζ ′2) ) K3 2 (α1a)K3 2 (α2a) +a2(1+2β−1 1 a−1) ( 1+4β−1 2 a−1(2ζ−2+ζ ′2) )( α1α 2 2K3 2 (α2a)K1 2 (α1a)−α2α 2 1K3 2 (α1a)K1 2 (α2a) ) +a 1 2β−1 1 ζ−2β−1 2 ( α3 2(α 2 1a 2−α4 1ζ −2a2)K1 2 (α2a)K3 2 (α1a)−α3 1(α 2 2a 2−α4 2ζ −2a2)K1 2 (α1a)K3 2 (α2a) ) a2ζ−2β−1 2 (α1α 3 2 − α3 1α2)(1 + 2β−1 1 a−1)K1 2 (α1a)K1 2 (α2a). Solving the system in the equations (38)-(40) algebraically, we get the value of the desired coefficient A needed in the drag expression (37). Fz 2 3πa 2 =− κ2Ua+ 3Uα2 2∆ −1a 3 2 (1 + 2β−1 1 a−1)(2α2 1(α 2 1ζ −2 − 1)− κ2) × (( 1 + 4β−1 2 a−1(2ζ−2 + ζ ′2) ) K3 2 (α1a)K3 2 (α2a) + α2ζ −2β−1 2 K3 2 (α1a)K1 2 (α2a) ) −3Uα2 1∆ −1a 3 2 (1 + 2β−1 1 a−1)(2α2 2(α 2 2ζ −2 − 1)− κ2) × (( 1 + 4β−1 2 a−1(2ζ−2 + ζ ′2) ) K3 2 (α1a)K3 2 (α2a) + α1ζ −2β−1 2 K3 2 (α2a)K1 2 (α1a) ) (41) The non-dimensional drag force of a rigid sphere is taken with respect to the drag force of a sphere translates through unbounded clear fluid which is given by F0 = −µπUa. 8. hpecial cases 8.1. No-slip and no-spin, β1 = β2 → ∞ The drag force acting on the rigid sphere moves through unbounded porous medium under the conditions,qr = U cos θ, qθ = −U sin θ, ωϕ = 0, we obtained the following: Aa−3 +Ba −3 2 K3 2 (α1a) + Ca −3 2 K3 2 (α2a) = −U, (42) Aa−3 +Ba −3 2 [ K3 2 (α1a) + α1aK1 2 (α1a) ] + Ca −3 2 [ K3 2 (α2a) + α2aK1 2 (α2a) ] = 2U, (43) Bα2 1a −3 2 K3 2 (α1a) + Cα2 2a −3 2 K3 2 (α2a) = 0. (44) The values of the constants A,B,C are given by: A = − ( a3 + 3aα−1 1 α−1 2 (α1a+ α2a+ 2) + 3aκ−2 + 3ζ−2κ−2(α1 + α2) ) , B = −3 expα1aU √ 2 πα1 (1+α2a) (α1−α2) , C = −3 expα2aU √ 2 πα2 (1+α1a) (α1−α2) . Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1576 The drag force in this case is obtained as: Fz F0 =− 2 3ζ −′2(α1 + α2)(1 + α1a)(1 + α2a) −1 9ζ −2 ( α2 1α 2 2a 3 + 3aα1α2(α1a+ α2a+ 2) + 3aζ2 + 3α1 + 3α2 ) (45) 8.2. Perfect slip and perfect spin,β1 = β2 = 0 The drag force acting on the rigid sphere moves through unbounded porous medium under the conditions,qr = U cos θ, Trθ = 0, MRϕ = 0, we obtained the following: Aa−3 +Ba −3 2 K3 2 (α1a) + Ca −3 2 K3 2 (α2a) = −U, (46) 3Aa−3+1 2Ba −3 2 [ (α2 1a 2 + 6− α4 1ζ −2a2)K3 2 (α1a) + 2α1aK1 2 (α1a) ] +1 2Ca −3 2 [ (α2 2a 2 + 6− α4 2ζ −2a2)K3 2 (α2a) + α2aK1 2 (α2a) ] = 0, (47) Bα2 1a −3 2 [ (2ζ−2 + ζ− ′2)K3 2 (α1a) + α1aζ −2K1 2 (α1a) ] +Cα2 2a −3 2 [ (2ζ−2 + ζ− ′2)K3 2 (α2a) + α2aζ −2K1 2 (α2a) ] = 0. (48) The values of the constants A,B,C in this case are given by: A = −Ua3 + 10Ua∆−1ζ2α2 2(α1a+ 1)(α2a+ 1)(ζ2 + (α2a+ 2)ζ ′2) − 10Ua∆−1ζ2α2 1(α1a+ 1)(ζ2(α1a+ 1) + (α2a 2 + 2α1a+ 2)ζ ′2), B = −10Ua∆−1 expα1a √ 2 πα1 ζ2α2 1α 2 2(α2a+ 1)(ζ2 + (α2a+ 2)ζ ′2), C = 10Ua∆−1 expα2a √ 2 πα2 ζ2α2 1α 2 2(ζ 2(α1a+ 1) + (α2 1a 2 + 2α1a+ 2)ζ ′2), where, ∆ = α2 1α 2 2 ( (α1a− 2α2a− 1)ζ4 + [ (α2a+ 2)(α2a 2 − α1α2a 2 + α1a− 3α2a− 1)ζ ′2 + (α2 1 − α2 2)(α1a+ 1)(α2a+ 1) ] ζ2 + (α2a+ 1) [ α3 1α2a 2 − α2 1α 2 2a 2 + 2α3 1a+ α2 1α2a − 2α1α 2 2a+ 2α2 1 − 2α2 ]) . The drag force is obtained as: Fz πUa2 =2 3∆ −1 ( κ2a−1 [ − a2∆+ 10ζ2α2 2(α1a+ 1)(α2a+ 1)(ζ2 + (α2a+ 2)ζ ′2) −10ζ2α2 1a(α1a+ 1)(ζ2(α1a+ 1) + (ζ2a2 + 2α2a+ 2)ζ ′2) ] −20α2 1α 2 2a 2(α2 1 − ζ2)(α1a+ 1)(α2a+ 1)(ζ2 + (α2a+ 2)ζ ′2) +20α2 1α 2 2a 2(α2 2 − ζ2)(α2a+ 1)(ζ2(α1a+ 1) + (α2a 2 + 2α1a+ 2)ζ ′2) ) . (49) Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1577 9. Results and discussions In the present section, we aim to represent numerically the normalized drag force Fz/F0 acting on a solid sphere and distributions of velocity components for different values of the physical parameters appearing in the governing equations in Figs 2-11 and Tables 1-3. All the results of drag force had been calculated for real part and MATLAB pro- gram neglect the imaginary part during the calculations. Fig. 2 shows the normalized drag force for different values of the couple stress parameter ζ = 0.01, 1.0, 10.0,∞ with ζ ′ and β1 aµ = aβ2 µ = 0.001, 1.0, 10.0,∞ which increases with the increase of the perme- ability parameter starts from low at κ → 0 for Stokes flow but at κ → ∞ gives Darcy flow. On the other hand, the normalized drag force decreases with the increase of the couple stress parameter. Fig. 3 presents the normalized drag force for different values β1 aµ = aβ2 µ = 0.0, 1.0, 10.0,∞ at the various values of the couple stress parameter ζ at ζ ′ which increases with the increase of the slip parameters. Table 1 represents the effects of slip and spin slip parameters with different values of the couple stress parameter. Fig. 4-5 exposes the velocity distribution for partial slip and partial spin slip for various values of the permeability and ζ = 4.0, ζ ′ = 0.01, β1 aµ = aβ2 µ = 2.0 the normal and tangential veloci- ties reduce with both the increase of the sphere radius and the increase of the permeability parameter. Figs. 6-8 and Table 2 show the special cases of no-slip and no-spin slip where the normalized drag force begins slowly at Stokes flow and increases gradually to reach high values for Darcy flow. In addition, the normal velocity decreases with the increase of radius r and reduces with the increase of the permeability while the tangential velocity reverse its improves the improve of the permeability parameter and near to r = 7.0 it behaves like the normal velocity. Figs. 9 and Table 3 illustrate the special cases of perfect slip and perfect spin slip for the normalized drag force gives the same effect for the above cases. Figs 10-11 explain the distributions of the streamlines for ζ = 10.0, ζ ′ = 0.1 for the three cases for the slip conditions with κ = 0.0, 1.0, 3.0, 4.0, 6.0, 10.0 and the couple stress parameter with the values of ζ ′ = 0.1, β1 aµ = aβ2 µ = 10.0 . It is noticed that the streamlines move uniformly at κ = 0.0 which agrees with the results of viscous fluid in the literature but for the increasing of the permeability, it appears as a loop around the sphere until reaching the Darcy flow the molecules are crowded. Moreover, it also clarifies in the case of the increase of the couple stress parameter with the rising in the values of κ. Shreen El-Sapa / Eur. J. Pure Appl. Math, 15 (4) (2022), 1566-1592 1578 Table 1: Normalized drag force for slip and spin slip with ζ ′ = 0.01. Fz/F0 β1 aµ = aβ2 µ κ ζ = 0.1 ζ = 0.1 ζ = 6.0 ζ = 10.0 ζ → ∞ 0.0 0.0 1.962039 1.000000 0.736977 0.709803 0.635865 0.2 4.444950 2.980156 1.826095 1.765162 1.646248 4.0 5.799853 4.888037 3.538837 3.348328 3.208901 6.0 8.037672 7.369523 6.134427 5.787377 5.558577 8.0 11.162395 10.626770 9.523765 9.147747 8.750814 10.0 15.174927 14.720016 13.735482 13.353335 12.806985 1.0 0.0 2.913268 1.199978 0.84000 0.804879 0.715290 2.0 6.439568 3.954626 2.238630 2.137213 1.946697 4.0 7.806085 6.226607 4.176816 3.904061 3.656288 6.0 10.052062 8.890593 6.973032 6.481510 6.104393 8.0 13.183798 12.255479 10.521991 9.976724 9.366142 10.0 17.202734 16.419035 14.860377 14.297513 13.473763 10.0 0.0 10.723054 1.714241 1.063293 1.007636 0.880511 2.0 24.330440 8.748913 3.697992 3.379155 2.850210 4.0 25.826824 14.363925 7.041395 6.316924 5.319628 6.0 28.157423 19.195222 11.255353 9.956683 8.440525 8.0 31.358927 24.030911 16.123972 14.486632 12.280256 10.0 35.440013 29.241834 21.640403 19.783658 16.882901 ∞ 0.0 50.690467 1.961480 1.153477 1.088268 0.944690 2.0 195.779204 14.324858 4.721100 4.186681 3.367234 4.0 200.897054 29.018703 9.890541 8.590854 6.566838 6.0 205.118892 43.957608 16.588341 14.128548 10.593281 8.0 209.642450 58.890966 24.546021 20.842851 15.449547 10.0 214.770348 73.744811 33.603516 28.680957 21.138434 Table 2: Normalized drag force for no-slip and no-spin slip with ζ ′ = 0.01. Fz/F0 κ ζ = 0.1 ζ = 0.1 ζ = 6.0 ζ = 10.0 ζ → ∞ 0.0 11.0 2.0 1.17 1.11 1.00 0.2 216.47 15.69 4.90 4.33 3.45 4.0 374.044 34.031 10.475 9.078 6.803 6.0 526.757 55.061 17.9566 15, 172 11.049 8.0 679.436 78.512 27.0281 22.626 16.192 10.0 833.66443 104.23 37.6196 31.4652 22.2316 100.0 11841.04 34461.03 20884.00 19090.00 14672.00 REFERENCES 1579 Table 3: Normalized drag force for perfect slip and perfect spin slip with ζ ′ = 0.01. Fz/F0 κ ζ = 0.1 ζ = 0.1 ζ = 6.0 ζ = 10.0 ζ → ∞ 0.0 31.00 18.85 13.89 13.38 11.98 0.2 80.121 56.155 34.412 33.26 31.024 4.0 107.00 92.110 66.692 63.103 60.476 6.0 149.556 138.882 115.614 109.075 104.765 8.0 208.563 200.278 179.498 172.414 164.936 10.0 284.198 277.433 258.000 251.6856 241.392 100.0 21020.3 21018.7 21016.0 21014.1 20980.9 10. Conclusion The present work investigated the basic relations of stresses and couple stresses also the solution of the problem for movements of a rigid sphere with a constant velocity embedded through an incompressible porous medium saturated with the couple stress fluid under the effects of slip and spin slip by using an analytical procedure. The normalized drag force represents numerically and graphically. On the other side, the velocity distributions are represented graphically. It is noticed that the normalized drag force increases with the effects of the slip and spin slip and improves with the increase of the permeability param- eter at constant values of the remaining parameters. In addition, there are improvements of the streamline’s distributions for the relevant parameters of the permeability and the couple stress parameters. References [1] V K Stokes. Effects of couple stresses in fluids on hydromagnetic channel flows. The Physics of Fluids, 9:1709–1715, 1966. 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Appendix Expressions of stresses and couple stresses for the couple stress fluids theory. (A.1): The strain tensor components are obtained by from equation (5) Err = ∂qr ∂r , APPENDIX 1582 Erθ = Eθr = 1 2 [1 r ∂qr ∂θ − qθ r + ∂qθ ∂r ] , Eθθ = [1 r ∂qθ ∂θ + qr r ] , Eθϕ = Eϕθ = 1 2 [sin θ r ∂ ∂θ ( qϕ sin θ ) + 1 r sin θ ∂qϕ ∂ϕ ] , Erϕ = Eϕr = 1 2 [ 1 r sin θ ∂qr ∂ϕ + r ∂ ∂r (qϕ r )] = 1 2 [ 1 r sin θ ∂qr ∂ϕ + ∂qϕ ∂r − qϕ r ] , Eϕϕ = [ 1 r sin θ ∂qϕ ∂ϕ + ∂qr ∂r + qθ cot θ r ] . (A.2): The couple stresses are calculated by from equation (6) Mrr = m+ 4(η + η′) ∂ωr ∂r , Mrθ = 4 [ η ∂ωθ ∂r + η′ 1 r (∂ωr ∂θ − ωθ )] , Mrϕ = 4 [ η ∂ωϕ ∂r + η′ 1 r ( 1 sin θ ∂ωr ∂ϕ − ωϕ )] , Mθr = 4 [ η 1 r (∂ωr ∂θ − ωθ ) + η′ ∂ωθ ∂r ] , Mθθ = m+ 4(η + η′) 1 r (∂ωθ ∂θ + ωr ) , Mθϕ = 4 [ η 1 r ∂ωϕ ∂θ + η′ 1 r ( 1 sin θ ∂ωθ ∂ϕ − cot θωϕ )] , Mϕr = 4 [ η 1 r ( 1 sin θ ∂ωθ ∂ϕ − cot θωϕ ) + η′ 1 r ∂ωϕ ∂θ ] , Mϕϕ = m+ 4(η + η′) 1 r [ 1 sin θ ∂ωϕ ∂ϕ + ωr + cot θωθ ] . (A.3): Used relations in the analysis I×∇.M = (−e⃗θe⃗ϕ + e⃗ϕe⃗θ) [∂Mrr ∂r + 2 r Mrr + cot θ r Mθr + 1 r ∂Mθr ∂θ + 1 r sin θ ∂Mϕr ∂ϕ − Mθθr r − Mϕϕ r ] , = (e⃗re⃗ϕ − e⃗ϕe⃗r) [∂Mrθ ∂r + 2 r Mrθ + cot θ r Mθθ + 1 r ∂Mθθ ∂θ + 1 r sin θ ∂Mϕθ ∂ϕ + Mθr r + cot θ r Mϕϕ ] , + (−e⃗re⃗θ + e⃗θe⃗r) [∂Mrϕ ∂r + 2 r Mrϕ + cot θ r Mθϕ + 1 r ∂Mθϕ ∂θ + 1 r sin θ ∂Mϕϕ ∂ϕ + Mϕr r + cot θ r Mϕθ ] . (A.4): The stresses are Trr = −P + 2µErr, (50) Trθ = 2µErθ − 1 2 [∂Mrϕ ∂r + 2 r Mrϕ + cot θ r Mθϕ + 1 r ∂Mθϕ ∂θ + 1 r sin θ ∂Mϕϕ ∂ϕ + Mϕr r + cot θ r Mϕθ ] , (51) APPENDIX 1583 Trϕ = 2µErϕ + 1 2 [∂Mrθ ∂r + 2 r Mrθ + cot θ r Mθθ + 1 r ∂Mθθ ∂θ + 1 r sin θ ∂Mϕθ ∂ϕ + Mθr r + cot θ r Mϕϕ ] , (52) Tθr = 2µEθr + 1 2 [∂Mrϕ ∂r + 2 r Mrϕ + cot θ r Mθϕ + 1 r ∂Mθϕ ∂θ + 1 r sin θ ∂Mϕϕ ∂ϕ + Mϕr r + cot θ r Mϕθ ] , (53) Tθθ = −p+ 2µEθθ, (54) Tθϕ = 2µEθϕ − 1 2 [∂Mrr ∂r + 2 r Mrr + cot θ r Mθr + 1 r ∂Mθr ∂θ + 1 r sin θ ∂Mϕr ∂ϕ − Mθθr r − Mϕϕ r ] , (55) Tϕr = 2µEθr − 1 2 [∂Mrθ ∂r + 2 r Mrθ + cot θ r Mθθ + 1 r ∂Mθθ ∂θ + 1 r sin θ ∂Mϕθ ∂ϕ + Mθr r + cot θ r Mϕϕ ] , (56) Tϕθ = 2µEθθ + 1 2 [∂Mrr ∂r + 2 r Mrr + cot θ r Mθr + 1 r ∂Mθr ∂θ + 1 r sin θ ∂Mϕr ∂ϕ − Mθθr r − Mϕϕ r ] , (57) Tθθ = −p+ 2µEϕϕ. (58) APPENDIX 1584 (a) β1 aµ = aβ2 µ = 0.001 (b) β1 aµ = aβ2 µ = 1.0 (c) β1 aµ = aβ2 µ = 10.0 (d) β1 aµ = aβ2 µ → ∞ Figure 2: Normalized drag force exerted on a rigid sphere of a unit radius for various values of the slip and spin slip, permeability, and the couple stress parameter with ζ ′ = 0.01 . APPENDIX 1585 (a) β1 aµ = aβ2 µ = 0.001 (b) β1 aµ = aβ2 µ = 1.0 (c) β1 aµ = aβ2 µ = 10.0 (d) β1 aµ = aβ2 µ → ∞ Figure 3: Normalized drag force exerted on a rigid sphere of a unit radius for various values of the spin slip, permeability, and the couple stress parameter with ζ ′ = 0.1 . APPENDIX 1586 Figure 4: Normal velocity distribution for partial slip and partial spin slip for various values of the permeability and ζ = 3.0, ζ ′ = 0.01, β1 aµ = aβ2 µ = 2.0 Figure 5: Tangential velocity distribution for partial slip and partial spin slip for various values of the permeability and ζ = 3.0, ζ ′ = 0.01, β1 aµ = aβ2 µ = 2.0 APPENDIX 1587 Figure 6: Normalized drag force exerted on a rigid sphere of a unit radius for no-slip and no-spin slip for various values of the permeability and the couple stress parameter. Figure 7: Normal velocity distribution for no-slip and no-spin slip for various values of the permeability and the couple stress parameter, ζ = 1.0. APPENDIX 1588 Figure 8: Tangential velocity distribution of a rigid sphere of a unit radius for no-slip and no-spin slip for various values of the couple stress parameter and the permeability, κ = 2.0. Figure 9: Normalized drag force exerted on a rigid sphere of a unit radius for perfect slip and perfect spin slip for various values of the permeability, the couple stress parameter and ζ ′ = 0.01. APPENDIX 1589 (a) β1 aµ = aβ2 µ = 0.0. APPENDIX 1590 (b) β1 aµ = aβ2 µ = 1.0. APPENDIX 1591 (c) β1 aµ = aβ2 µ → ∞. Figure 10: htreamlines distribution for various parameters ζ = 10.0, ζ ′ = 0.1. APPENDIX 1592 (a) ζ = 0.001 (b) ζ = 0.1 (c)ζ = 0.1 (d)ζ = 10.0 Figure 11: htreamlines distribution for various parameters β1 aµ = aβ2 µ = 10.0, ζ ′ = 0.1.