A Novel Design of Three-Dimensional MHD Flow of Second-Grade Fluid past a Porous Plate
In this study, a novel theoretical model for three-dimensional (3D) laminar magnetohydrodynamic (MHD) flow of a non-Newtonian second-grade fluid along a plate of semi-infinite length is developed based on slightly sinusoidal transverse suction velocity. The suction velocity involves a steady distribution with a low superimposed perpendicularly varying dispersion. The strength of the uniform magnetic field is incorporated in the normal direction to the wall. The variable suction transforms the fluidic problem into a 3D flow problem because of variable suction velocity in the normal direction to the plane wall. The proposed mathematical modeling and its dynamical analysis are prescribed for the boundary layer flow keeping the magnetic effects without taking into consideration the induced magnetic field. An analytical perturbation technique is employed for the series solutions of the system of ordinary differential equations of velocity profile and pressure. Graphical illustrations are used to analyze the behavior of different proficient parameters of interest. The magnetic parameter is responsible for accelerating the main flow velocity, while controlling the cross flow velocities.
The electrically conducting fluid flows due to the effect of the transverse magnetic field have received attention of a large section of research community because of their use in engineering, astrophysics, and geophysics and in the field of aerodynamics to control the boundary layer. In industries, the induced magnetic field-based procedures are used for excursion, heat exchanger, pump, and levitate liquid metals. Gersten and Gross  investigated the heat transfer effects on the three-dimensional fluid flows based on main and cross flow velocity components along a porous plane wall based on the transverse sinusoidal suction velocity distribution. Singh et al.  analyzed the problem of the 3D porous medium based on convective flow. The authors also incorporated heat transfer effects in the fluidic problem. Furthermore, Singh [3, 4] analyzed the hydromagnetic and magnetohydrodynamic impacts on 3D free convective fluid flow. The authors considered a vertical porous wall for the impact of porosity on the flow problem. Furthermore, the effect of the magnetic field with unvarying strength is incorporated perpendicular to the free convective fluid flow which is electrically conducting through a semi-infinite plate . Later on, Helmy  discussed the heat transfer impact on the flow of an electrically conducting fluid across an infinite plane wall with variable suction.
Researchers have extensively exploited the field of magnetohydrodynamics with reference to geometrical configuration, various types of formulations, and different dimensions using analytical and numerical methods [7–19]. The authors investigated the influence of the magnetic field on different structures of flow dimensions for distributions of concentrations and temperatures. Moreover, the magnetic parameter is proved to be a controlling parameter to restrain the fluid and heat flows under the closed spaces. Different types of non-Newtonian fluids bargain applications in many magnetohydrodynamic devices and in power generation as well. Magnetohydrodynamics with different types of convective flows, heat generation/absorption, and transfer analysis [20–25] have been investigated in detail.
Gupta and Johari  presented the analysis of 3D magnetohydrodynamic flow across a porous plane wall, and the fluid taken into account is laminar having the power of conducting heat. The authors considered a magnetic flux in the normal direction to the plate. Moreover, Guria and Jana  considered a vertical porous wall for the three-dimensional hydrodynamic fluid flow problem. Furthermore, Greenspan and Carrier , Rossow , and Singh  presented their studies extensively on the magnetohydrodynamic impacts on the flow across a plane wall based on injection or suction. There are some non-Newtonian models presenting the three-dimensional fluidic problems though a porous wall with variable suction [31–35]. Periodic suction has received very much attention in the field of aerodynamics [36–38]. The porous medium has very much importance in the fluid dynamics [39–45]. Abbas et al.  studied 3D peristaltic flow fluid with hyperbolic tangent in the nonuniform channel along flexible walls. Bhatti and Lu  studied analytical analysis of the head-on collision mechanism among hydroelastic solitary waves with uniform current. Jhorar et al.  analyzed the microfluid in the channel for the electroosmosis-modulated biomechanical transport.
The motivation of this study is to analyze the 3D MHD flow of the simplest non-Newtonian second-grade fluidic model through a semi-infinite wall which is based on succession of waves or curves with fluctuating velocity distribution. A uniform suction velocity along the surface of the plane wall transforms the problem into a 2D asymptotic suction velocity solution ; however, because of the variable suction velocity distribution in the normal direction, the fluidic system turns out to be 3D. The analytical perturbation technique is incorporated for finding the series solution of this problem. The proposed outcomes are estimated for different parameters of interest such as the suction parameter , second-grade parameter , Reynolds number , and Hartmann number . This article is organized as follows: Section 2 consists of the statement of the problem, Section 3 presents the perturbation method, Section 4 specifies the design of the problem, Sections 5 and 6 describe analytical solutions, Section 7 integrates results and discussion, and Section 8 contains conclusions. In addition to these, appendices and nomenclature are given thereafter.
2. Statement of the Problem
In this problem, the 3D steady, laminar magnetohydrodynamic flow of an incompressible non-Newtonian second-grade fluid passing through an infinite plate is considered. The plane is considered where -axis is normal to the plane (see Figure 1). A time-independent distribution is a basic steady distribution , in which l and indicate the wavelength and amplitude of the variable suction velocity, respectively. Thus, variable suction velocity has the following form:
The fluidic system becomes two-dimensional because of constant suction velocity, whereas it is three-dimensional in case of variable suction velocity . A constant magnetic field in the normal direction to the wall is applied. Also, the following are considered: (i) the fluid has electric conduction; (ii) the fluid has steady and laminar flow; (iii) the fluid has uniform free stream velocity; (iv) the magnetic Reynolds number is at small scale, and also the induced magnetic field is inconsiderable; (v) Hall and polarization effects are neglected; and (vi) all physical properties of the parameters are independent of because of the infinite extended length of the plate in the direction, but the flow remains 3D because of the variable suction velocity (1).
3. Perturbation Method
Perturbation methods [50–52] are strong mathematical tools to find the series/approximate solutions of those problems whose analytic or exact solutions are not possible or hard to find. These techniques have frequently been used for the problems arising in the fields of engineering and science. The function is convoluted in physical problems, and then it can be shown mathematically by the differential equationsubject to the boundary conditionwhere is a vector or scalar independent variable and is a parameter. One cannot solve this problem exactly, in general. However, if there exists an ( can be scaled so that ) for which the above problem can be solved exactly, then one explores to obtain the solution for small in the formwhere does not depend on and is the solution of the problem for . One then substitutes this expansion into equations (2) and (3), which expands for small and collects coefficients of each power of . Since these equations must hold for all values of , each coefficient of ϵ must vanish independently because sequences of ϵ are linearly independent. These usually are simpler equations governing which can be solved successively.
4. Design of the Problem
The equations of continuity and momentum are presented in the following way:with the boundary conditions 
Now the following dimensionless parameters are introduced :
Since is a small number, solutions are assumed as follows:
For , the problem becomes two-dimensional because of constant suction velocity given in equation (1), which is resulted as follows:subject to boundary conditions
Consider the following form of the solution:where is a small elastic parameter. Using equation (20) in equations (18) and (19) and correlating the coefficients of and , the following boundary value problems are obtained:
The corresponding conditions on the boundary (15) take the form
5. Cross Flow Solution
The cross flow velocity components and along with pressure are considered and presented in the following way:
The conditions on the boundary of the plate become
We assume that
Then, the corresponding conditions on the boundary take the form
6. Main Flow Solution
Then, the conditions on the boundary of the plate are reduced to
Furthermore, it is assumed that
Then, the analogous boundary conditions (30) are
It should be noted that the limiting velocity as and differs from that computed by Gersten and Gross . This is because of some calculation mistakes in their work.
7. Results and Discussion
The 3D steady, laminar MHD flow of incompressible second-grade fluid across a horizontal plate with infinite length subjected to variable suction is analyzed. A well-known perturbation technique is employed to solve the governing equations for the velocity profile and pressure. Graphical and tabular illustrations are used to analyze the behavior of different proficient parameters of interest.
7.1. Main Flow Velocity Field
The velocity profiles are presented for dimensionless parameters for the dynamics of the present flow problem such as the suction parameter α, second-grade parameter , Reynolds number , and magnetic parameter . The ranges of the parameters of interest appearing in the model are considered according to the adjustment of physical quantities in the present fluidic problem. The values of the suction parameter are small because of the boundary layer region which is close to the plane wall. Since the holes in the semi-infinite plate vary in size and shape, variable suction velocity distribution is considered close to the region of the plate, but the value of suction velocity becomes uniform when one moves in the region away from the plane wall. These proposed variations are presented in Figures 2–5. The impact of the suction variable α on the main velocity component u is shown in Figure 2. The component of velocity u decreases with the increase of α. Figure 3 shows the influence of the second-grade parameter on the velocity in the main flow direction . It is shown that the magnitude of this flow velocity increases near the plate, but a reverse trend is noticed when one goes away from the plane wall. Figure 4 exhibits the impact of the magnetic parameter on the velocity component based on the main flow direction . In Figure 4, it can be seen that the velocity based on the main flow direction is accelerating function of the magnetic parameter . Figure 5 depicts that the main flow velocity component retards in the neighborhood of the plate as increases, and a reverse trend is seen as its position moves away from the plate. Furthermore, as .
7.2. Cross Flow Velocity Field
The velocity profile in the direction of cross flow is presented for dimensionless parameters for the dynamics of the present flow problem such as the suction parameter α, second-grade parameter , Reynolds number , and magnetic parameter . These proposed variants are presented in Figures 6–9. The impact of the suction variable α on the cross flow velocity component is shown in Figure 6. The component of velocity decreases near the surface of the plate, but a reverse impact is observed when one enters the region away from the plate because of the suction velocity parameter α. Figure 7 shows the impact of the elastic parameter K on the velocity in the cross flow direction . It is shown that the dominant impact of the second-grade parameter K in the region close to the plate is seen, and it is also observed that cross flow velocity is decreasing function of the non-Newtonian parameter . It is interesting to see that Figures 8 and 9 reflect almost a similar impact of the magnetic parameter and Reynolds number on the cross velocity component. In both figures, the cross flow velocity accelerates as one moves in the region away from the plate. The impact of the suction parameter α, second-grade parameter K, and Hartmann number on the velocity component based on the cross flow direction is presented in Table 1. It depicts that increases as α increases. Also, the effect of on is noted. It decreases in the region close to the wall but increases away from the plate, and opposite behavior of cross flow velocity is observed for different values of Hartmann number. However, it decreases in the y direction.
8. Concluding Remarks
The 3D steady, laminar magnetohydrodynamic flow of an incompressible non-Newtonian second-grade fluid subjected to variable suction velocity is investigated. The key outcomes of this analysis are as follows:(i)The velocity component based on the main flow direction decreases with the increase of the suction parameter α.(ii)It is shown that the magnitude of the velocity component based on the main flow direction increases near the plate, but the main flow velocity decreases when one goes away from the plate.(iii)The main flow velocity is increasing function of the magnetic parameter .(iv)The limiting result of the velocity components as and is look-alike to that observed by Gersten and Gross  and also that computed by Singh  in the case of time independence.(v)The Newtonian outcomes  are retrieved when and .(vi)The main flow velocity of the fluidic system u declines near the plane wall as increases, and it accelerates as one moves away at a distance from the wall. Furthermore, as .(vii)The component of velocity decreases near the surface of the plate, but a reverse effect is seen when one enters the region away from the plate because of the suction velocity parameter α.(viii)A similar impact of the magnetic parameter and Reynolds number on the velocity component based on cross flow is observed.
|L:||Wavelength of suction velocity distribution|
|:||Uniform magnetic field applied in the direction|
|U:||Free stream velocity|
|:||The dimensional velocity components along directions|
|μ:||Coefficient of viscosity|
All the data used to support the findings of this research work are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
P. Singh, J. K. Mishra, and K. A. Narayan, “Three-dimensional convective flow and heat transfer in a porous medium,” Indian Journal of Pure and Applied Mathematics, vol. 19, no. 11, pp. 1130–1135, 1988.View at: Google Scholar
K. D. Singh, “Three-dimensional MHD free convection flow along a vertical porous plate,” Proceedings-Indian National Science Academy. Part A, Physical Sciences, vol. 57, no. 4, pp. 547–552, 1991.View at: Google Scholar
K. D. Singh, “Hydromagnetic free convective flow past a porous plate,” Indian Journal of Pure and Applied Mathematics, vol. 22, no. 7, pp. 591–599, 1991.View at: Google Scholar
A. A. A. A. Al-Rashed, K. Kalidasan, L. Kolsi et al., “Three-dimensional investigation of the effects of external magnetic field inclination on laminar natural convection heat transfer in CNT-water nanofluid filled cavity,” Journal of Molecular Liquids, vol. 252, pp. 454–468, 2018.View at: Publisher Site | Google Scholar
A. J. Chamkha and A. R. A. Khaled, “Hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid-saturated porous medium,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 10, no. 5, pp. 455–477, 2000.View at: Publisher Site | Google Scholar
P. S. Reddy, P. Sreedevi, and A. J. Chamkha, “MHD boundary layer flow, heat and mass transfer analysis over a rotating disk through porous medium saturated by Cu-water and Ag-water nanofluid with chemical reaction,” Powder Technology, vol. 307, pp. 46–55, 2017.View at: Publisher Site | Google Scholar
R. Tajammal, M. A. Rana, N. Z. Khan, and M. Shoaib, “Slip effect on combined heat and mass transfer in three dimensional MHD porous flow having heat,” in Proceedings of the 2018 15th International Bhurban Conference on Applied Sciences and Technology (IBCAST), pp. 635–644, IEEE, Islamabad, Pakistan, January 2018.View at: Google Scholar
Y. Swapna, M. C. Raju, R. P. Sharma, and S. V. K. Varma, “Chemical reaction, thermal radiation and injection/suction effects on MHD mixed convective oscillatory flow through a porous medium bounded by two vertical porous plates,” Bulletin of the Calcutta Mathematical Society, vol. 109, no. 3, pp. 189–210, 2017.View at: Google Scholar
K. Chand and N. Thakur, “Effects of rotation, radiation and Hall current on MHD flow of A viscoelastic fluid past an infinite vertical porous plate through porous medium with heat absorption, chemical reaction and variable suction,” The Journal of the Indian Mathematical Society, vol. 85, no. 1-2, pp. 16–31, 2018.View at: Publisher Site | Google Scholar
A. Chamkha, “MHD flow of a micropolar fluid past a stretched permeable surface with heat generation or absorption,” Nonlinear Analysis: Modelling and Control, vol. 14, no. 1, pp. 27–40, 2009.View at: Google Scholar
G. D. Gupta and R. Johari, “MHD three dimensional flow past a porous plate,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 3, pp. 377–386, 2001.View at: Google Scholar
V. J. Rossow, “On flow of electrically conducting fluids over a flat plate in the presence of a transverse magnetic field,” 1958, NACA Report No. 1358.View at: Google Scholar
K. D. Singh, “Three dimensional MHD oscillatory flow past a porous plate,” ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift fr Angewandte Mathematik und Mechanik, vol. 71, no. 3, pp. 192–195, 1991.View at: Google Scholar
M. Shoaib, M. A. Rana, and A. M. Siddiqui, “The effect of slip condition on the three-dimensional flow of Jeffrey fluid along a plane wall with periodic suction,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 39, no. 7, pp. 2495–2503, 2017.View at: Publisher Site | Google Scholar
S. Koganezawa, A. Mitsuishi, T. Shimura, K. Iwamoto, H. Mamori, and A. Murata, “Pathline analysis of traveling wavy blowing and suction control in turbulent pipe flow for drag reduction,” International Journal of Heat and Fluid Flow, vol. 77, pp. 388–401, 2019.View at: Publisher Site | Google Scholar
H. Schlichting, Boundary Layer Theory, Mcgraw-Hill Book Co., New York, NY, USA, 1968.
A. W. Bush, Perturbation Methods for Engineers and Scientists, CRC. Library of Engineering Mathematics, Boca Raton, FL, USA, 1992.
D. Armbruster, Perturbation Methods Bifurcation Theory and Computer Algebraic, Springer, Berlin, Germany, 1987.
M. H. Holmes, Introduction to Perturbation Methods, Springer-Verlag, Berlin, Germany, 2013.