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 number have opposite effect on the velocity profile. It is also observed that Casson parameter shows opposite effect on the velocity profile in case of positive and negative squeeze number . MHD parameter and permeability constant 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 , and . 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.
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, Section 2 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 distance . 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 . implies a receding motion of the plates. With these conditions, the non-Newtonian Casson fluid, using [32, 33], is defined aswhere 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 aswhere and are the velocity components in and directions, is the pressure, and are the viscosity and kinematic viscosity of the fluid, 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 getwhereThe similarity transform for a two-dimensional flow [34] iswhere . 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. corresponds to the plates moving apart, while corresponds to the collapsing movement. Using (8), the boundary conditions for the problem are reduced toWhen 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 .
3. 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 is the boundary of the domain . A homotopy is then constructed which satisfieswhere is an embedding parameter and is the initial guess which satisfies the boundary conditions. From (14), we have Thus, as varies from to , the solution approaches from to . To obtain an approximate solution, we expand in a Taylor series about as follows: Setting , the approximate solution of (25) would beSubstituting (17) into (13) will give If , then 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 Solution of the zeroth order problem is
First-order problem isSolution of the first-order problem is
Second-order problem is
Solution of the second-order problem is where and 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 , and , the third-order solution is therefore
The residual error of the problem is
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 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, , , and the squeeze number is varied as . 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 and decreases when .
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
(a) Normal component of velocity
(b) Radial component of velocity
In the next three figures, the effect of various fluid parameters on the velocity profile with is depicted. First, the effect of Casson parameter is shown in Figure 4, showing that the normal velocity increases as is increased, whereas 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 and increases when . 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 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 and increases in . 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.
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.
Appendix
are the coefficients of various powers of in first-order solution: are the coefficients of various powers of in second-order solution:
Competing Interests
There are no competing interests regarding the publication of this paper.