Using group-theoretic method, an analysis is presented for a similarity solution of boundary layer equations which represents an unsteady two-dimensional separated stagnation-point (USSP) flow of an incompressible fluid over a porous plate moving in its own plane with speed u0(t). It is observed that the solution to the governing nonlinear ordinary differential equation for the USSP flow admits of two solutions (in contrast with the corresponding steady flow where the solution is unique): one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result of the analysis is that in the case of stationary plate (u0(t) = 0), after a certain value of the magnitude of the blowing d (<0) at the plate, only the AFS exists and the solution becomes unique. For a stationary plate (u0(t) = 0), the USSP flow is found to be separated for all values of d in both the cases of AFS and RFS. It is also observed that when u0(t) = 0, in the RFS flow with wall suction d (>0), there are two stagnation-points in the flow but in the presence of blowing d (<0), there is only one stagnation-point in the flow which moves further and further up with increase in |d|. Suction is shown to increase the wall shear stress while blowing has an opposite effect. Streamlines for an USSP flow when u0(t) ≠ 0 are also plotted. It is found that in this case, the USSP flow is not in general separated.

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

In this paper, we have investigated theoretically linear as well as weakly nonlinear stability of a viscous liquid film flowing down an inclined or vertical plane under the action of gravity. The classical momentum-integral method, which is applicable for small as well as large values of Reynolds number Re, has been used to formulate the single nonlinear free surface equation in terms of the dimensionless perturbed film thickness η(x,t). Using sinusoidal perturbation in the linearized part of the surface evolution equation, we obtain the stability criterion and the critical value of the wave number kc which conceives the physical parameters Re, inclination angle θ and Weber number We. However, the linear stability analysis reveals the stabilizing influence of We as well as the destabilizing influence of Re and θ on this flow dynamics. The multiple-scale analysis has been used to derive the complex Ginzburg–Landau type nonlinear equation for investigating the weakly nonlinear stability analysis. We demarcate all the four states of the flow in the Re-k (or θ-k)-plane which are found after the critical value of Rec (or θc) depending upon the values of the other parameters. A novel result which emerges from the nonlinear stability analysis is a simple relationship among the parameters k, Re, We, and θ. This relationship essentially gives us the conditions needed for the existence of an explosive unstable zone when (3Re−3cot θ−4ReWek2)= 0; otherwise, the flow system will be free from this zone. Indeed, this zone decreases with the increase in We, whereas it increases with the increase in Re and θ confirming the stabilizing role of We and destabilizing role of Re and θ as found in linear stability analysis.

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In this paper we have investigated the boundary layer analysis of an unsteady separated stagnation-point (USSP) flow of an incompressible viscous fluid over a flat plate, moving in its own plane with a given speed u t ( ) 0