Unsteady separated stagnation-point flow of an incompressible viscous fluid on the surface of a moving porous plate

Abstract

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.

A simplified approach to the minimally invasive antral membrane elevation technique utilizing a viscoelastic medium for hydraulic sinus floor elevation

“…The continuous flow of the viscous biomaterial results in the gradual and continuous elevation of the membrane. The rheological mechanism that aids in the expression of lateral forces that reflect the membrane, instead of vertical forces that could potentially lead to perforation of the membrane, is the establishment of a stagnation point at the point where the infractured bony floor of the sinus comes in contact with the viscous putty [18].…”

Unsteady separated stagnation-point flow with suction towards a stretching sheet

“…Under these assumptions, the simplified two-dimensional boundary layer equations of this problem can be written as (see Ma and Hui [9] and Dholey and Gupta [15])…”

“…There are quite a number of studies on unsteady flow over a stretching sheet in Newtonian and non-Newtonian fluids, see for example [10][11][12][13][14], but none of them investigated the problem of unsteady separated stagnation-point flow. It has to be mentioned here that Dholey and Gupta [15] studied the unsteady separated stagnation-point flow of moving plate with suction. Its surface is moving with speed that is solely depended on time, whereas in our case, the sheet is stretched from the origin of the surface, where the velocity is in terms of time and spatial (the direction is along the surface).…”

Unsteady boundary layer separated stagnation-point flow towards a permeable shrinking sheet

“…Under these assumptions the basic equations of this problem can be written as (see Ma and Hui [2] and Dholey and Gupta [5] Here, u and v are the velocity components along the x-and y-axes, is the kinematic viscosity of the fluid, and ) , ( t x v w is the mass flux velocity that will be defined later. Following Ma and Hui [2], and Dholey and Gupta [5], we introduce the following similarity transformations…”

“…Fan et al [4] studied the unsteady stagnation flow and heat transfer problem towards a shrinking sheet but they did not study the case of separation flow. Recently, Dholey and Gupta [5] studied the unsteady separated stagnation-point flow of moving plate with suction where the velocity is depended on time. For our case, the velocity of the shrinking sheet is a function of two independent variables, i.e.…”