Multiple Solutions for an Unsteady Stretching Cylinder
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MULTIPLE SOLUTIONS FOR AN UNSTEADY STRETCHING CYLINDER G. D. Tabassuma,∗ , A. Mehmooda, M. Usmana , and A. Darb
UDC 532.6
Abstract: The current study is devoted to investigating the duality of the solution for an unsteady stretching cylinder flow subjected to wall normal suction. It is demonstrated by using numerical methods that the dual solutions exist for various values of the curvature parameter regardless of the presence or absence of wall suction. Keywords: dual solutions, stretching cylinder, unsteady viscous flow, suction, injection. DOI: 10.1134/S0021894420030165 INTRODUCTION The boundary-layer flow developed due to the motion of a continuous surface is an important class of boundary-layer flows, which makes a complete counterpart of viscous flows past finite-length bodies. Important contributions of Blasius [1] and Falkner and Skan [2] to the latter class helped a lot in fundamental understanding of boundary-layer flows. Sakiadis [3, 4] studied boundary-layer flows induced by moving continuous surfaces and experimentally proved that they also have a boundary-layer character. Contributions worth noting in this regard can be found in [5–8]. The most attractive feature of the flows due to moving continuous surfaces is the case of a variable wall velocity, which is impossible to consider unless the moving continuous surface is assumed to be flexible and stretchable. Because of this fact, these flows are involved in several technological applications in many production processes, e.g., in polymer industry. These flows enjoyed far more attention and praise when a reciprocal of the stretching surface flow was first considered by Miklavcic and Wang [9] in 2005. They assumed the wall velocity of the form uw (x) = −ax (a > 0) and named it as the shrinking sheet flow. The most interesting and attracting outcomes of that study [9] were the non-existence of the solution in the absence of sufficient wall suction and the existence of multiple solutions upon the provision of sufficient wall suction. With these surprising findings for a shrinking surface flow, several other researchers performed investigations in this field and produced hundreds of research papers, where the basic results of [9] were confirmed. However, we believe that such situations are not particular for the shrinking surface flows only; rather these facts can equally be seen in the stretching surface flows as well. Moreover, the non-existence of the solution for a shrinking surface flow is totally wrong. Recently, Mehmood [8] explored this fact and concluded that the solution for a stretching/shrinking surface flow ceases to exist in the situation where the stretching/shrinking wall velocity becomes sufficiently decelerated so that it is unable to cause any convective transport in the vicinity of the stretching/shrinking surface. More recently Mehmood and Usman [10] also explored a mathematical argument regarding the production of multiple solutions by the self-similar ordinary differential equations concerning these flows. They pointed out that the power-law exponent m of the stre
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