The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation

  Citation: Alam, M.K.; Bibi, K.; Khan, A.; Fernandez-Gamiz, U.; Noeiaghdam, S. The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation. Energies 2022,15, 2473. https://doi.org/10.3390/en15072473 Academic Editor: Dmitry Eskin Received: 3 February 2022 Accepted: 22 March 2022 Published: 28 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation Muhammad Kamran Alam 1,*, Khadija Bibi 1, Aamir Khan 1, Unai Fernandez-Gamiz 2and Samad Noeiaghdam 3,4 1Department of Mathematics & Statistics, The University of Haripur, Haripur 22620, Pakistan; [email protected] (K.B.); aamir[email protected] (A.K.) 2 Nuclear Engineering and Fluid Mechanics Department, University of the Basque Country UPV/EHU, Nieves Cano 12, 01006 Vitoria-Gasteiz, Spain; [email protected] 3Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia; [email protected] or [email protected] 4Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia *Correspondence: [email protected] Abstract: In this paper, the 3-D squeezing flow of viscous incompressible fluid between two parallel plates rotating at the same rate is investigated. The flow is observed under the influence of the varying magnetic field. The flow phenomena are modeled by utilizing the basic governing equations, i.e., equation of continuity, coupled Navier Stokes, and Magnetic Field equations. Using appropriate similarity transformations, the resultant partial differential equations are then transformed into a system of ordinary differential equations. The computational technique is developed via the Homotopy Analysis Method (HAM) to obtain the solution of transformed systems of ordinary differential equations. The influence of several engineering fluid parameters, such as squeeze Reynolds number, magnetic field strength parameter, and magnetic Reynolds number, on velocity and magnetic field components, are observed from different graphs. It has been investigated that by increasing the squeeze Reynolds number, fluid velocity in the yand zdirections will be increased as well. On the magnetic field component along the y-axis, an increasing influence of squeezing Reynolds number is also noticed. Similarly, raising the magnetic Reynolds number increases the velocity along the y-axis, whereas the inverse relationship is found for magnetic field components. Furthermore, for each flow phenomenon, an error analysis is also presented. Keywords: MHD; Homotopy Analysis Method; heat and mass transfer; time dependent squeeze phenomenon; variable magnetic filed; viscous fluid 1. Introduction When a fluid is squeezed between two parallel plates approaching one other, it is called a squeeze flow. The unsteady squeezing flow between two plates rotating at different angular velocities is regarded as one of the most important study subjects due to its extensive applications in science and technology. Among these are hydrodynamic lubrication, polymer technology, biomechanics, the petroleum sector, and aerodynamic heating. The interaction of conducting fluids with electromagnetic fields is widely known as MagentoHydro Dynamics (MHD). The use of an MHD fluid as a lubricant in industrial applications is appealing because it prevents the unanticipated variation of lubricant viscosity with temperature under such high working conditions. Many experts are showing interest in this field; for example, the unsteady squeezing flow between parallel plates was considered for viscous MHD fluid by Siddiqui et al. [ 1 ]. Further, Erik Sweet [ 2 ] investigated the analytical solution for a viscous fluid flow between moving parallel plates in an unstable MHD flow. Energies 2022,15, 2473. https://doi.org/10.3390/en15072473 https://www.mdpi.com/journal/energies Energies 2022,15, 2473 2 of 21 They used the Homotopy Analysis Method to find the solution, which indicated that the magnetic field’s strength has a significant impact on the flow. Later on, Murty et al. [ 3 ] observed the electrically conducting fluid in a two-phase MHD convective flow under the action of a constant transverse magnetic field through an inclined channel in a rotating system. Onyango et al. [ 4 ] experimented on an unsteady MHD flow of viscous fluid between two parallel plates under a constant pressure gradient. Khan et al. [ 5 ] observed the flow of a viscous fluid between compressing parallel plates under the influence of a varying magnetic field. They investigated the entropy generation due to magnetic fields, fluid friction, and heat transfer in a two-dimensional flow problem. Muhammad et al. [6] discussed the squeezing MHD flow between two parallel plates using Jeffrey fluid. MHD fluid flow between two parallel plates was investigated by Verma et al. [ 7 ]. Later on, Hayat et al. [ 8 ] analytically treated the squeeze flow of MHD nanofluids between two parallel plates. Furthermore, Linga Raju [ 9 ] discussed the MHD two-fluid flow of ionized gases and investigated the effect of hall current on temperature distribution. The effect of magnetohydrodynamics on a fluid film was then observed by Hamza [ 10 ], who studied the squeezed flow between two surfaces while rotation was added to the surfaces. Unsteady Couette flow was then studied by Das et al. [ 11 ] where the flow was unsteady, and the MHD effect was added. The flow was observed in a rotating system. A viscous fluid flow between rotating parallel plates with varying but constant angular velocities was investigated by Parter et al. [ 12 ]. In addition, [ 13 ] also added remarks on the flow when the viscous fluid is flowing between two parallel rotating plates. Further on, Rajagopal [ 14 ] also studied second ordered fluid flowing in a rotating system. Later on, the MHD double-diffusive flow of nanofluids was studied by Tripathi et al. [ 15 ], the flow was observed in a rotating channel with viscous dissipation and hall effect. The MHD flow of viscous fluids in a rotating frame was also studied in cylindrical coordinates, as was discussed by Hughes et al. [ 16 ]. They examined the lubrication flow of such viscous fluids between rotating parallel disks. In addition, Elshekh et al. [ 17 ] talked about the film of a fluid squeezed between rotating parallel disks where an external magnetic field was applied. The influence of a changing magnetic field on the unsteady squeezing flow of viscous fluids between rotating discs was also examined by Shah et al. [ 18 ]. The squeezing unsteady flow of MHD fluid between two disks was also discussed by Ganji et al. [19] , they observed the flow with suction or injection involved. Between squeezing discs moving at various velocities, the effects of MFD viscosity and magnetic field-based (MFD) thermosolutal convection of the fluid dynamics were examined by Khan et al. [20]. The unsteady squeeze flow of viscous fluids is also observed in three-dimensional rotating systems. Recently, Munawar et al. [ 21 ] studied the squeeze flow of viscous fluids in a three-dimensional rotating system. The flow was considered between parallel plates with the lower stretching plate kept porous. Further on, Alzahrani et al. [ 22 ] numerically treated the squeezed flow of viscous fluid between rotating parallel plates in a threedimensional system and examined the effect of Dufour and Soret number. Similar work has been done on third-grade nanofluids in a three-dimensional rotating system, where the thermophoresis effect and Brownian motion were observed by Shah et al. [ 23 ]. In addition, the thin-film flow of Darcy Forchheimer hydromagnetic nanofluid between rotating parallel disks in a three-dimensional system was discussed by Riasat et al. [ 24 ]. They examined the importance of the Magnetic Reynolds number in such a system. Moreover, Fiza et al. [ 25 ] examined the flow of Jeffrey fluid in a three-dimensional rotating system. Very recently, for different fluid flow phenomena, the well-known HAM method was utilized by different authors [ 26 – 30 ]. They utilized the HAM method to examine the behavior of their study and predict the behavior of different problems. The above existing literature witnessed that no study in past has been conducted so far on the 3-dimensional squeeze flow of viscous fluids between two parallel plates under the influence of the variable magnetic field, while both the plates have some angular velocity. Hence, the suggested work is the best approach toward such problems and is a way of Energies 2022,15, 2473 3 of 21 motivation for researchers bringing a new idea of studying the flow between unsteady rotating parallel plates. 2. Modeling and Formulation of the Physical Problem The incompressible viscous fluid flow between two horizontal squeezing plates separated by a distance D(t) = l(1−βt)1/2 (Please see Figure 1), where l is the spacing between plates at time t= 0. The upper plate rotates with an angular velocity of Ωu , whereas the lower plate moves with angular velocity Ωl . The effect of variable magnetic field M is added externally, which produces the induced magnetic field B with the following components, Bx,By, and Bz. Figure 1. Geometry of the flow problem The system of coordinates selected, is Cartesian coordinates. The origin is placed in the lower plate’s center, in which the x-axis is taken along the horizontal axis and the z-axis is at a right angle to both the plates (along the vertical axis). The rotation of plates is along the y-axis. The flow between the plates occurs due to the motion of plates towards each other, i.e., the squeezing effect. The effect of gravity on the fluid is negligible. Now we observe the velocity profile of the given fluid, and the effect of the magnetic field on the velocity of fluid for these viscous fluids in a three-dimensional system. Since the coordinates of the flow are in such a way that the x -component is along the direction of the fluid and the z -component is normal to the direction of flow; thus, the component form of equation of continuity, Navier–Stokes, and magnetic field equation are, Continuity equation: ∂u ∂x+∂v ∂y+∂w ∂z=0, (1) Navier–Stokes equation x-component: ρ∂u ∂t+u∂u ∂x+v∂u ∂y+w∂u ∂z=−∂P ∂x+µ∂2u ∂x2+∂2u ∂y2+∂2u ∂z2+ 1 µ2Bz∂Bx ∂z−Bz∂Bz ∂x−By ∂By ∂x+By∂Bx ∂y. (2) Navier–Stokes equation y-component: ρ∂v ∂t+u∂v ∂x+v∂v ∂y+w∂v ∂z=−∂p ∂y+µ∂2v ∂x2+∂2v ∂y2+∂2v ∂z2+ 1 µ2Bx ∂By ∂x−Bx∂Bx ∂y−Bz∂Bz ∂y+Bz ∂By ∂z. (3) Energies 2022,15, 2473 4 of 21 Navier–Stokes equation z-component: ρ∂w ∂t+u∂w ∂x+v∂w ∂y+w∂w ∂z=−∂p ∂z+µ∂2w ∂x2+∂2w ∂y2+∂2w ∂z2+ 1 µ2By∂Bz ∂y−By ∂By ∂z−Bx∂Bx ∂z+Bx∂Bz ∂x. (4) Magnetic field equation x-component: ∂Bx ∂t=u∂By ∂y+By∂u ∂y−v∂Bx ∂y−Bx∂v ∂y−w∂Bx ∂z−Bx∂w ∂z+u∂Bz ∂z+Bz∂u ∂z+ 1 δµ2∂2Bx ∂x2+∂2Bx ∂y2+∂2Bx ∂z2. (5) Magnetic field equation y-component: ∂By ∂t=v∂Bz ∂z+Bz∂v ∂z−w∂By ∂z−By∂w ∂z−u∂By ∂x−By∂u ∂x+v∂Bx ∂x+Bx∂v ∂x+ 1 δµ2"∂2By ∂x2+∂2By ∂y2+∂2By ∂z2#. (6) Magnetic field equation z-component: ∂Bz ∂t=w∂Bx ∂x+Bx∂w ∂x−u∂Bz ∂x−Bz∂u ∂x−v∂Bz ∂y−Bz∂v ∂y+w∂By ∂y+By∂w ∂y+ 1 δµ2∂2Bz ∂x2+∂2Bz ∂y2+∂2Bz ∂z2. (7) 3. Boundary Conditions The boundary conditions for the above fluid flow are given as: u=0, v=Ωlx 1−βt,w=0, Bx=By=Bz=0, at z=0. u=0, v=Ωux 1−βt,w=dD(t) dt ,Bx=0, By=xN0 1−βt,Bz=−βM0 (1−βt)1/2 , at z=D(t)where D(t) = l(−βt)1/2. Here, ρ is the density of fluid, P is pressure, and B is the induced magnetic field. Now, using the following transformation to convert the above partial differential equations to ordinary differential equations: u=βx (1−βt)f0(η),v=Ωlx (1−βt)g(η),w=−βl (1−βt)1/2 f(η),Bx=βxM0 l(1−βt)m0(η), By=xN0 (1−βt)n(η),Bz=−βM0 (1−βt)1/2 m(η),η=z l(1−βt)1/2 . After non-dimensionlizing the above equations and given boundary conditions will be converted to the following O.D.E.’s: f0000 =Szh3f00 +(η−2f)f000 −2f0f00i+ 2SzM2 xh2Rmmm0+ηmm00 −f mm00 +m2f00−m0m00i,(8) g00 =Szh2g+ηg0+2f0g−2g0fi−2SzMxMyhm0n−n0mi, (9) m00 =Rmhm+ηm0−2m0f+2f0mi, (10) Energies 2022,15, 2473 5 of 21 n00 =Rm2n+ηn0−2n0f+2f0n−2Mx Myg0m−m0g,(11) where Sz=βl2 2ν , denotes the Squeezing Reynolds number, Mx=M0 l√ρµ2 , represents the Magnetic field strength along the x-axis , My=N0 Ωl√ρµ2 is the Magnetic field strength along x-axis and Rm=SzBt which is given by, Rm=βl2 2ν(νσµ2) is the Magnetic Reynolds number, and the boundary conditions become of the form: f(0) = 0, f0(0) = 0, g(0) = 1, m(0) = 0, n(0) = 0. f(1) = 1 2,f0(1) = 0, g(1) = Ωu Ωl =S,m(1) = 1, n(1) = 1. 4. Method of Solution An analytical technique was used to find the solution of Equations (8)–(11), known as the Homotopy Analysis Method. We express the functions f , g , m , and n (where f , g , m , and nare the functions of η,ηK,K≥0) as a set of base functions: fn= ∞ ∑ K=0 aKηr(12) gn= ∞ ∑ K=0 bKηr(13) mn= ∞ ∑ K=0 cKηr(14) nn= ∞ ∑ K=0 dKηr(15) where the constant co-efficients aK,bK,cK, and dkare to be determined. Initial approximations are chosen as follows: f0=1.5 ∗η2−η3; (16) g0= (S−1)∗η+1; (17) m0=η; (18) n0=η(19) now to choose the auxiliary operators: £f=∂4 ∂η4,£g=∂2 ∂η2,£m=∂2 ∂η2,£n=∂2 ∂η2(20) with the following properties £f(k1∗η3+K2∗η2+K3∗η+K4∗) = 0 (21) £g(K5∗η+K6∗) = 0 (22) £m(K7∗η+K8∗) = 0 (23) £n(K9∗η+K10∗) = 0 (24) where K1∗,K2∗,K3∗,K4∗,K5∗,K6∗,K7∗,K8∗,K9∗, and K10∗are arbitrary constants. Energies 2022,15, 2473 6 of 21 We can obtain the Zeroth order deformation as: (1−s)£f[f(η;s)−f0(η)] = s¯hfℵf[f(η;s),m(η;s)] (25) (1−s)£g[g(η;s)−g0(η)] = s¯hgℵg[f(η;s),g(η;s),m(η;s),n(η;s)] (26) (1−s)£m[m(η;s)−m0(η)] = s¯hmℵm[f(η;s),m(η;s)] (27) (1−s)£n[n(η;s)−n0(η)] = s¯hnℵn[f(η;s),g(η;s),m(η;s),n(η;s)] (28) From Equations (14)–(17), the nonlinear operators are define as: ℵf[f(η;s),m(η;s)] = ∂4f(η;s) ∂η4−Sz3∂2f(η;s) ∂η2+(η−2f)∂3f(η;s) ∂η3 −2∂f(η;s) ∂η ∂2f(η;s) ∂η2−2SzM2 x M∂3M(η;s) ∂η3−∂M(η;s) ∂η ∂2M(η;s) ∂η2 (29) ℵg[f(η;s),g(η;s),m(η;s),n(η;s)] = ∂2g(η;s) ∂η2−Sz2g+η∂g(η;s) ∂η +2∂f(η;s) ∂η g −2∂g(η;s) ∂η f−2SzMxMy ∂m(η;s) ∂η n−m∂n(η;s) ∂η  (30) ℵm[f(η;s),m(η;s)] = ∂2m(η;s) ∂η2−Rmm+η∂m(η;s) ∂η −2 f∂m(η;s) ∂η −m∂f(η;s) ∂η  (31) ℵn[f(η;s),g(η;s),m(η;s),n(η;s)] = ∂2n(η;s) ∂η2−R2n+η∂n(η;s) ∂η −2∂n(η;s) ∂η f−n∂f(η;s) ∂η +2∂Mx ∂My ∂g(η;s) ∂η m−g∂m(η;s) ∂η  (32) where s is a fixed parameter, nonlinear parameters are ℵf , ℵg , ℵm and ℵn , while ¯hf , ¯hg , ¯hm , and ¯hnare the nonzero auxiliary parameters. For s=0 and s=1, we have: f(η, 0) = fo,f(η, 1) = f(η) g(η, 0) = go,g(η, 1) = g(η) m(η, 0) = mo,m(η, 1) = m(η) n(η, 0) = no,n(η, 1) = n(η) (33) as s varies from 0 to 1, exact solutions of f(η) , g(η) , n(η) , and n(η) can be obtained from initial guesses of f0,g0,m0, and n0, respectively. For these functions, the Taylor’s series are given by: f(η;s) = f0+ ∞ ∑ n=1 qnfn(η)(34) Energies 2022,15, 2473 7 of 21 g(η;s) = g0+ ∞ ∑ n=1 qngn(η)(35) m(η;s) = m0+ ∞ ∑ n=1 qnmn(η)(36) n(η;s) = n0+ ∞ ∑ n=1 qnnn(η)(37) fn(η) = 1 n! ∂nf(η;s) ∂ηns=0 ,gn(η) = 1 n! ∂ng(η;s) ∂ηns=0 mn(η) = 1 n! ∂nm(η;s) ∂ηns=0 ,nn(η) = 1 n! ∂nn(η;s) ∂ηns=0 (38) It can be noted that in the above series convergence strongly depends upon ¯hf , ¯hg , ¯hm , and ¯hn. Assuming that these nonzero auxiliary parameters are chosen so that the equations converge at s=1, one can obtain: f(η) = f0+ ∞ ∑ n=1 fn(η)(39) g(η) = g0+ ∞ ∑ n=1 gn(η)(40) m(η) = m0+ ∞ ∑ n=1 mn(η)(41) n(η) = n0+ ∞ ∑ n=1 nn(η)(42) Differentiating the Equations (28)–(31) n-times with respect to s and putting s= 0, we have: £f[fn(η)−χnfn−1(η)] = ¯hfRf,n(η)(43) £g[gn(η)−χngn−1(η)] = ¯hgRg,n(η)(44) £m[mn(η)−χnmn−1(η)] = ¯hmRm,n(η)(45) £n[nn(η)−χnnn−1(η)] = ¯hnRn,n(η)(46) with the given boundary conditions, fn(0) = 0, f0 n(0) = 0, gn(0) = 1, mn(0) = 0, nn(0) = 0 fn(1) = 0.5, f0 n(1) = 0, gn(1) = S,mn(1) = 1, nn(1) = 1(47) Rf,n(η) = f0000 n−1(η)−Sz3f00 n−1(η) + (η)f000 n−1(η)−2f0 n−1(η)f00 n−1(η) −2 n−1 ∑ j=0 fj(η)f000 n−j−1(η)+2SzM2 x2Rm n−1 ∑ j=0 mj(η)m0 n−j−1(η) +ηm00 n−j−1(η) + mn−j−1(η)f00 n−j−1(η)−m0 n−1(η)m00 n−1(η) (48) Energies 2022,15, 2473 8 of 21 Rg,n(η) = g00 n−1(η)−Sz2gn−1(η) + (η)g0 n−1(η) + 2 n−1 ∑ j=0 gj(η)f0 n−j−1(η)−fj(η)g0 n−j−1(η)+2SzMxMy n−1 ∑ j=0nj(η)m0 n−j−1(η)−nj(η)m0 n−j−1(η) (49) Rm,n(η) = m00 n−1(η)−Rmmn−1(η) + (η)m0 n−1(η)+ 2 n−1 ∑ j=0mj(η)f0 n−j−1(η)−fj(η)m0 n−j−1(η) (50) Rn,n(η) = n00 n−1(η)−Rm2nn−1(η) + (η)n0 n−1(η)+ 2 n−1 ∑ j=0nj(η)f0 n−j−1(η)−fj(η)n0 n−j−1(η) −2Mx Mymj(η)g0 n−j−1(η)−gj(η)m0 n−j−1(η) (51) and χn=1, i f n >1, and 0, i f n =1. Finally, the general solution of (41–43) can be written as: fn(η) = Zη 0Zη 0Zη 0Zη 0¯hfRf,n(z)dzdzdzdz +χnfn−1+K1∗η3+K2∗η2+K3∗η+K4∗(52) gn(η) = Zη 0Zη 0¯hgRg,n(z)dzdz +χngn−1+K5∗η+K6∗(53) mn(η) = Zη 0Zη 0¯hmRm,n(z)dzdz +χnmn−1+K7∗η+K8∗(54) nn(η) = Zη 0Zη 0¯hnRn,n(z)dzdz +χnnn−1+K9∗η+K10∗(55) and so for f(η),g(η),m(η), and n(η), the exact solution becomes: f(η)≈ n ∑ m=0 fm(η) g(η)≈ n ∑ m=0 gm(η) m(η)≈ n ∑ m=0 mm(η) n(η)≈ n ∑ m=0 nm(η). (56) 5. Optimal Convergence Control Parameters It sholud be noted that the nonzero auxiliary parameters ¯hf , ¯hg , ¯hm , and ¯hn contained in the series solutions (41–43), through which the rate of the homotopy series solutions and convergence region can be determined. Average residual errors were used to obtain the optimal values of ¯hf, ¯hg, ¯hm, and ¯hn: εf n=1 K+1 K ∑ j=0ℵfn ∑ i=0 f(η), n ∑ i=0 m(η)m=jδm2 dη(57) Energies 2022,15, 2473 9 of 21 εg n=1 K+1 K ∑ j=0ℵgn ∑ i=0 f(η), n ∑ i=0 g(η), n ∑ i=0 m(η), n ∑ i=0 n(η)m=jδm2 dη(58) εm n=1 K+1 K ∑ j=0ℵmn ∑ i=0 f(η), n ∑ i=0 m(η)m=jδm2 dη(59) εn n=1 K+1 K ∑ j=0ℵnn ∑ i=0 f(η), n ∑ i=0 g(η), n ∑ i=0 m(η), n ∑ i=0 n(η)m=jδm2 dη(60) Furthermore, εt n=εf n+εg n+εm n+εn n(61) where the total squared residual error is εt n . We can minimize the total average squared residual error by applying the Mathematica package BVPh 2.0. To acquire the local optimal convergence control parameters, the command Minimize was used. 6. Error Analysis Taking 10 −40 as a maximum residual error, the problem was solved with the HAM BVPh 2.0 package. An investigation was made using 40th-order approximations. The provision of error analysis supports the authentication of results for many relevant physical parameters in Figure 2and from results given in Table 1. Figure 2. Total residual error, with Sz=−0.1, Mx=0.1, My=0.3, Rm=0.01, and S=1. Table 2is provided to determine the equations’ inaccuracy from the Navier–Stokes and magnetic field equations. An increase in the order of approximation can be seen, the solution obtained from these equations converges to the exact analysis. Table 1. Estimating the total residual error with fixed values of Sz=− 0.1, Mx= 0.1, My= 0.3, Rm=0.01, and S=1, for different orders of approximations. mεf mεg mεm mεn mCPU Time 1 5.56818 ×10−71.50838 ×10−52.75325 ×10−83.42075 ×10−80.42 s 5 1.04968 ×10−24 4.31742 ×10−19 7.60909 ×10−23 1.89839 ×10−18 4.19 s 10 1.48326 ×10−30 9.10580 ×10−33 1.64090 ×10−35 4.81645 ×10−34 18.5 s 15 1.47350 ×10−30 9.86847 ×10−33 1.86358 ×10−35 1.38805 ×10−35 33.5 s 20 1.48626 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.50240 ×10−35 61.6 s 25 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 106.08 s 30 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 163.13 s 35 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 240.95 s 40 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 456.57 s Energies 2022,15, 2473 16 of 21 Figure 9. Impact of magnetic field strength Mx on velocity component f0 , keeping Sz=− 2, My= 3, Rm=1, and S=1. Figure 10. Impact of Magnetic field strength Mx on velocity component g and magnetic field component n, Keeping Sz=−0.25, My=3, Rm=0.5 and S=1. Energies 2022,15, 2473 17 of 21 Figure 11. Impact of magnetic field strength Mx on magnetic field component m , keeping Sz=− 1.75, My=3, Rm=1, and S=1. Figure 12. Impact of magnetic field strength My on velocity component g and magnetic field component n , keeping Sz=− 0.5, Mx=− 0.5(for g), Mx=− 1.5 (for n), Rm=− 2 (for g), Rm=− 1 (for n), and S=1. Energies 2022,15, 2473 18 of 21 Figure 13. Impact of magnetic Reynolds number Rm on velocity component f and f0 , keeping Sz=−1.5, Mx=−1.5, My=3, and S=1. Figure 14. Impact of magnetic Reynolds number Rm on velocity component g , keeping Sz=− 1.5, Mx=−0.75, My=1, and S=1. Energies 2022,15, 2473 19 of 21 Figure 15. Impact of magnetic Reynolds number Rm on magnetic field components m and n , keeping Sz=−1.5, Mx=−0.75, My=1, and S=1. 8. Conclusions In this paper, the 3-D squeezing MHD flow of viscous fluids was considered between two parallel plates, where both the plates are rotating with the same angular velocities. The effect of the variable magnetic field was applied and the phenomenon was modeled using the coupled governing equations, i.e., continuity, Navier–Stokes, and the magnetic field equation. Furthermore, by using the suitable similarity transformation, the modeled equations of the flow phenomena were transformed to the ordinary differential Equations (8)–(11) and were solved by using the analytical technique (HAM) using Mathematica package BVPh 2.0. The error analysis was carried out up to 10 −40 th order, and the effect of different parameters on the velocity and magnetic field were observed through graphs and tables. Given below are some conclusions made from the above analysis: • It was observed that increasing the squeeze effect on the upper plate causes an increase in the flow velocity along the y - and z -direction, while along the x -direction, the velocity increase initially, but a decrease in the velocity has been observed in the upper domain (η→1). • It was also investigated and concluded that, by increasing the squeeze Reynolds number, the magnetic field component decreased the effect of the magnetic field along the z-component, whereas the effect increased along the y-component. • Furthermore, from the above problem, it was observed that increasing the magnetic field strength parameter Mx , which is the strength of the magnetic field along the x -axis, increases the fluid velocity along the z -axis; however, velocity along the y -axis showed a gradual decrease by increasing Mx . Moreover, along the x -axis, first an increase in the velocity component was observed, but as η→ 1 the velocity started decreasing. • An inverse relation was observed between the magnetic field strength parameter Mx and the magnetic field component along the z -axis, i.e., increasing the value of Mx showed a decreasing effect in the value of the magnetic field along the z -component, and a direct relation could be seen along the y-axis. • Furthermore, it was seen that an increasing value of the magnetic field strength parameter along the y -component caused a decrease in the velocity of the fluid along the y-axis and the effect of the magnetic field along the y-axis. Energies 2022,15, 2473 20 of 21 • It is concluded that for the magnetic Reynolds number Rm , a decrease in the flow velocity along the z -axis was observed with increasing Rm . On the other hand, velocity along the y -axis showed an increasing effect by increasing the Rm flow velocity along the x -axis; this showed a decreasing pattern initially, but as η→ 1, an increasing effect was observed. • It was also observed that for the magnetic field, increasing the magnetic Reynolds number showed a decrease in the value of the magnetic field along both the y - and z-axes. Author Contributions: Conceptualization, M.K.A.; Data curation, M.K.A.; Formal analysis, M.K.A. and U.F.G.; Funding acquisition, S.N.; Investigation, K.B.; Methodology, M.K.A. and U.F.-G.; Resources, U.F.-G.; Writing—review & editing, A.K. and U.F.-G. All authors have read and agreed to the published version of the manuscript. Funding: The work of U.F.-G. was supported by the government of the Basque Country for the ELKARTEK21/10 KK-2021/00014 and ELKARTEK20/78 KK-2020/00114 research programs, respectively. Institutional Review Board Statement: Not applicable. 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