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Abstract

The magnetohydrodynamic flow of an incompressible, steady and electrically conducting fluid is considered in double lid-driven square cavity under an inclined magnetic field. The non-dimensional governing equations are solved numerically using a radial basis function (RBF) approximation meshless method in order to obtain the flow patterns. Hartmann number $(10 \leq M \leq 80)$ and the inclination angle of magnetic field $(0 ° \leq α \leq 90 °)$ are taken as a problem variables that produce various flow transformations by their variation for $S = - 1, 1$ . Streamlines are examined topologically, and it has been observed that the inclination angle and strength of magnetic field have different influence on the bifurcation of flow structures.

Keywords

Flow bifurcation MHD flow double-lid driven cavity radial basis functions (RBF)

Introduction

Lid-driven cavity problem is studied for many researchers as a benchmark problem, which can be classified in two types: single-lid or two sliding sided. Ghia et al.¹ employed the implicit multi-grid method to the lid-driven cavity problem, which is modeled by the Navier-Stokes (NS) equation. A novel fully implicit finite volume method is implemented to solve both steady and unsteady NS equations Sahin and Owens.² Choi and Balaras³ presented the application of dual reciprocity boundary element method to NS equations in terms of primitive variables. Turkyilmazoglu⁴ considered a driven flow motion by a dually moving lid (two active/passive stirrers) of a square cavity. Numerical simulations are carried out to investigate the streamlines and pressure contours and the mixing properties of the flow exploring a broad parameter space. In the double sided wall cavity problems, Kuhlmann et al.⁵ investigated the steady flow in a two sided lid-driven rectangular cavity. They found that two separate co-rotating vortices to the moving walls for low Reynolds number. The flow behavior in a two sided lid-driven cavity is simulated using Lattice Boltzmann method by Arun and Satheesh.⁶ They showed the impacts of Reynolds number on streamlines, velocity profiles and pressure contours. Chowdhury and Kumar⁷ investigated the flow behavior of non-newtonian fluids in a two sided lid-driven cavity with the use of stabilized finite element method. They observed that the aspect ratio of the domain and the direction of the moving lid affect vortices. Moreover, heat transfer in a double lid-driven cavity is conducted by many researchers. Oztop and Dagtekin⁸ studied the problem of steady state mixed convection in an enclosure with vertical two-sided lid-driven and differentially heated square cavity. They found that both Richardson number and direction of lid affect the fluid flow and heat transfer. Azizul et al.⁹ examined the heat line visualization of the mixed convection mechanism in a double lid-driven square cavity having a heated wavy wall. The numerical results are obtained for several values of Reynolds number, Richardson number, Prandtl number and oscillation number.

The effect of magnetic field on heat transfer in a two sided lid-driven cavity is analyzed by Sajjadi et al.¹⁰ They reported that the heat transfer declines with an increment of the Hartmann number for various Richardson numbers. Sivasankaran et al.¹¹ simulated magnetohydrodynamic (MHD) and heat transfer of Cu-water nanofluid in a porous cavity with a partial slip. They observed that the direction of moving wall is not influenced by flow and heat transfer subject to the slip parameter. Soomro et al.¹² examined the flow behavior of MHD mixed convection inside a water filled semicircular cavity containing a cylindrical obstacle. The results demonstrated that the presence of cold obstacle causes the splitting of water flow in two parts. The impact of inclination angle of the magnetic field on heat transfer is also analyzed by many researchers. Hussain et al.¹³ performed numerical simulations to determine the effect of inclination angle on the heat transfer of Al2O3–water nanofluid for mixed convection flows in a partially heated double lid driven inclined cavity. Kaladhar et al.¹⁴ figured out the inclined magnetic field effects on mixed convection flow by using spectral quasilinearization method. Finite element method is applied to solve laminar governing equations with buoyancy term by Hussain and Öztop.¹⁵ It is concluded that the inclination angle can be used as a control parameter for heat and mass transfer with different power law index. The comparison of single sided mowing lid and double sided moving lid is discussed in the impact of copper-water nanofluid.¹⁶ They implemented finite element method to find solutions for various ranges of distance among fins, Hartmann number, length of fins, Richardson number, and magnetic field inclination. Al Khasawneh et al.¹⁷ numerically surveyed on the gaseous flow in a rectangular micro channel subject to the inclined magnetic field. The results revealed that the temperature increases as Hartmann number increases, but decreases as Darcy number increases. Mansour and Bakier¹⁸ investigated the mixed convection phenomenon of hybrid nanofluids within an undulating porous cavity due to presence of double-moving lid. They concluded that the driven lids mounting at top and bottom surfaces enforces the forced convection dominance.

There are many studies on the investigation of flow bifurcations in different shape of cavities. Gürcan et al.¹⁹ considered Stokes flow in a rectangular driven cavity with double-lid to analyze eddy genesis and transformation by variation of the cavity aspect ratio A and speed ratio S. Gürcan²⁰ also investigated the effect of Reynolds numbers $(Re)$ on the streamline patterns and their bifurcations in a double-lid-driven cavity with free surfaces. Zhang²¹ analyzed the streamline topologies of Bingham fluid in a rectangular double-lid-driven cavity. He determined that in addition to the reduction in aspect ratio as in Newtonian flows, Bingham number B also governs the bifurcation of fluid. Sidik and Razali²² considered the incompressible, laminar flow inside a square cavity with two-sided moving in parallel to examine the effects of different speed ratios and Reynolds numbers on eddy genesis. Bilgil and Gürcan²³ employed the finite element method to the NS equations to analyze the flow structures in a lid-driven sectorial cavity. It is found that increase in Reynolds number changes the position of the bifurcation points. Sthavishtha et al.²⁴ depicted the structures of the flow in double sided cross-shaped lid-driven cavity. The movement and formation of vortices are captured with the variation of Reynolds number and oscillating wall motions. Flow pattern and eddy genesis in Z-shaped cavity is studied by Çelik.²⁵ He also analyzed the effect of height of the channel on Stokes flow behavior. It is clear from all these studies that flow bifurcations depend on both geometry of the problem and Reynolds number.

Radial basis function (RBF) approximation is a power-full numerical method to find the solutions of various types of PDEs such as flow problems, elasticity applications. Besides its efficiency,²⁶ its applicability is quite easy. Indeed, this method does not require lattice-type node layout and can be easily applied to regions with both uniform and curved boundaries.²⁷ Therefore, the RBF method has been attracted the attention of many researchers. In 1990, Kansa²⁸ introduced the RBF collocation method. Chinchapatnam et al.²⁹ solved the stream function-vorticity form of the Navier-Stokes equations by using the RBF approximation. They considered three model problems: lid-driven square cavity, rectangular cavity and backward-facing step channel. Colaço et al.³⁰ implemented RBF approximation to the MHD convection flow in a square cavity. Gürbüz and Tezer-Sezgin³¹ investigated the effects of applied magnetic field on Stokes flow in lid-driven cavity and backward-facing step channel. They also employed the RBF approximation to MHD convection flow in different channels.^32,33 Geridonmez and Oztop³⁴ adopted RBF to MHD natural convection in a cavity subjected to cross partial magnetic field. They reported that the augmentation of Lorentz force decreases the convective heat transfer. RBF-FD is implemented to steady MHD duct flow in a square annulus duct with an obstacle by Prasanna Jeyanthi and Ganesh.³⁵ They also analyzed the effects of size and shape of the obstacle. Moreover, RBF based algorithm is extended to three-dimensional application for different boundary value problems in Refs.^36–38

Based on the literature survey, we are motivated by the lack of study on the flow of an electrically conducting fluid under the uniform inclined magnetic field in a double-lid driven enclosure. The governing equations are solved iteratively by using the very popular meshless method RBFs. Then, the flow patterns for different lid case $(S = - 1, 1)$ are presented and the effects of controlling parameters such as strength of magnetic field $(M)$ and its orientation $(α)$ on the flow characteristic are investigated in detailed.

Governing equations with boundary conditions

The schematic representation of the problem is depicted, and the boundary values are presented in Figure 1.

Figure 1.

Schematic view of the flow field.

The cavity under the magnetic field with the inclination angle $α$ respect to the x-axis is double-lid. While the upper-lid moves to the right at a constant speed $u = 1$ , the lower one moves in the same direction (S = 1) or in the opposite direction (S = −1) at the same speed. We assume that the flow is steady and two dimensional; the fluid is Newtonian, incompressible and electrically conducting. This problem is modeled by the continuity and Navier-Stokes equations containing the Lorentz force term³⁹ which are

\nabla \cdot u = 0

(1)

Re (u \cdot \nabla) u = - \nabla p + \nabla^{2} u + M^{2} (J \times H)

(2)

where $u$ , $H$ , $J$ and p are the velocity field, magnetic field, current density and pressure, respectively. Non-dimensional variables $x \to xl, y \to yl, u \to u u_{l}, v \to v u_{l}, p \to p ρ ν u_{l} / l$ and the non-dimensional parameters are Reynolds number $Re = l u_{l} / ν$ and the Hartmann number $M = l μ H_{0} (σ / ρ ν)^{1 / 2}$ where l, $u_{l}$ , $H_{0}$ , $μ$ , $σ$ , $ρ$ and $ν$ are characteristic length, characteristic velocity (mean axis velocity), externally applied magnetic field intensity, magnetic permeability, electric conductivity, density and kinematic viscosity of the fluid, respectively. Stream function equation $\nabla^{2} ψ = - ω$ is derived from the continuity equation (1) by using the definition of stream function $u = \frac{\partial ψ}{\partial y}$ , $v = - \frac{\partial ψ}{\partial x}$ and the vorticity definition $(ω = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$ . The equation for the vorticity is obtained by differentiating of equation (2) with respect to x and y, and subtracting them each other. Thus, MHD equations (1) and (2) are also expressed in stream function $ψ$ and vorticity $ω$ form as

\nabla^{2} ψ = - ω

(3)

\begin{matrix} \nabla^{2} ω = Re (u \frac{\partial ω}{\partial x} + v \frac{\partial ω}{\partial y}) \\ + M^{2} (- \frac{\partial u}{\partial y} \overset{2}{\sin} α + (\frac{\partial v}{\partial y} - \frac{\partial u}{\partial x}) \sin α \cos α \\ + \frac{\partial v}{\partial x} \overset{2}{\cos} α) \end{matrix}

(4)

The unknown vorticity boundary conditions are obtained from the vorticity definition by using coordinate matrix $F$ which is explained in the numerical method section.

Numerical method: Radial basis function (RBF) approximation

RBF approximation^29,40,41 is a meshless method to find the approximate solution of the differential equation $Lu (x) = f (x)$ on a domain $Ω$ with boundary condition $Bu (x) = g (x)$ on $\partial Ω$ , where L is Laplacian and B is a boundary operator, respectively. The problem is discretized on a global set of collocation points $x_{1}, . . ., x_{N_{b}}$ on $\partial Ω$ , $x_{N_{b + 1}}, . . ., x_{N}$ in $Ω$ . In this method, the function f is approximated by RBFs as

f (x) \approx \sum_{j = 1}^{N} β_{j} ϕ (‖ x - x_{j} ‖)

(5)

and the approximate solution u is written as a summation of $Ψ_{j}$ ’s linked through $L Ψ = ϕ$

u (x) \approx \sum_{j = 1}^{N} β_{j} Ψ (‖ x - x_{j} ‖),

(6)

which satisfies the boundary condition

g (x) \approx \sum_{j = 1}^{N} β_{j} B Ψ (‖ x - x_{j} ‖) .

(7)

where ‖·‖ represents Euclidean distance between collocation points. The matrix-vector form of (5) and (7) is $A β = d$ where

[\begin{matrix} B Ψ {(‖ x - x_{j} ‖) |}_{x = x_{1}, . . ., x_{N_{b}}} \\ \begin{matrix} ϕ {(‖ x - x_{j} ‖) |}_{x = x_{N_{b + 1}}, . . ., x_{N}} \end{matrix} \end{matrix}] [\begin{matrix} β \end{matrix}] = [\begin{matrix} g \\ f \end{matrix}] .

(8)

Substituting $β$ into (6), the approximate solution is expressed $u = U A^{- 1} d$ where $U_{ij} = Ψ_{j} (‖ x_{i} - x_{j} ‖)$ , $1 \leq i, j \leq N$ . To avoid repetitive solving of large linear system, the solution can be rearrange as summation of the boundary condition and contribution of interior solution:

u = U A^{- 1} d

(9)

= [\begin{matrix} R_{1} & R_{2} \end{matrix}] [\begin{matrix} u_{bc} \\ f \end{matrix}]

(10)

= R_{1} u_{bc} + R_{2} f

(11)

where ${(R_{1})}_{N \times N_{b}}$ and ${(R_{2})}_{N \times N_{i}}$ are split matrices of $U A^{- 1}$ . To evaluate the contribution, we set $K = [\begin{matrix} 0 & R_{2} \end{matrix}]$ and $\hat{f} = {f_{i}}_{i = 1}^{N}$ . Thus, the approximated solution of the function is given as

u = R_{1} u_{bc} + K \hat{f}

(12)

The MHD equations (3) and (4) include space derivatives of the unknown functions which are approximated by similar procedure in (5)

u = F \tilde{β}

(13)

where the coordinate matrix $F_{ij} = ϕ_{j} (‖ x_{i} - x_{i} ‖), 1 \leq i, j \leq N$ . Differentiating both sides of (13) with respect to $x and y$ yields

\frac{\partial u}{\partial x} = \frac{\partial F}{\partial x} \tilde{β} and \frac{\partial u}{\partial y} = \frac{\partial F}{\partial y} \tilde{β}

(14)

Substituting the unknown coefficient which is obtained from (13) into (14) gives the approximation of the desired derivatives of unknowns as

\frac{\partial u}{\partial x} = \frac{\partial F}{\partial x} F^{- 1} u and \frac{\partial u}{\partial y} = \frac{\partial F}{\partial y} F^{- 1} u .

(15)

In the literature, different types of radial basis functions are used. Some of them and their corresponding particular solutions for $L = \nabla^{2}$ are given in the Table 1. Multiquadratics, inverse multiquadratics and Gaussian, which are the most preferred functions in calculations, contain predefined shape parameters. It is known that the solution of the RBF approximation depends on the shape parameter.²⁹ Some special algorithms are required to determine the optimum shape parameter for the desired solution. To avoid this problem, Gürbüz⁴¹ used polynomial RBF to find solution of flow problem. It is deduced that polynomial RBF gives the similar results obtained by multiquadratics RBF and takes less computational time. For that reason in this study, $ϕ = 1 + r$ , and $Ψ = \frac{r^{2}}{4} + \frac{r^{3}}{9}$ are given as basis functions for the non-homogeneous part and the solution, respectively.

Table 1.

RBFs and their corresponding particular solutions^40,41 (c is pre-defined constant).

$ϕ$	$Ψ$
(Multiquadratics) $\sqrt{r^{2} + c^{2}}$	$\frac{1}{9} (4 c^{2} + r^{2}) \sqrt{r^{2} + c^{2}} - \frac{c^{3}}{3} \ln (c + \sqrt{r^{2} + c^{2}})$
(Inverse MQ $1) / \sqrt{r^{2} + c^{2}}$	$\sqrt{r^{2} + c^{2}} - c \ln (c^{2} + c \sqrt{r^{2} + c^{2}})$
(Gaussian) $e^{- c^{2} r^{2}}$	$\frac{1}{4 c^{2}} \int_{c^{2} r^{2}}^{\infty} \frac{e^{- t}}{t} dt$

The iterative method is used for the solution of RBF system of the coupled equations (3) and (4). The computation starts with initial guess for $ω = ω_{0}$ , then the solution for the stream function $ψ^{n + 1}$ is obtained. The unknown boundary values of vorticity $ω_{bc}$ are obtained from the definition of the vorticity using the coordinate matrix F as $ω = \frac{\partial F}{\partial x} F^{- 1} v - \frac{\partial F}{\partial y} F^{- 1} u$ . Next, we solve the equation (4) with obtained boundary conditions for vorticity to get its new value $ω^{n + 1}$ . Relaxation parameter $δ = 0.01$ is used to accelerate convergence with

ω^{n + 1} = (1 - δ) ω^{n} + δ ω^{n + 1} .

(16)

Iteration continues until the relative error $\frac{{‖ ξ^{n + 1} - ξ^{n} ‖}_{\infty}}{{‖ ξ^{n + 1} ‖}_{\infty}} < ε$ holds for $ω$ and $ψ$ where $ε$ is assigned $ε = 10^{- 5}$ .

Preliminary on streamline topologies

In this section, streamline topology in two-dimensional incompressible flows is briefly explained. Streamlines are derived from the stream function ( $ψ$ ) such that

\overset{\cdot}{x} = u = \frac{\partial ψ}{\partial y} \overset{\cdot}{y} = v = - \frac{\partial ψ}{\partial x} .

(17)

In order to have local information about the flow structure around the critical point, which is often thought of $(0, 0)$ , $ψ$ is expanded to the power series:

ψ = \sum_{i + j = 1}^{\infty} a_{i, j} x^{i} y^{j}

(18)

Linear approximation of (17)−(18) leads the following dynamical system for the streamlines

(\begin{matrix} \overset{\cdot}{x} \\ \overset{\cdot}{y} \end{matrix}) = (\begin{matrix} a_{0, 1} \\ - a_{1, 2} \end{matrix}) + (\begin{matrix} a_{1, 1} & 2 a_{0, 2} \\ - 2 a_{2, 0} & - a_{1, 1} \end{matrix}) (\begin{matrix} x \\ y \end{matrix})

which has the determinant of the Jacobian matrix $| J | = 4 a_{2, 0} a_{0, 2} - a_{1, 1}^{2}$ . $| J |$ ’s positivity or negativity provides information about the structure of the stagnation point such that If $| J | > 0$ , the origin is the center, if $| J | < 0$ , it is the saddle (see Figure 2).

Figure 2.

Non-degenerate critical points: (a) center and (b) saddle.

Flow bifurcation is called a change in the flow’s structure or in the number of critical points in the flow pattern (e.g. from center to saddle or vice versa). Brøns and Hartnack⁴² investigated the bifurcation in view of a topological point near the degenerate critical point away from the boundary. They used normal form transformations to examine the higher terms of (18) and constructed bifurcation diagrams according to degrees of normal form. Some of them are shown in Figure 3. Saddle-node bifurcation, also called cusp, is the bifurcation of the degenerate point into a saddled center (Figure 3(a)). The term pitchfork bifurcation refers to the transformation of critical points from the center to the saddle (Figure 3(b)). To have more information and related procedures, see Brøns and Hartnack⁴² and the others.^43,44

Figure 3.

Bifurcation diagram for: (a) normal form of order 3 and (b) symmetric normal form of order 4.

Numerical results

Validation case: Single lid-driven cavity

This subsection presents the numerical results of the test problem using the RBF scheme with polynomial basis function. Obviously, the coupled equations (3) and (4) reduces to the Navier-Stokes equations by taking $M = 0$ . The code is validated by comparing the results for $Re = 100$ and $Re = 400$ with those obtained by Ghia et al.¹ They applied implicit multi-grid method to Navier-Stokes equations.

The flow domain $[0, 1] \times [0, 1]$ is discretized by uniform collocation points set in size of $41 \times 41$ . Figure 4 shows the streamlines and velocity profiles along horizontal and vertical center of the cavity for $Re = 100$ . The solution obtained for the $41 \times 41$ uniformly distributed collocation points is in good agreement with those for the set of $129 \times 129$ points by Ghia et al.¹ This shows the effectiveness of the RBF meshless method. In the case of $Re = 400$ , the velocity profile comparison in Figure 5 shows that more points may be needed compared to $Re = 100$ to have better output in the numerical process. However, the main vortex and Moffatt vortices⁴⁵ located at both corners of flow domain satisfactorily captured. Thus, the code validation is presented by the results that are in good agreement with the benchmark case.

Figure 4.

Single lid-driven cavity for $Re = 100$ : (a) streamlines and (b) comparison of the centerline velocities obtained from the present numerical simulation with those of Ghia et al.¹

Figure 5.

Single lid-driven cavity for $Re = 400$ : (a) streamlines and (b) comparison of the centerline velocities obtained from the present numerical simulation with those of Ghia et al.¹

In the test case, calculations are carried out by using the radial basis function without shape parameters. It can be derived from the results, even if the distance function with shape parameter is not used, the results show that the polynomial function can be preferred in the RBF approximation for flow problems with moderate Reynolds numbers. Thus, the rest of the computations in this study are carried out with polynomial RBF.

Double lid-driven cavity

Streamlines of the double lid-driven MHD flow exposed to the magnetic field at different angles are presented. Flow is driven by translating lids in the confined region. Then the results are analyzed in two cases: the lids move in opposite direction, i.e. $S = - 1$ , and in the same direction, that is, $S = 1$ with a fixed Reynolds number ( $Re = 100$ ) because of the domination of the Hartmann number in equation (4). In both cases, the ranges for the problem variables are $(10 \leq M \leq 80)$ and ( $0^{°} \leq α \leq 90^{°}$ ). The grid size is $41 \times 41$ for $M = 10, 30, 50$ , but more points are taken to obtain smooth flow pattern as the Hartmann number increases. Therefore, $51 \times 51$ collocation points are considered for $M = 80$ .

Case 1: Lids moving in the opposite

Figure 6 shows the effect of the Hartmann number and the angle of magnetic field on the streamlines. When the magnetic field is applied horizontally $(α = 0^{°})$ , for $M = 10$ the cavity consists of an unsymmetrical separatrix with two sub-eddies and saddle point. As the Hartmann number increases, a new vortex emerges in the center of the cavity. With a further increase in Hartmann number $(M = 80)$ , two outer eddies slide toward the lids, thus the central vortex gets larger and turns into a sub-eddies with saddle point by pitchfork bifurcation.

Figure 6.

Streamlines for different values of Hartmann number, inclination angle of magnetic field and $S = - 1$ .

The exposure of the flow field to the magnetic field at different angles has a strong effect on the flow patterns. Indeed, the streamlines are distorted by the change of the angle, but the flow structures remain topologically the same except for $M = 80, α = 30^{°}$ . Two separation bubbles form asymmetrically on the vertical walls. This structure appears frequently in cavity-type flows.^25,46 In such bifurcation, two saddle points on the wall and a center point in the flow is connected with each other by heteroclinic trajectory. When $α = 90^{°}$ , the effect of the angle decreases on the flow behavior. As the Hartmann number gradually increases, the separatrix first turns into a full recirculating vortex in clockwise, and then it bifurcates into a separatrix lying in the middle of the cavity because of the pitchfork bifurcation.

Computations are also carried out for different values of Reynolds number for fixed $α = 0 °$ and $M = 30$ to analyze the effect of Re. Figure 7 shows that flow structures are the same for values of Reynolds number increasing up to 500. Only difference is that the eddies near the top and bottom walls shift through right and left corners, respectively, due to the effect of the direction of walls. Since the flow topology remains the same for $Re \leq 500$ and our aim is to analyze the impacts of Hartmann number and inclination angle on the flow, we fixed $Re = 100$ in our computations.

Figure 7.

Flow structures for $Re = 200$ and $Re = 500$ with $S = - 1$ and fixed $α = 0^{°}$ , $M = 30$ .

Case 2: Lids moving in the same

In Figure 8, the variation of the flow patterns in the cavity under the effect of inclination angle of the magnetic field and Hartmann number is presented. For $α = 0^{°}$ and $M = 10$ , the flow is symmetric at its center, and there are two main vortices in the cavity isolated by the $ψ = 0$ streamline. They circulate in opposite directions relative to each other. As the strength of the magnetic field increases, for example, for $M = 50 and M = 80$ , the flow region contains four eddies by forming new eddies above and below the separation line. It is clear that the symmetric position of the flow exposed at different angles to the magnetic field is disturbed due to the Lorentz force term containing the angular terms in the governing equations. As this effect increases, it is observed that the flow weakens and fluctuates. Although the flow subject to either horizontal or vertical magnetic field has a similar flow topology for $M = 10$ , the bifurcation of the flow varies with the increasing Hartmann number. It is seen that the flow is squeezed through the moving walls and has a tendency to flatten with the increasing number of M as one should expect. In addition, when $α = 90^{°}$ , the center point on both sides of the cavity becomes a saddle point with two sub-eddies by the cusp bifurcation. Particularly for $M = 80$ , the flow is confined to the narrow space and the structure of the critical point is difficult to determine and requires special work. However, previous studies^42,47 shown that flow patterns turn into a separatrix enclosing three sub-eddies after sequence of pitchfork bifurcation.

Figure 8.

Streamlines for different values of Hartmann number inclination angle of magnetic field and $S = 1$ .

Figures 9 and 10 illustrate the influence of the inclination angle on the u- and v-profiles along the mid-points of the cavity. Along $x = 0.5$ , the velocity component u drops dramatically as the magnetic field rotates for both $S = - 1$ and $S = 1$ . However, the velocity component v increases as the magnetic field rotates from $0^{°}$ to $30^{°}$ at $y = 0.5$ . Especially, v takes its maximum value at $α = 45^{°}$ for $S = 1$ . Then, the velocity component in the y-direction approaches zero when the magnetic field is parallel to the y-axis. It is observed that the effect of inclination angle of the magnetic field on v velocity profile is more pronounced.

Figure 9.

Effect of inclination angle on velocity profile with fixed parameters $S = - 1, Re = 100, M = 50$ .

Figure 10.

Effect of inclination angle on velocity profile with fixed parameters $S = 1, Re = 100, M = 50$ .

Conclusion

A numerical study of MHD flow in a cavity which has two sliding edges (opposite $S = - 1$ and same $S = 1$ ) is conducted to investigate the influence of problem variables on the transformation of flow patterns by using the RBFs meshless method. The variation of flow structure is presented for several values of the Hartmann number and inclination angle of magnetic field. It is found that both the inclination angle and the strength of the magnetic field determine the type of bifurcation of the critical point. The rotation of the magnetic field affects the velocity components. The velocity component in the $y -$ direction attains maximum value at $α = 45^{°}$ . However, $u -$ velocity values become almost zero at the center of the cavity regardless of the direction of the magnetic field.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Ebutalib Çelik

Merve Gürbüz-Çaldağ

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Streamline analysis of MHD flow in a double lid-driven cavity

Abstract

Keywords

Introduction

Governing equations with boundary conditions

Numerical method: Radial basis function (RBF) approximation

Preliminary on streamline topologies

Numerical results

Validation case: Single lid-driven cavity

Double lid-driven cavity

Case 1: Lids moving in the opposite

Case 2: Lids moving in the same

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iDs

References