Existence of multiple solutions for magnetohydrodynamic flows of second-grade and Walter’s B fluids due continuously contracting flat sheet with partial slip

Abstract

In the present investigation, an attempt is performed to analyse the existence of multiple exact solutions for magnetohydrodynamic flows of second-grade and Walter’s B fluids due continuously contracting flat sheet with partial slip condition at the boundary. After transforming basic governing equations by suitable similarity transformations, single and double closed-form exact solutions of the flow are achieved on certain restrictions on flow variables of two aforesaid fluids. The impacts of magnetic field and partial slip on existence and non-existence of solutions are revealed. The changes of flow surface drag and velocity with various involved parametric values are also explored. For both viscoelastic type fluids, i.e., second-grade and Walter’s B fluids, double solutions exist for certain flow conditions. In second-grade case, if viscoelastic parameter exceeds a definite value, then double similarity solutions are achieved for any type of flow, i.e., slip flow or no-slip flow. For Walter’s B fluid similar results are obtained with unique similarity slip flow solution if magnitude of viscoelastic parameter exceeds certain fixed value. Importantly, the magnetic field and its MHD effect stabilize the uncertainty situation in existence of similarity solution for both fluids and produce unique solution for a suitable choice.

Keywords

• Multiple solutions
• Magnetohydrodynamic flow
• Second-grade and Walter’s B fluids
• Continuously contracting flat sheet
• Partial slip