Unsteady Self-similar Flow over an Impulsively Started Shrinking Sheet: Flow Augmentation with No Separation

Abstract

Abstract The unsteady self-similar flow due to a permeable shrinking sheet is analyzed in this investigation. The current theoretical study is enacted in the light of correct self-similar formulation proposed by (Mehmood. A. Viscous flows: Stretching and shrinking of surfaces, Springer, 2017). For the existence of a meaningful solution the retarded boundary-layer developed due to retarded shrinking wall velocity u w x = ± x t \left( {{u_w}\left( x \right) = {{ \pm x} \over t}} \right) is supported by the sufficient mass suction introduced at the shrinking surface. Main theme of this paper is to utilize the correct mathematical formulation for the shrinking sheet flow in order to investigate the existence of dual solutions of the considered problem. By using suitable self-similar transformations, the nonlinear Navier–Stokes equations are converted to corresponding non-linear ordinary differential equations. A reliable numerical technique (shooting method) is applied to solve the resulting ODEs for the involved physical parameters. Particular to this problem, it is observed that a sufficient amount of wall suction is mandatory S < S c ; S c ≤ 0 \left( {S \lt {S_c}; {S_c} \le 0} \right) for the existence of meaningful solutions; and also the solutions are non-unique. As the magnitude of wall suction is reduced (i. e. S → S c S \to {S_c} ) the solution seizes to exist by having non-zero coefficient of wall skin-friction, reflecting the absence of any flow separation. On the other hand, a stronger flow augmentation is noted for the increasing values of the wall suction parameter S S . The results are presented through graphs as well as in tabulated form. It is noticed that dual solutions exist for sufficiently large magnitude of the suction parameter S > S c \left( { \left| S \right| \gt \left| {{S_c}} \right| } \right) .