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

An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing through porous medium channel ismodeled and investigated. Similarity transformations are used to convert the partial differential equations (PDEs) of nonNewtonian 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 number Sq 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 constantMp 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, andMp. Besides this, analysis of skin friction coefficient has also been presented. It is observed that squeeze number,MHDparameter, and permeability parameter have direct relationship while Casson parameter has inverse relationship with skin friction coefficient.


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 wellknown 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

Mathematical Formulation
An incompressible flow of Casson fluid is considered between two parallel plates that have been separated by a distance  = ±(1−) 1/2 = ±ℎ().Here,  is the initial gap between the two plates at time , and  is the squeezing motion of both plates.Both plates touch one another at  = /. < 0 implies a receding motion of the plates.With these conditions, the non-Newtonian Casson fluid, using [32,33], is defined as where   is the (, )th component of the stress tensor,  =     ,   being the (, )th component of the deformation rate,   is the critical value of the material,   is plastic dynamic viscosity, and   is the yield stress of the fluid.A constant magnetic field of strength   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 where   and   are the velocity components in  and  directions,  is the pressure,  and ] are the viscosity and kinematic viscosity of the fluid,  =   √2/  is the Casson fluid parameter,  is the magnitude of the imposed magnetic field, and  is the permeability constant.The boundary conditions for the problem are given as follows: Cross differentiating (3) and (4) and by introducing the vorticity function , we get where The similarity transform for a two-dimensional flow [34] is where  = /[(1 − ) 1/2 ].Substituting ( 8) into ( 6) using (7) gives the following nonlinear differential equation that describes Casson's fluid flow: where ) is the nondimensional squeeze number that describes movement of the plates.  > 0 corresponds to the plates moving apart, while   < 0 corresponds to the collapsing movement.Using (8), the boundary conditions for the problem are reduced to When   =   = 0 and  → ∞, the current problem is reduced to the problem discussed in [34].The skin friction coefficient is defined as [35] In terms of (8), we have where

Basic Theory of Homotopy Perturbation Method
The basic theory of HPM can be exhibited using the following differential equation: where  is an unknown and () is a known function, , ,  are linear, nonlinear, and boundary operators, and Υ is the boundary of the domain Ω.A homotopy (, ) : Ω×[0, 1] → R is then constructed which satisfies where  ∈ [0, 1] is an embedding parameter and  0 is the initial guess which satisfies the boundary conditions.From (14), we have Thus, as  varies from 0 to 1, the solution (, ) approaches from  0 () to w().To obtain an approximate solution, we expand (, ) in a Taylor series about  as follows: Setting  = 1, the approximate solution of (25) would be Substituting ( 17) into (13) will give If  = 0, then Ũ will be the exact solution but usually this does not happen in nonlinear problems.

Implementation of HPM to Squeezing Flow of Casson Fluid
Using ( 9) and ( 10), various-order problems are presented as follows.
Zeroth-order problem is Solution of the zeroth order problem is

Mathematical Problems in Engineering
First-order problem is Solution of the first-order problem is Second-order problem is Solution of the second-order problem is where Φ 1  and Φ 2  are the coefficients of various powers of .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, By fixing values of ,   ,   , and   in ( 25) polynomial solution can be found.For instance, when  = 0.01,   = −0.2,  = 0.5, and   = 0.5, the third-order solution is therefore The residual error of the problem is

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 number   , Casson parameter , MHD parameter   , and the permeability parameter   .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 Table 2. 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,   = 0.5,  = 0.05, and the squeeze number is varied as −0.2 ≤   ≤ 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.
Numerical values of skin friction coefficients corresponding to various fluid parameters are given in Table 1.Analysis of these numerical quantities show that increase in  decreases the skin friction coefficient.Furthermore, increase in   ,   , and   increases the skin friction coefficient.It is also observed that increase in   and   increases the skin friction coefficient.
The effects of identified parameters on the velocity profile are illustrated graphically in Figures 3-10.The effect of negative   on the velocity profile is shown in Figure 3, where it is shown that the normal velocity profile increases with the increase in negative   .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 <  ≤ 0.45 and decreases when 0.45 <  ≤ 1.
In the next three figures, the effect of various fluid parameters on the velocity profile with   < 0 is depicted.First, the effect of Casson parameter  is shown in Figure 4, showing that the normal velocity increases as  is increased,  S q = −0.1 S q = −3.0S q = −6.0S q = −8.0whereas the radial velocity increases near the lower plates and terminates near the upper plate.The effect when  is increased is similar to that of increasing negative   .Secondly, the effect of MHD parameter   is depicted in Figure 5. Here, it is seen that the normal component of the velocity decreases with the increase in   , while the radial component decreases near the lower plate and terminates near the upper plate.Moreover, the radial component of velocity decreases when 0 <  ≤ 0.45 and increases when 0.45 <  ≤ 1. Lastly, the effect of permeability parameter   is shown in Figure 6.Here, the normal velocity decreases when   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   and   is similar on the velocity profile in case of negative squeeze number.
The effect of   > 0 on the velocity profile is also investigated.In Figure 7, it is shown that the normal component of velocity decreases as   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 <  ≤ 0.45 and increases in 0.45 <  ≤ 1.The next three figures show the effect of ,   , and   in case of positive squeeze number.First, the effect of  is illustrated in Figure 8, showing that the normal component of velocity decreases as  is increased, while the radial component decreases near the lower plate and terminates near the upper plate.The effect of   is shown in Figure 9, while that of   is shown in Figure 10.In both cases, a similar effect as that of  can be observed.
In summary, it can be observed that positive and negative   have opposite effects on the velocity profile.Moreover,  shows opposite effect on the velocity profile in case of positive and negative   .However,   and   have similar effects on the velocity profile irrespective of the sign of the squeeze number.It is also observed that   ,   , and   have similar effect while  has opposite effect on the skin friction coefficient.

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 fourthorder 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.

Figure 3 :
Figure 3: Effect of negative squeeze number   on the velocity profile.

Figure 4 :
Figure 4: Effect of Casson parameter  on the velocity profile when   is negative.
1.0 M g = 4.0 M g = 8.0 M g = 12 (b) Radial component of velocity

Figure 5 :
Figure 5: Effect of MHD parameter   on the velocity profile when   is negative.
0.1 M p = 5.0 M p = 10 M p = 15 (b) Radial component of velocity

Figure 6 :
Figure 6: Effect of permeability constant   on the velocity profile when   is negative.
1.0 S q = 4.0 S q = 8.0 S q = 12.0 (a) Normal component of velocity Radial component of velocity

Figure 7 :
Figure 7: Effect of positive squeeze number   on the velocity profile.

Figure 8 :
Figure 8: Effect of Casson parameter  on the velocity profile when   is positive.

Figure 9 :
Figure 9: Effect of MHD parameter   on the velocity profile when   is positive.
Radial component of velocity

Figure 10 :
Figure 10: Effect of permeability constant   on the velocity profile when   is positive.

Table 1 :
Skin friction coefficient for various values of fluid parameters.

Table 2 :
Various-order homotopy perturbation solutions along with absolute residual errors for fixed values of fluid parameters.

Table 3 :
Comparison of analytical and numerical solution for various   .