# Unsteady Squeezing Flow of Casson Fluid with Magnetohydrodynamic Effect and Passing through Porous Medium

Authors: Hamid Khan; Mubashir Qayyum; Omar Khan; Murtaza Ali
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing through porous medium channel is modeled and investigated. Similarity transformations are used to convert the partial differential equations (PDEs) of non-Newtonian fluid to a highly nonlinear fourth-order ordinary differential equation (ODE). The obtained boundary value problem is solved analytically by Homotopy Perturbation Method (HPM) and numerically by explicit Runge-Kutta method of order 4. For validity purpose, we compare the analytical and numerical results which show excellent agreement. Furthermore, comprehensive graphical analysis has been made to investigate the effects of various fluid parameters on the velocity profile. Analysis shows that positive and negative squeeze numberSq have opposite effect on the velocity profile. It is also observed that Casson parameter β shows opposite effect on the velocity profile in case of positive and negative squeeze number Sq. MHD parameter Mg and permeability constant Mp have similar effects on the velocity profile in case of positive and negative squeeze numbers. It is also seen that, in case of positive squeeze number, similar velocity profiles have been obtained for β,Mg, and Mp. Besides this, analysis of skin friction coefficient has also been presented. It is observed that squeeze number, MHD parameter, and permeability parameter have direct relationship while Casson parameter has inverse relationship with skin friction coefficient.


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## Body

## 1. Introduction

Squeezing flow between parallel plates is an important problem in the area of fluid dynamics. The problem can be described akin to the principle of moving pistons, where the squeezing behavior of two parallel plates produces a flow that is normal to the plates. Applications of the problem are found in hydraulic machinery and tools, electric motors, food industry, bioengineering, and automobile engines. Other simpler but equally important examples are flow patterns occurring in syringes and compressible tubes. In these applications, flow patterns can be classified into laminar, turbulent, and transitional flows on the basis of the well-known Reynold’s number. From an industrial perspective, it is necessary to study the effect of these different behaviors for non-Newtonian fluids, the mechanics of which have proved to be a significant challenge to the research community. The non-Newtonian fluid model being considered in our case is that of Casson [1, 2] as it is able to capture complex rheological properties of a fluid, unlike other simplified models like the powerlaw [3] and grade-two or grade-three [4] models. Concentrated fluids like sauces, honey, juices, blood, and printing inks [5] can be well described using this model. More formally, Casson fluid can be defined as a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [6]. Application of Casson fluid for flow between two rotating cylinders is performed in [7]. In some industrial applications, the model has to deal with conducting fluids which exhibit different behaviors under the influence of a magnetic field. In these cases, the magnetohydrodynamic (MHD) aspect of the flow needs to be considered. In this article, we investigate this particular case for a porous medium channel and present a comprehensive analysis. To the best of our knowledge, this particular case has not been addressed before. A porous medium, identified as a material that contains fluid-filled pores, is always characterized by properties such as porosity and permeability. Porosity defines the quantity of fluid that can be held by the material, whereas permeability is the amount of fluid that can pass through it. Various applications include ground water hydrology, chemical reactors, irrigation, drainage, seepage, and recovery of crude oil from pores of reservoir rocks [8–12]. These applications can specifically be classified to engineering fields such as petroleum, reservoir, and chemical engineering.Due to the nonlinearity of the model under consideration, exact solutions are rarely found in body of literature and, if found, involve simplified assumptions. For this purpose, many analytical approximation techniques are used instead [13, 14]. The usual approach for boundary value problems is the usage of perturbation techniques. However, due to assumptions of small or large parameters, this is not sufficient. In this regard, a seminal work that combined these perturbation techniques with homotopy was proposed as the Homotopy Perturbation Method (HPM) in [15–17]. Since its introduction, the method has been applied to different nonlinear equations [18–22]. Specifically in the case of fluid dynamics, the method has been applied in [18–20, 23]. The classical HPM has also been modified by few researchers [24, 25]. Other approximation techniques that have been used for the case of fluid dynamics include the Homotopy Analysis Method (HAM) [26], Optimal Homotopy Asymptotic Method (OHAM) [27], Adomian Decomposition Method (ADM) [28], and Variational Iteration Method (VIM) [29]. In addition to these analytical approaches, various numerical schemes can also be used to solve these problems. Examples are the family of Runge-Kutta [30], finite difference [31], and wavelet methods.In the remaining part of the manuscript, Section2 includes mathematical formulation of the problem. Sections 3 and 4 present the basic theory of HPM and its application to Casson fluid model. Section 5 comprises the results and discussion. Finally, conclusion is presented in Section 6.


## 2. Mathematical Formulation

An incompressible flow of Casson fluid is considered between two parallel plates that have been separated by a distancez=±l1-αt1/2=±ht. Here, l is the initial gap between the two plates at time t, and α is the squeezing motion of both plates. Both plates touch one another at t=a/α. α<0 implies a receding motion of the plates. With these conditions, the non-Newtonian Casson fluid, using [32, 33], is defined as(1)τij=2μB+Py2πeij,π>πc,2μB+Py2πceij,πc>π,where τij is the (i,j)th component of the stress tensor, π=eijeij,eij being the (i,j)th component of the deformation rate, πc is the critical value of the material, μB is plastic dynamic viscosity, and Py is the yield stress of the fluid. A constant magnetic field of strength Mg is applied perpendicularly and relatively fixed to the walls. It is assumed that the intensity of the effective field produced due to the conducting fluid is negligible and that there is no other external electric field. The governing relation for flow under these assumptions is given as(2)∂ux∂x+∂uy∂y=0,(3)∂ux∂t+ux∂ux∂x+uy∂ux∂y=-1ρ∂p∂x+ν1+1β2∂2ux∂x2+∂2ux∂y2+2∂2uy∂y∂x-σB2ρux-μρkux,(4)∂uy∂t+ux∂uy∂x+uy∂uy∂y=-1ρ∂p∂y+ν1+1β2∂2uy∂x2+∂2uy∂y2+2∂2ux∂y∂x-μρkuy,where ux and uy are the velocity components in x and y directions, p is the pressure, μ and ν are the viscosity and kinematic viscosity of the fluid, β=μB2π/Py is the Casson fluid parameter, B is the magnitude of the imposed magnetic field, and k is the permeability constant. The boundary conditions for the problem are given as follows:(5)ux=0,uy=vw=dhdtaty=ht,∂ux∂y=0,uy=0aty=0. Cross differentiating (3) and (4) and by introducing the vorticity function ω, we get(6)∂ω∂t+ux∂ω∂x+uy∂ω∂y=ν1+1β∂2ω∂x2+∂2ω∂y2-σB2ρ∂ux∂y-μρkω,where(7)ω=∂uy∂x-∂ux∂y.The similarity transform for a two-dimensional flow [34] is(8)ux=αx21-αtU′η,uy=-αl21-αt1/2Uη,where η=y/l(1-αt)1/2. Substituting (8) into (6) using (7) gives the following nonlinear differential equation that describes Casson’s fluid flow:(9)1+1βd4Uηdη4-SqηUη+3d2Uηdη2+dUηdηd2Uηdη2-Uηd3Uηdη3-Mgd2Uηdη2-Mpd2Uηdη2=0,where Sq=αl2/2ν is the nondimensional squeeze number that describes movement of the plates. Sq>0 corresponds to the plates moving apart, while Sq<0 corresponds to the collapsing movement. Using (8), the boundary conditions for the problem are reduced to(10)U0=0,U′′0=0,U1=1,U′1=0.When Mg=Mp=0 and β→∞, the current problem is reduced to the problem discussed in [34].The skin friction coefficient is defined as [35](11)Cf=ν1+1β∂ux/∂yy=htvw2. In terms of (8), we have(12)l2x21-αtRexCf=1+1βU′′1, where Rex=2lvw2/νx(1-αt)1/2.


## 3. Basic Theory of Homotopy Perturbation Method

The basic theory of HPM can be exhibited using the following differential equation:(13)Lw+Nw-gr=0,r∈Ω,Bw,dwdn=0,r∈Υ,where w is an unknown and g(r) is a known function, L, N, B are linear, nonlinear, and boundary operators, and Υ is the boundary of the domain Ω. A homotopy θ(r,p):Ω×0,1→R is then constructed which satisfies(14)ψθ,p=1-pLθ-Lw0+pLθ+Nθ-gr=0,r∈Ω,where p∈[0,1] is an embedding parameter and w0 is the initial guess which satisfies the boundary conditions. From (14), we have (15)ψθ,0=Lθ-Lw0=0,ψθ,1=Lθ+Nθ-gr=0.Thus, as p varies from 0 to 1, the solution θ(r,p) approaches from w0(r) to w~(r). To obtain an approximate solution, we expand θ(r,p) in a Taylor series about p as follows:(16)θr,p=θ0+∑k=1∞θkpk. Setting p=1, the approximate solution of (25) would be(17)U~=limp→1⁡θr,p=∑k=1∞θk.Substituting (17) into (13) will give(18)Rx=LU~x+NU~x-fx. If R=0, then U~ will be the exact solution but usually this does not happen in nonlinear problems.


## 4. Implementation of HPM to Squeezing Flow of Casson Fluid

Using (9) and (10), various-order problems are presented as follows.Zeroth-order problem is(19)U0ivη+1βU0ivη=0,U00=0,U0′′0=0,U01=1,U0′1=0. Solution of the zeroth order problem is(20)U0η=123η-η3.First-order problem is(21)1βU1ivη+U1ivη-SqηU0η-MpU0′′η-MgU0′′η-3SqU0′′η-SqU0′ηU0′′η+SqU0ηU0′′′η=0,U10=0,U1′′0=0,U11=0,U1′1=0.Solution of the first-order problem is(22)U1η=-167201+βΦ11η-Φ12η3+Φ13η5-Φ14η6-Φ15η7+Φ16η8.Second-order problem is(23)U2ivη+1βU2ivη-SqηU1η-SqU1′ηU0′′η-MpU1′′η-MgU1′′η-3SqU1′′η-SqU0′ηU1′′η+SqU1ηU0′′′η+SqU0ηU1′′′η,U20=0,U2′′0=0,U21=0,U2′1=0.Solution of the second-order problem is(24)U2η=-196864768001+β2Φ21η+Φ22η3+Φ23η5+Φ24η6+Φ25η7+Φ26η8+Φ27η9+Φ28η10-Φ29η11+Φ210η12+Φ211η13, where Φ1i and Φ2j are the coefficients of various powers of η. These coefficients are given in the Appendix for the reader convenience.In a similar way, higher order problems can be formulated and solved. These approximations have been excluded from the manuscript for brevity purpose.Considering the third-order solution,(25)U~η=∑j=03Ujη.By fixing values of β,Sq,Mg, and Mp in (25) polynomial solution can be found. For instance, when β=0.01,Sq=-0.2,Mg=0.5, and Mp=0.5, the third-order solution is therefore(26)U~η=0.0+1.49988η-0.499762η3-0.0000989628η5-8.25014×10-6η6-7.08874×10-6η7+5.87168×10-7η8-2.33686×10-9η9+1.61604×10-10η10+4.40191×10-11η11-4.96322×10-12η12-3.44636×10-14η13-3.72105×10-15η14-1.04633×10-15η15+1.29806×10-16η16+2.11403×10-19η17+1.83374×10-21η18.The residual error of the problem is(27)RE=1+1βd4U~dη4-SqηU~+3d2U~dη2+dU~dηd2U~dη2-U~d3U~dη3-Mgd2U~dη2-Mpd2U~dη2.


## 5. Results and Discussion

In this article, an unsteady squeezing flow of Casson fluid having MHD effect and passing through porous medium channel is considered. Four parameters are considered here: the squeeze numberSq, Casson parameter β, MHD parameter Mg, and the permeability parameter Mp. The resulting boundary value problem is solved for various values of the mentioned parameters using HPM and the results are compared with numerical solutions obtained using explicit Runge-Kutta method of order 4 (ERK4). Tables 4–7 shows the comparison of analytical and numerical solutions along with residual errors for various values of fluid parameters. A quick analysis of these tables reveals that the results from HPM are consistent and in good agreement with the numerical results.The convergence of the homotopy solution is confirmed by finding various-order solutions along with absolute residual errors in Table2. Here, it can be observed that the HPM solution improves considerably as the order of approximation is increased. The validity of analytical solution based on HPM is checked by comparing it with numerical solutions of ERK4 in Table 3. Here, Mg=0.5, β=0.05, and the squeeze number is varied as -0.2≤Sq≤0.6. Validity is confirmed for all variations of squeeze number. Both the convergence and validity are also demonstrated graphically in Figures 1 and 2.Figure 1
Convergence of homotopy perturbation solution.Figure 2
Comparison of analytical and numerical solutions.Numerical values of skin friction coefficients corresponding to various fluid parameters are given in Table1. Analysis of these numerical quantities show that increase in β decreases the skin friction coefficient. Furthermore, increase in Sq, Mg, and Mp increases the skin friction coefficient. It is also observed that increase in Mg and Mp increases the skin friction coefficient.Table 1
Skin friction coefficient for various values of fluid parameters.
Sq β M g M p 1 + 1 β U ′ ′ ( 1 ) - 8.0 0.5 1.0 1.0 - 2.144 - 6.0  1.0 1.0 - 4.919 - 3.0  1.0 1.0 - 7.728 - 0.6 0.05 0.5 0.5 - 62.922 - 0.2   1.5 - 63.307 1.7 - 63.347 0.01  0.5 - 303.108 0.1   - 33.107 0.3   - 13.108 0.4   - 10.608 0.5   - 9.108 0.8   - 6.858 1.0   - 6.108 0.05 0.1  - 63.563 0.7  - 63.147 1.2  - 63.247 0.2 0.01 0.5  - 303.292 0.3   - 13.287 0.4   - 10.786 0.05  1.4 - 63.470 1 - 63.390 0.6   0.5 - 63.473 0.8    - 63.563 1.0 0.5 1.0 1.0 - 9.785 2.0    - 10.130 4.0    - 10.756 1.0   - 7.731 3.0   - 5.969 5.0   - 5.722 8.0   - 5.620 8.0 0.5   - 12.310 3.0 1.0 1.0  - 7.381 4.0  - 7.862 8.0  - 8.542 5.0  1.0 0.1 - 7.998 5.0 - 8.972 10 - 10.146Table 2
Various-order homotopy perturbation solutions along with absolute residual errors for fixed values of fluid parameters.
η Zeroth order First order Second order Third order Solution Error Solution Error Solution Error Solution Error 0.0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0.1 0.1495 0.12395 0.149488 5.58 × 1 0 - 5 0.149488 6.38 × 1 0 - 9 0.149488 8.92 × 1 0 - 13 0.2 0.296 0.25664 0.295977 1.03 × 1 0 - 4 0.295977 7.98 × 1 0 - 9 0.295977 2.45 × 1 0 - 12 0.3 0.4365 0.40239 0.436469 1.30 × 1 0 - 4 0.436469 1.13 × 1 0 - 9 0.436469 4.63 × 1 0 - 12 0.4 0.568 056384 0.567965 1.25 × 10 - 4 0.567965 1.52 × 10 - 8 0.567965 6.20 × 10 - 12 0.5 0.6875 074375 0.687464 7.15 × 10 - 5 0.687464 3.76 × 10 - 8 0.687464 4.79 × 10 - 12 0.6 0.792 094464 0.791969 5.16 × 10 - 5 0.791969 5.52 × 10 - 8 0.791969 1.91 × 10 - 12 0.7 0.8785 1.16879 0.878477 2.67 × 1 0 - 4 0.878477 4.70 × 1 0 - 8 0.878477 1.37 × 1 0 - 11 0.8 0.944 1.41824 0.943987 6.05 × 1 0 - 4 0.943987 2.20 × 1 0 - 8 0.943987 2.36 × 1 0 - 11 0.9 0.9855 1.69479 0.985496 1.09 × 1 0 - 3 0.985496 2.05 × 1 0 - 7 0.985496 1.05 × 1 0 - 11 1.0 1.0 2.0 1.0 1.78 × 10 - 3 1.0 5.77 × 10 - 7 1.0 6.93 × 10 - 11Table 3
Comparison of analytical and numerical solution for variousSq.
η E R K 4 - H P M M p = M g = 0.5 ,   β = 0.05 S q = - 0.2 S q = - 0.4 S q = - 0.6 S q = 0.2 S q = 0.4 S q = 0.6 0.0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0.1 8.32 × 1 0 - 15 5.77 × 1 0 - 14 6.37 × 1 0 - 12 2.12 × 1 0 - 11 9.52 × 1 0 - 11 2.80 × 1 0 - 10 0.2 1.38 × 1 0 - 14 1.10 × 1 0 - 13 1.20 × 1 0 - 11 3.94 × 1 0 - 11 1.77 × 1 0 - 10 5.23 × 1 0 - 10 0.3 1.38 × 1 0 - 14 1.52 × 1 0 - 13 1.63 × 1 0 - 11 5.21 × 1 0 - 11 2.36 × 1 0 - 10 6.97 × 1 0 - 10 0.4 7.38 × 1 0 - 15 1.78 × 1 0 - 13 1.88 × 1 0 - 11 5.77 × 1 0 - 11 2.63 × 1 0 - 10 7.80 × 1 0 - 10 0.5 4.44 × 1 0 - 15 1.85 × 1 0 - 13 1.90 × 1 0 - 11 5.56 × 1 0 - 11 2.55 × 1 0 - 10 7.61 × 1 0 - 10 0.6 1.68 × 1 0 - 14 1.70 × 1 0 - 13 1.70 × 1 0 - 11 4.65 × 1 0 - 11 2.16 × 1 0 - 10 6.48 × 1 0 - 10 0.7 2.37 × 1 0 - 14 1.32 × 1 0 - 13 1.29 × 1 0 - 11 3.27 × 1 0 - 11 1.54 × 1 0 - 10 4.65 × 1 0 - 10 0.8 1.97 × 1 0 - 14 7.87 × 1 0 - 14 7.63 × 1 0 - 12 1.75 × 1 0 - 11 8.42 × 1 0 - 11 2.55 × 1 0 - 10 0.9 7.67 × 1 0 - 15 2.53 × 1 0 - 14 2.49 × 1 0 - 12 5.13 × 1 0 - 12 2.50 × 1 0 - 11 7.68 × 1 0 - 11 1.0 1.24 × 1 0 - 19 1.10 × 1 0 - 16 1.11 × 1 0 - 16 2.57 × 1 0 - 21 6.86 × 1 0 - 22 4.98 × 1 0 - 19Table 4
Solutions along with residual errors for variousSq when Mg=Mp=0.5,β=0.05.
Parameter η Solution Residual error HPM ERK4 HPM ERK4 S q = - 0.2 0 . 0 0 . 0 0 . 0 0 . 0 1.40 × 1 0 - 7 0.1 0.149441 0.149441 9.93 × 1 0 - 11 6.59 × 1 0 - 9 0.2 0.29589 0.29589 2.73 × 1 0 - 10 - 1.35 × 1 0 - 9 0.3 0.436351 0.436351 5.16 × 1 0 - 10 2.71 × 1 0 - 10 0.4 0.567829 0.567829 6.89 × 1 0 - 10 - 2.04 × 1 0 - 10 0.5 0.687329 0.687329 5.33 × 1 0 - 10 2.41 × 1 0 - 10 0.6 0.791849 0.791849 - 2.13 × 1 0 - 10 - 8.42 × 1 0 - 11 0.7 0.878387 0.878387 - 1.53 × 1 0 - 9 - 5.29 × 1 0 - 10 0.8 0.943935 0.943935 - 2.63 × 1 0 - 9 2.65 × 1 0 - 9 0.9 0.985479 0.985479 - 1.17 × 1 0 - 9 - 1.08 × 1 0 - 8 1.0 1.0 1.0 7.72 × 1 0 - 9 - 1.50 × 1 0 - 7  S q = - 0.4 0 . 0 0 . 0 0 . 0 0 . 0 7.25 × 1 0 - 7 0.1 0.1495 0.1495 - 5.03 × 1 0 - 11 3.68 × 1 0 - 8 0.2 0.295999 0.295999 - 9.77 × 1 0 - 11 - 8.75 × 1 0 - 9 0.3 0.436497 0.436497 - 1.52 × 1 0 - 10 2.33 × 1 0 - 9 0.4 0.567995 0.567995 - 2.62 × 1 0 - 10 - 7.32 × 1 0 - 10 0.5 0.687493 0.687493 - 5.45 × 1 0 - 10 - 5.65 × 1 0 - 12 0.6 0.791991 0.791991 - 1.15 × 1 0 - 9 6.91 × 1 0 - 10 0.7 0.878491 0.878491 - 2.09 × 1 0 - 9 - 1.86 × 1 0 - 9 0.8 0.943994 0.943994 - 2.85 × 1 0 - 9 6.26 × 1 0 - 9 0.9 0.985498 0.985498 - 1.81 × 1 0 - 9 - 2.54 × 1 0 - 8 1.0 1.0 1.0 4.49 × 1 0 - 9 - 4.34 × 1 0 - 7  S q = - 0.6 0 . 0 0 . 0 0 . 0 0 . 0 1.61 × 1 0 - 6 0.1 0.149558 0.149558 - 1.24 × 1 0 - 8 8.03 × 1 0 - 8 0.2 0.296108 0.296108 - 2.49 × 1 0 - 8 - 1.93 × 1 0 - 8 0.3 0.436644 0.436644 - 3.79 × 1 0 - 8 5.07 × 1 0 - 9 0.4 0.568161 0.568161 - 5.28 × 1 0 - 8 - 1.55 × 1 0 - 9 0.5 0.687657 0.687657 - 7.21 × 1 0 - 8 3.17 × 1 0 - 10 0.6 0.792134 0.792134 - 9.99 × 1 0 - 8 1.06 × 1 0 - 9 0.7 0.878596 0.878596 - 1.42 × 1 0 - 7 - 4.49 × 1 0 - 9 0.8 0.944053 0.944053 - 2.06 × 1 0 - 7 1.61 × 1 0 - 8 0.9 0.985561 0.985561 - 3.01 × 1 0 - 7 - 6.40 × 1 0 - 8 1.0 1.0 1.0 - 4.33 × 1 0 - 7 - 1.05 × 1 0 - 6Table 5
Solutions along with residual errors for variousβ when Mg=Mp=0.5,Sq=-0.2.
Parameter η Solution Residual error HPM ERK4 HPM ERK4 β = 0.01 0 . 0 0 . 0 0 . 0 0 . 0 - 1.76 × 1 0 - 7 0.1 0.149488 0.149502 8.92 × 1 0 - 13 - 8.33 × 1 0 - 9 0.2 0.295977 0.296004 2.45 × 1 0 - 12 4.75 × 1 0 - 9 0.3 0.436469 0.436506 4.63 × 1 0 - 12 - 2.87 × 1 0 - 9 0.4 0.567965 0.568006 6.20 × 1 0 - 12 - 3.59 × 1 0 - 10 0.5 0.687464 0.687506 4.79 × 1 0 - 12 2.45 × 1 0 - 9 0.6 0.791969 0.792005 - 1.91 × 1 0 - 12 - 1.50 × 1 0 - 9 0.7 0.878477 0.878504 - 1.37 × 1 0 - 11 - 1.89 × 1 0 - 9 0.8 0.943987 0.944002 - 2.36 × 1 0 - 11 8.25 × 1 0 - 9 0.9 0.985496 0.985501 - 1.05 × 1 0 - 11 - 2.78 × 1 0 - 8 1.0 1.0 1.0 6.93 × 1 0 - 11 - 3.31 × 1 0 - 7  β = 0.1 0 . 0 0 . 0 0 . 0 0 . 0 3.51 × 1 0 - 7 0.1 0.149388 0.149522 6.91 × 1 0 - 10 1.83 × 1 0 - 8 0.2 0.295789 0.296041 1.90 × 1 0 - 9 - 4.39 × 1 0 - 9 0.3 0.436215 0.436554 3.59 × 1 0 - 9 1.08 × 1 0 - 9 0.4 0.567674 0.568059 4.79 × 1 0 - 9 - 2.68 × 1 0 - 10 0.5 0.687173 0.687557 3.71 × 1 0 - 9 - 2.80 × 1 0 - 11 0.6 0.791712 0.792048 - 1.48 × 1 0 - 9 2.99 × 1 0 - 10 0.7 0.878285 0.878533 - 1.06 × 1 0 - 8 - 8.24 × 1 0 - 10 0.8 0.943876 0.944018 - 1.83 × 1 0 - 8 2.81 × 1 0 - 9 0.9 0.985461 0.985505 - 8.20 × 1 0 - 9 - 1.16 × 1 0 - 8 1.0 1.0 1.0 5.37 × 10 - 8 - 1.95 × 10 - 7  β = 0.2 0 . 0 0 . 0 0 . 0 0 . 0 6.28 × 1 0 - 7 0.1 0.149295 0.14954 4.25 × 1 0 - 9 3.03 × 1 0 - 8 0.2 0.295614 0.296075 1.17 × 1 0 - 8 - 7.38 × 1 0 - 9 0.3 0.435977 0.436599 2.21 × 1 0 - 8 2.07 × 1 0 - 9 0.4 0.567403 0.568109 2.95 × 1 0 - 8 - 7.36 × 1 0 - 10 0.5 0.686901 0.687605 2.28 × 1 0 - 8 1.63 × 1 0 - 10 0.6 0.791472 0.792087 - 9.14 × 1 0 - 9 4.70 × 1 0 - 10 0.7 0.878105 0.878561 - 6.57 × 1 0 - 8 - 1.87 × 1 0 - 9 0.8 0.943773 0.944032 - 1.12 × 1 0 - 7 6.77 × 1 0 - 9 0.9 0.985428 0.985509 - 5.05 × 1 0 - 8 - 2.73 × 1 0 - 8 1.0 1.0 1.0 3.31 × 10 - 7 - 4.47 × 10 - 7Table 6
Solutions along with residual errors for variousMg when Mp=0.5,Sq=-0.2,β=0.05.
Parameter η Solution Residual error HPM ERK4 HPM ERK4 M g = 0.1 0 . 0 0 . 0 0 . 0 0 . 0 1.73 × 1 0 - 7 0.1 0.149372 0.149488 - 3.30 × 1 0 - 8 1.73 × 1 0 - 7 0.2 0.29576 0.295977 - 6.29 × 1 0 - 8 - 2.10 × 1 0 - 9 0.3 0.436177 0.436469 - 8.69 × 1 0 - 8 4.10 × 1 0 - 10 0.4 0.567634 0.567964 - 1.02 × 1 0 - 7 - 7.13 × 1 0 - 11 0.5 0.687137 0.687463 - 1.08 × 1 0 - 7 5.99 × 1 0 - 11 0.6 0.791684 0.791966 - 1.02 × 1 0 - 7 - 1.65 × 1 0 - 11 0.7 0.878267 0.878474 - 8.46 × 1 0 - 8 - 3.51 × 1 0 - 10 0.8 0.943868 0.943985 - 5.38 × 1 0 - 8 1.99 × 1 0 - 9 0.9 0.985459 0.985459 - 9.68 × 1 0 - 9 - 9.11 × 1 0 - 9 1.0 1.0 1.0 4.83 × 10 - 8 - 1.24 × 10 - 7  M g = 0.4 0 . 0 0 . 0 0 . 0 0 . 0 8.57 × 1 0 - 8 0.1 0.149338 0.149453 - 7.57 × 1 0 - 8 1.30 × 1 0 - 9 0.2 0.295695 0.295912 - 1.43 × 1 0 - 7 - 2.14 × 1 0 - 10 0.3 0.436089 0.43638 - 1.95 × 1 0 - 7 7.59 × 1 0 - 11 0.4 0.567534 0.567863 - 2.26 × 1 0 - 7 - 3.50 × 1 0 - 10 0.5 0.687037 0.687362 - 2.31 × 1 0 - 7 5.60 × 1 0 - 10 0.6 0.791597 0.791878 - 2.08 × 1 0 - 7 - 2.96 × 1 0 - 10 0.7 0.878202 0.878409 - 1.55 × 1 0 - 7 - 7.90 × 1 0 - 10 0.8 0.943831 0.943947 - 7.53 × 1 0 - 8 3.92 × 1 0 - 9 0.9 0.985447 0.985483 3.09 × 1 0 - 8 - 1.56 × 1 0 - 8 1.0 1.0 1.0 1.59 × 10 - 7 - 2.04 × 10 - 7  M g = 1.0 0 . 0 0 . 0 0 . 0 0 . 0 1.28 × 1 0 - 7 0.1 0.149268 0.149383 - 2.69 × 1 0 - 7 4.96 × 1 0 - 9 0.2 0.295564 0.29578 - 5.05 × 1 0 - 7 - 1.04 × 1 0 - 9 0.3 0.435913 0.436203 - 6.78 × 1 0 - 7 3.68 × 1 0 - 10 0.4 0.567334 0.567662 - 7.64 × 1 0 - 7 - 4.53 × 1 0 - 10 0.5 0.686833 0.687162 - 7.49 × 1 0 - 7 2.64 × 1 0 - 10 0.6 0.791422 0.791703 - 6.25 × 1 0 - 7 3.42 × 1 0 - 10 0.7 0.878073 0.878279 - 3.99 × 1 0 - 7 - 8.95 × 1 0 - 10 0.8 0.943757 0.943873 - 8.61 × 1 0 - 8 1.28 × 1 0 - 9 0.9 0.985424 0.98546 2.89 × 1 0 - 7 - 2.79 × 1 0 - 9 1.0 1.0 1.0 6.93 × 10 - 7 - 9.05 × 10 - 8Table 7
Solutions along with residual errors for variousMp when Mg=0.5,Sq=-0.2,β=0.05.
Parameter η Solution Residual error HPM ERK4 HPM ERK4 M p = 1.0 0 . 0 0 . 0 0 . 0 0 . 0 1.28 × 1 0 - 7 0.1 0.149383 0.149383 - 2.87 × 1 0 - 9 4.96 × 1 0 - 9 0.2 0.29578 0.295978 - 4.06 × 1 0 - 9 - 1.04 × 1 0 - 9 0.3 0.436203 0.436203 - 2.56 × 1 0 - 9 3.68 × 1 0 - 10 0.4 0.567662 0.567662 1.37 × 1 0 - 9 - 4.53 × 1 0 - 10 0.5 0.687162 0.687162 6.02 × 1 0 - 9 2.64 × 1 0 - 10 0.6 0.791703 0.791703 8.52 × 1 0 - 9 3.42 × 1 0 - 10 0.7 0.878279 0.878279 6.00 × 1 0 - 9 - 8.95 × 1 0 - 10 0.8 0.943873 0.943873 - 2.25 × 1 0 - 9 1.28 × 1 0 - 9 0.9 0.98546 0.98546 - 1.14 × 1 0 - 8 - 2.79 × 1 0 - 9 1.0 1.0 1.0 - 6.12 × 10 - 9 - 9.05 × 10 - 8  M p = 1.5 0 . 0 0 . 0 0 . 0 0 . 0 5.71 × 1 0 - 8 0.1 0.149325 0.149325 - 2.60 × 1 0 - 8 1.22 × 1 0 - 10 0.2 0.295671 0.295671 - 4.28 × 1 0 - 8 1.00 × 1 0 - 10 0.3 0.436056 0.436056 - 4.40 × 1 0 - 8 8.10 × 1 0 - 11 0.4 0.567495 0.567495 - 2.82 × 1 0 - 8 - 3.44 × 1 0 - 10 0.5 0.686995 0.686995 - 3.16 × 1 0 - 10 4.62 × 1 0 - 10 0.6 0.791558 0.791558 2.87 × 1 0 - 8 - 2.07 × 1 0 - 10 0.7 0.878171 0.878171 4.50 × 1 0 - 8 - 6.54 × 1 0 - 10 0.8 0.943812 0.943812 3.74 × 1 0 - 8 3.13 × 1 0 - 9 0.9 0.985441 0.985441 7.51 × 1 0 - 9 - 1.22 × 1 0 - 8 1.0 1.0 1.0 - 1.60 × 10 - 8 - 1.62 × 10 - 7  M p = 1.7 0 . 0 0 . 0 0 . 0 0 . 0 1.58 × 1 0 - 7 0.1 0.149302 0.149302 - 4.79 × 1 0 - 8 1.08 × 1 0 - 8 0.2 0.295627 0.295627 - 8.04 × 1 0 - 8 - 2.34 × 1 0 - 9 0.3 0.435997 0.435997 - 8.64 × 1 0 - 8 2.54 × 1 0 - 10 0.4 0.567428 0.567428 - 6.27 × 1 0 - 8 1.88 × 1 0 - 10 0.5 0.686929 0.686929 - 1.60 × 1 0 - 8 - 7.14 × 1 0 - 11 0.6 0.791499 0.791499 3.70 × 1 0 - 8 - 8.03 × 1 0 - 11 0.7 0.878128 0.878128 7.43 × 1 0 - 8 - 1.19 × 1 0 - 10 0.8 0.943787 0.943787 7.58 × 1 0 - 8 1.11 × 1 0 - 9 0.9 0.985433 0.985433 3.80 × 1 0 - 8 - 4.87 × 1 0 - 9 1.0 1.0 1.0 - 5.44 × 10 - 9 - 6.74 × 10 - 8The effects of identified parameters on the velocity profile are illustrated graphically in Figures3–10. The effect of negative Sq on the velocity profile is shown in Figure 3, where it is shown that the normal velocity profile increases with the increase in negative Sq. On the other hand, the radial velocity increases near the lower plate and terminates near the upper plate. It can also be observed that, for fixed values of fluid parameters, the normal velocity monotonically increases while the radial velocity monotonically decreases. Moreover, the radial velocity increases when 0<η≤0.45 and decreases when 0.45<η≤1.Figure 3
Effect of negative squeeze numberSq on the velocity profile.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 4
Effect of Casson parameterβ on the velocity profile when Sq is negative.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 5
Effect of MHD parameterMg on the velocity profile when Sq is negative.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 6
Effect of permeability constantMp on the velocity profile when Sq is negative.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 7
Effect of positive squeeze numberSq on the velocity profile.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 8
Effect of Casson parameterβ on the velocity profile when Sq is positive.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 9
Effect of MHD parameterMg on the velocity profile when Sq is positive.
(a)
Normal component of velocity (b)
Radial component of velocityFigure 10
Effect of permeability constantMp on the velocity profile when Sq is positive.
(a)
Normal component of velocity (b)
Radial component of velocityIn the next three figures, the effect of various fluid parameters on the velocity profile withSq<0 is depicted. First, the effect of Casson parameter β is shown in Figure 4, showing that the normal velocity increases as β is increased, whereas the radial velocity increases near the lower plates and terminates near the upper plate. The effect when β is increased is similar to that of increasing negative Sq. Secondly, the effect of MHD parameter Mg is depicted in Figure 5. Here, it is seen that the normal component of the velocity decreases with the increase in Mg, while the radial component decreases near the lower plate and terminates near the upper plate. Moreover, the radial component of velocity decreases when 0<η≤0.45 and increases when 0.45<η≤1. Lastly, the effect of permeability parameter Mp is shown in Figure 6. Here, the normal velocity decreases when Mp is increased, while the radial velocity decreases near the lower plate and terminates near the upper plate. It can therefore be concluded from these observations that the effect of Mp and Mg is similar on the velocity profile in case of negative squeeze number.The effect ofSq>0 on the velocity profile is also investigated. In Figure 7, it is shown that the normal component of velocity decreases as Sq is increased, while the radial component of velocity decreases near the lower plate and terminates near the upper plate. Moreover, the radial velocity decreases in the interval 0<η≤0.45 and increases in 0.45<η≤1. The next three figures show the effect of β, Mg, and Mp in case of positive squeeze number. First, the effect of β is illustrated in Figure 8, showing that the normal component of velocity decreases as β is increased, while the radial component decreases near the lower plate and terminates near the upper plate. The effect of Mg is shown in Figure 9, while that of Mp is shown in Figure 10. In both cases, a similar effect as that of β can be observed.In summary, it can be observed that positive and negativeSq have opposite effects on the velocity profile. Moreover, β shows opposite effect on the velocity profile in case of positive and negative Sq. However, Mg and Mp have similar effects on the velocity profile irrespective of the sign of the squeeze number. It is also observed that Sq, Mg, and Mp have similar effect while β has opposite effect on the skin friction coefficient.


## 6. Conclusion

This article presents a similarity solution for an unsteady squeezing flow of the non-Newtonian Casson fluid with MHD effect and passing through porous medium. The PDEs were reduced to a highly nonlinear fourth-order ODE by applying similarity transformations and then solved using the Homotopy Perturbation Method (HPM) and the fourth-order explicit Runge-Kutta method. Convergence and validity of the obtained solution was confirmed and found to be in good agreement. A comprehensive analysis was also performed to investigate the effect of various fluid factors like squeeze number, Casson, permeability, and MHD parameters on the velocity profile.



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*Source: 4293721-2016-12-29.xml*