Exploration of Unsteady Squeezing Flow through Least Squares Homotopy Perturbation Method

Squeezing flow has many applications in different fields including chemical, mechanical, and electrical engineering as these flows can be observed in many hydrodynamical tools and machines. Due to importance of squeezing flow, in this paper, an unsteady squeezing flow of a viscous magnetohydrodynamic (MHD) fluid which is passing through porous medium has been modeled and analyzed with and without slip effects at the boundaries. +e least squares homotopy perturbation method (LSHPM) has been proposed to determine the solutions of nonlinear boundary value problems. To check the validity and convergence of the proposed scheme (LSHPM), the modeled problems are also solved with the Fehlberg–Runge–Kutta method (RKF45) and homotopy perturbationmethod (HPM) and residual errors are compared with LSHPM. To the best of the authors’ knowledge, the current problems have not been attempted before with LSHPM.Moreover, the impact of different fluid parameters on the velocity profile has been examined graphically in slip and no-slip cases. Analysis shows that the Reynolds number, MHD parameter, and porosity parameter have opposite effects in case of slip and no slip at the boundaries. It is also observed that nonzero slip parameter accelerates the velocity profile near the boundaries. Analysis also reveals that LSHPM provides better results in terms of accuracy as compared to HPM and RKF45 and can be effectively used for the fluid flow problems.


Introduction
In squeezing flow, fluid is squeezed between two parallel objects.
is phenomenon can be observed in many hydrodynamical machines and tools. One can observe the application of these flows in food industry and chemical engineering [1][2][3][4]. Few common examples are polymer handling, compression and injection molding, and modeling of lubrications. Modeling of squeezing flow was initiated around nineteenth century, and from then onward, it got attention because of its endless implementations in various fields. e initial studies in these flows were done by Stefan [5], and he worked on Newtonian fluid and found an asymptotic solution.
e study of fluids with electromagnetic field is an exceptionally interesting area of magnetohydrodynamics (MHD). Utilizing MHD fluid as lubricant is intriguing, since under certain extreme conditions, it averts the unpredicted change of lubricant viscosity with temperature. Maki et al. [6] investigated the MHD fluid as lubricant in an externally pressurized thrust bearing. Different authors have examined the effects of magnetic field in the flow of fluids [7][8][9][10]. e squeezing flow between two disks together with the presence of a magnetic field was studied in [11], and that between the rotating disks was studied in [12,13]. A material that has fluid-filled pores is known as porous medium. It is described by properties which are permeability and porosity. e amount of fluid which passes through the porous medium is known as permeability, while porosity is characterized as the sum of fluid held by the fabric. Porous medium has several applications in the field of petroleum, reservoir, and chemical engineering [14][15][16][17][18].
Slip is difficult to calculate or observe directly in a porous medium [19]. us, most observations rely upon theoretical analysis and numerical methods. e affirmation of slip has been confirmed experimentally through indirect approaches [20]. Considering slip results in increasing error and computational time, which is why it is ignored in many theoretical analyses [21,22].
At a fluid-solid interface, the effect of slip is an established boundary condition. It has various applications in science and technology, for instance, material processing, rheometric measurements, and fluid transportation [20,23]. e boundary condition of a viscous fluid at a macroscopic level is considered to be no-slip condition [21] which means that the instantaneous velocity would be zero at the boundary. Nonetheless, noninstantaneous slip conditions because of shear stress have also been investigated by the implication of unexpected decrease in viscosity or increase in flow rates [20]. is slip condition is known as the apparent slip in fluid mechanics.
Mooney presented the pioneering work in slip analysis [24]. He determined explicit formulas for fluidity and slip. e techniques for dealing with slip condition are significant for investigation of fluid flow; however, it is generally underestimated or ignored in the investigations including complex fluid systems [25].
In this paper, numerical and seminumerical solutions of an unsteady squeezing flow of viscous fluid with MHD and porosity effects have been analyzed with slip and no-slip conditions at the boundary. Problems have been solved through LSHPM which is the modification of well-known HPM [26]. LSHPM is the coupling of HPM along with least squares optimizer [27][28][29]. e main feature of LSHPM is accelerated convergence with less computational cost. For validation of the LSHPM results, these problems were also solved with HPM and Fehlberg-Runge-Kutta method (RKF45), and the results were compared with LSHPM. Error analysis is performed in this study via tables and graphs. e remaining sections are organized as follows. Mathematical formulation is given in Section 2. e basic concept of LSHPM is given in Section 3. e application of LSHPM on modeled problems is given in Section 4. Results and discussion are given in Section 5. Finally, the conclusion is presented in Section 6.

Problem Formulation
Let us consider a rectilinear, unsteady magnetohydrodynamic squeezing flow of an incompressible viscous fluid passing through a porous medium between two infinite parallel plates. e distance between the plates is always 2α(t) at any time t. e x-axis is considered the central axis of the channel, and the y-axis is perpendicular to the channel. β � β(0, β 0 , 0) is a uniform magnetic field that acts along the y-axis. e induced magnetic field is considered negligible, and the direction of the magnetic field is perpendicular to the flow of the fluid and has a constant strength H 0 . In fact, β 0 � H 0 μ 0 , where μ 0 is the magnetic permeability. Furthermore, the plate is considered to move symmetrically about the central axis of the channel.
Following are the mass and momentum conservation equations presenting the unsteady flow [30]: where u and v are the velocity components along x-axis and y-axis, respectively, and σ, ρ, and ] represent the electric conductivity, density, and kinematic viscosity, respectively. Let us define the vorticity function ω as and the generalized pressure h as By plugging (4) and (5) in (1)-(3), the above mass and momentum equations turn out be zh zy After eliminating pressure gradient using (7) and (8), the following is obtained: along with boundary conditions where v w (t) � dα/dt represents the velocity of plates. Let a dimensionless variable ξ � y/α(t), where 2α(t) is the distance between the plates at any time t. Here equations (6) and (9) have the following form: On u(x, ξ, t) and v(x, ξ, t), the boundary conditions are as follows: Velocity components are defined as [30,31] u By plugging (16) in (12) and (13), the continuity equation is identically satisfied and (13) becomes where the primes are representing the derivative of f with respect to ξ and M g and M p are the MHD parameter and the porosity parameter, respectively. We determine the boundary conditions from (14)- (16): erefore, for a similarity solution, let us define the following: where R and Q are the functions of t, but these functions are taken to be constants for a similarity solution. By integrating αv w /] � R, we obtain It follows from (13) and (14) that Q � −1, so (10) becomes where R denotes the Reynolds number. e boundary conditions in case of no slip subject to (18) and (19) are and in case of slip at the boundary, the corresponding conditions are where c is the slip parameter.

Basic Idea of Least Squares Homotopy Perturbation Method
Let us consider a general differential equation where L, N, and B represent the linear and nonlinear parts and boundary operator, respectively, and φ represents an unknown function while g(x) is known function. First of all, we construct a homotopy for (25) such that where p ∈ [0, 1] is the embedding parameter and L(φ 0 ) is the initial guess.
As p varies from 0 to 1, then φ 0 approaches φ(r). Taylor series expansion of η(r, p) about p is Journal of Mathematics Setting p � 1, the approximate solution of (25) would be After reassigning the dummy coefficients c i′ s in obtained series solution (29), we substitute the approximate solution φ back in equation (25) to get the residual function: Now we will compute the sum of squares of residuals: After computing J(c i ), we find the optimal values of c i ′ s from system of equations obtained from zJ/zc i � 0.
Putting these optimal values back into series (29), we will get our final series form solution of LSHPM.

Application of LSHPM to Squeezing Flow
Let us construct a homotopy for the given problem as follows: Using (32) and (23), various order problems along with solutions are given as follows.
e zeroth-order problem is e solution of (33) is e first-order problem is e solution of (35) is e second-order problem is Journal of Mathematics Now, 2 i�0 f i (ξ) gives a series form solution f(ξ) as It follows that (39) consists of ξ, ξ 3 , ξ 5 , ξ 7 , ξ 9 , ξ 11 . Let us assume that Now, we find the optimal coefficients of ξ i ′ s by using the least squares optimizer. By applying boundary conditions (23), the problem reduces to the following form: Putting c 1 and c 2 in f(ξ) gives Next, putting f(ξ) in (22) will gives us the following residual function.
After computing J(c 3 , c 4 , c 5 , c 6 ) � 1 0 R 2 (c 3 , c 4 , c 5 , c 6 )dξ, optimal values of c i ′ s can be obtained from zJ/zc i � 0. e same procedure is applied to (22) and (24) to calculate the second-order solution in case of slip at the boundary.

Journal of Mathematics
In addition, a graphical study has also been conducted to check the impact of different fluid parameters on the normal and axial velocity. Figures 5 and 6 show the effect of R on the velocity profile in case of no slip and slip at the boundary. As the Reynolds number is the ratio of inertial forces to viscous forces within a fluid, it has the ability to calculate scaling effects and can be used to help predict fluid behavior on a larger scale. Due to this, normal velocity has shown an increase with an increase in R, while axial velocity decreases near the central axis and increases near the wall when no-slip boundary condition is employed. R shows opposite effect when slip occurs at the boundary. Figures 7 and 8 present the effect of M g and M p on velocity profile in case of no slip. M g plays the role of a resistance contributed to by the magnetic pressure field component of Lorentz force. Due to this, the normal velocity decreases with an increase in M g , whereas the axial velocity increases near the wall but decreases near the central axis of the channel. M p has an inverse relationship with the permeability constant k, and hence it shows similar behavior as M g when keeping other parameters fixed. M g and M p show opposite effect in slip case which can be seen in Figures 9-11.

Conclusion
In this article, an unsteady squeezing flow of MHD fluid passing through a porous medium has been modeled and analyzed considering the cases of slip and no slip at the boundaries using LSHPM. For validation purpose, the modeled nonlinear problems are also solved through HPM and Fehlberg-Runge-Kutta method. e obtained results show the effectiveness of LSHPM over other schemes. Obtained solutions clearly show that LSHPM is a more consistent scheme in terms of accuracy with less computational cost as compared to the other stated schemes and can be employed in various fields of science and engineering.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.