Integral Transform Method for a Porous Slider with Magnetic Field and Velocity Slip

: Current research is about the injection of a viscous fluid in the presence of a transverse uniform magnetic field to reduce the sliding drag. There is a slip-on both the slider and the ground in the two cases, for example, a long porous slider and a circular porous slider. By utilizing similarity transformation Navier-Stokes equations are converted into coupled equations which are tackled by Integral Transform Method. Solutions are obtained for different values of Reynolds numbers, velocity slip, and magnetic field. We found that surface slip and Reynolds number has a substantial influence on the lift and drag of long and circular sliders, whereas the magnetic effect is also noticeable.

Therefore, the goal of the current work is to examine the impact of slip and Reynolds number when there is a transverse magnetic field on the performance of the porous slider. Literature survey clearly indicates that no solutions have been given for the threedimensional flows of this type with slip and a uniform magnetic field. Hence, the goal of the current research is to analyze the performance of the porous slider in the presence of slip and Reynolds number with a constant magnetic field and to assess their effects on the components of velocity lift and drag. Structure of the article is arranged as follows: In the introduction, we presented a brief history of the problem of the porous slider and its application. In second and third sections we discussed the formulation of the problem and the method to be used to solve the resulting problems of the long and the CPS. This paper adopted an Integral Transform Method (ITM) [Faraz, Khan, Lu et al. (2019)] that comprises both Variation Iteration Algorithm-II (VIM-II) and Adomian Decomposition Method (ADM) to reduce the computational work. VIM-II is the improved form of Variation Iteration Algorithm-I (VIM-I), proposed by Faraz in 2010 [Faraz, Khan and Austin (2010)]. Apparently, the final formulation of the proposed method has great symmetry with the existing methods such as VIM-I [El-Sayed and El-Mongy (2018); Wazwaz and Kaur (2019)] and ADM [Patel and Meher (2016); de Vargas Lisbôa and Marczak (2018); Turkyilmazoglu (2019c)] but this method does not need to calculate the Lagrange multiplier separately and gives a direct formulation of VIM-II, which avoids the unnecessary calculations, which is one of the goals of our study is to introduce a new method based on integral transformation to cover the shortcomings of the VIM-I, VIM-II, and ADM for solving nonlinear boundary value problems given in Eqs. (7), (8), (9), (12), (28), (29) and (32). The LPS solved by ] involves unnecessary and repeated calculations. Similarly, section four five and six are related to problem formulation and solution of the long and the CPS respectively. Section seven is based on results and discussion and finally, the last part is the conclusion.

Problem formulation
As discussed above, in this study we will consider velocity slip condition. Navier introduced the slip condition for the first time as follows: In Eq. (1) tangential velocity u is proportional to the shear stress and tangential velocity are directly proportional to each other with H as the constant of proportionality which is actually a slip coefficient. In order to ignore the end effects, it is assumed that the gap between slider and ground is quite small as compared to the slider's lateral dimension. We tried to study both the CPS and LPS. with velocity components and elevated because of injection of fluid from its bottom with a magnetic field which is applied externally. In order to avoid the induced magnetic field formed by the movement of fluid, it is assumed that the magnetic Reynolds number is not very big. Furthermore, the induced and imposed electric field are supposed to be negligible, therefore the electromagnetic body force per unit volume simplifies B= 0, 0, B is the magnetic field. Under the above-stated assumptions and conditions, Navier Stokes equations take the following form: where ( , , ) u v w are the velocity components in Cartesian coordinate ( ) , , , x y z where ρ , p and υ are density, pressure and kinematic viscosity respectively. Law of conservation of mass is as follows: The system of equations in a steady, compressible, laminar boundary layer is composed of two fundamental equations. Those are the continuity equation and the momentum equation. The solutions of these equations, when solved simultaneously for a twodimensional boundary layer, are the velocity in the , x y and z direction ( ) , , u v w . The system of equations is a system of partial differential equations (PDE) and is usually difficult to solve. Therefore, sophisticated transformation methods, called similarity transformations are introduced to convert the original partial differential equation set to a simplified ordinary differential equation (ODE) set. To do so we introduce the following similarity transformations [Faraz, Khan, Lu et al. (2019)].
Here 1 H , 2 H and µ ρυ = are slip coefficients and viscosity respectively. Eqs. (10) and (11) gives where 1 are slip factors. Eqs. (7)-(9) and (12) will be solved by using ITM. We can deduce the expression for pressure from Eqs. (2)-(4) as follows: If we take 2l as the width of the slider with ambient pressure 0 ρ , then Eq. (13) gives The relationship between depth and lift can be expressed as follows: is a normalized factor. The relationship between depth and drag in the 1

Integral transform method
Let us assume the general nonlinear second-order differential equation Integrating Eq. (19) with respect toη from 0 α toη twice yields By using the second boundary condition we can evaluate Substituting Eqs. (24) into (22) results as ,

Circular porous slider
From Fig. 1(b), we can see the circular slider, where L the radius of the slider which we assume comparatively is bigger than the width. Since slider is levitated so we fix our axes on the slider such that the ground is moving with a velocity component in x − direction.
7 Results and discussion ITM has been applied to compute the solution of the problems given in Eqs. (7)-(9), (12),     For strip slider, the effect of Reynolds number, in case of slip and the magnetic field is shown in Figs. 5 to 13. It is observed that the velocity profile is very much changed. Figs. 6, 7, 9 and 10, 12 and 13 display that slips near the ground reduce the lateral velocity much more than slip on the slider. Moreover, increasing the magnetic parameter decreases the lateral velocity components further. Similarly, the effects of Reynolds number on typical velocity distribution is displayed in Figs. 14 to 23 for the circular slider. The behaviour of velocity profiles is similar for stipe and circular slider in case of no-slip (see Figs. 2,3,14 & 15). Also, velocity profiles are behaving in a similar fashion as stipe slider i.e., parabolic or linear for low Reynolds number and for large Reynolds number boundary layer formed near the surface. Figs. 16 to 23 determine the effect of the slip parameter on the velocity components corresponding to different values of the Reynolds number. These pictorial descriptions demonstrate that velocity profiles decrease with an increase in slip parameters and this decrease become even further after applying the magnetic field. This is due to the fact that slip hinders the fluid particles and displays the motion in the vicinity. These results qualitatively agree with expectation since the application of a transverse magnetic field normal to the lateral flow directions has a tendency to create a drag-like Lorentz force. This force decreases the lateral velocity components. Lift and drag components are important physical quantities for a porous slider. It is interesting to note that the lift is free of translation, but the drag components and depend on the crossflow.
The effectiveness of a porous slider can be enhanced by making the ratio of friction force to lift smaller. As pointed out by ], porous slider should be operated at cross-flow Reynolds number less than unity for optimum efficiency. Tab. 1 shows that the fact that porous sliders should be operated at small values of still remains valid even when an external uniform magnetic field is applied. Moreover, from the optimum efficiency point of view, it is more efficient to move a flat slider on a fluid subject to a magnetic field with high intensity.

Conclusions
In this study, we have compiled different studies altogether. Different researchers have analyzed fluid flow on stripe slider, without slip and some were interested only in circular slider without slip. Wang presented the comparative study of both stripe and circular slider and added velocity slip but did not cover the effects of the magnetic field. We were concerned with a theoretical investigation of the steady three-dimensional flow of a viscous fluid between a porous slider and ground in the presence of a transverse uniform magnetic field with velocity slip. The effects of values physical parameter like Reynolds number and magnetic parameter on the lateral velocity profiles, lift and drag components were presented in graphical and tabular form in the presence of velocity slip. It is hoped that the results of the present study would be useful for the understanding of various technological problems related to porous sliders, where magnetic and velocity slip are the main physical parameters.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.