Magnetohydrodynamic unsteady separated stagnation‐point flow of a viscous fluid over a moving plate

Abstract

An analysis has been made for the unsteady separated stagnation-point (USSP) flow of an incompressible viscous and electrically conducting fluid over a moving surface in the presence of a transverse magnetic field. The unsteadiness in the flow field is caused by the velocity and the magnetic field, both varying continuously with time t. The effects of Hartmann number M and unsteadiness parameter β on the flow characteristics are explored numerically. Following the method of similarity transformation, we show that there exists a definite range of β(< 0) for a given M, in which the solution to the governing nonlinear ordinary differential equation divulges two different kinds of solutions: one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). We also show that below a certain negative value of β dependent on M, only the RFS occurs and is continued up to a certain critical value of β. Beyond this critical value no solution exists. Here, emphasis is given on the point as how long would be the existence of RFS flow for a given value of M. An interesting finding emerges from this analysis is that, after a certain value of M dependent on β (< 0), only the AFS exists and the solution becomes unique. Indeed, the magnetic field itself delays the boundary layer separation and finally stabilizes the flow since the reverse flow can be prevented by applying the suitable amount of magnetic field. Further, for a given positive value of β and for any value of M, the governing differential equation yields only the attached flow solution.S. Dholey: Magnetohydrodynamic unsteady separated stagnation-point flow of a viscous fluid over a moving plate Mathematical formulationWe consider an unsteady two-dimensional hydromagnetic laminar boundary layer flow of an incompressible viscous and electrically conducting fluid over a surface moving continuously in its own plane with a uniform speed u 0 (t). The physical model of this problem is shown in Fig. 1. Here the moving plate emerges from a virtual wall. This fixes the origin of the coordinate system, x-axis runs along the plate and y-axis is normal to it. The corresponding components of velocity are denoted, respectively by u and v, and the time is denoted by t. We allow the magnetic field applied in the direction normal to the plate satisfy a certain functional form for which this USSP flow will exist. Here the applied magnetic field is B(t 1 ) = B 0 / (t 0 − βt 1 ) j, where t 0 (= νt ref /l 2 ) and t 1 (= νt/ l 2 ) are the dimensionless forms of the constant reference C

Unsteady hydromagnetic flow and heat transfer of a viscous fluid near a suddenly accelerated flat surface

“…Here we assume that the magnetic Reynolds number R m is very small (i.e., R m ( 1). As a consequence, the applied magnetic field contributes only to the x-component of Lorentz force (ÀrB 2 u=q) in the momentum equation (see Dholey [19]), where r and q are electrical conductivity and density of the fluid, respectively. Finally, we assume that the entire flow field is exposed to thermal radiation.…”

Magnetohydrodynamics unsteady separated stagnation‐point (USSP) flow of a hybrid nanofluid on a moving plate

“…The Cartesian coordinates are embodied by axes 𝑥 (along the plate) and 𝑦 (normal to the plate). This plate continuously moves with velocity 𝑢 0 (𝑡) = 𝜕𝑥 0 (𝑡)∕𝜕𝑡 which turn up from a virtual wall where 𝑥 0 (𝑡) and 𝑡 refer to the displacement of the plate and time, respectively [8]. Further assumption is 𝑢 𝑒 (𝑥, 𝑡) denotes the free stream velocity which locates outside the boundary layer and parallel to the plate.…”

“…In addition, the denominator in the magnetic field expression must satisfy 𝑡 0 − 𝛽𝑡 1 > 0 or 𝑡 1 < (𝑡 0 ∕𝛽). Considering all the assumptions stated above and following the mathematical formulations given by Dholey [8], Waini et al [31] and Khashi'ie et al [32], the governing boundary layer equations are…”

“…Following Dholey [8], the definition of the similarity functions 𝑓(𝜂), 𝜃(𝜂) and the similarity variable 𝜂 to generate the similarity solution(s) for Equations ( 2)-( 4…”

“…Unlike the steady flow, Dholey [7] found F I G U R E 1 Physical model and Coordinate system the solution for the unsteady separated stagnation point (USSP) flow admits two solutions which are the attached flow solution (AFS) and the reverse flow solution (RFS). Later, Dholey [8] found that the magnetic field could stabilize the RFS whilst delay the separation of boundary layer of USSP flow. Some of the works on the topic of SSP flow with different surfaces are found in the papers published by [9][10][11][12][13][14].…”