Exact solutions for 2D boundary layer flow of two types of viscoelastic fluids and heat transfer on a permeable shrinking sheet with thermal radiation and variable surface temperature: existence of multiple solutions

Abstract

A study with existence of multiple exact solutions has its own importance, and if the study is performed for a viscoelastic fluid, then the attraction becomes very high. So, here an attempt is performed, where the flow of two types of viscoelastic fluids and heat transfer due to shrinking of a permeable sheet is described. In the energy flow, the impacts of thermal radiation and variable temperature of the wall are simultaneously undertaken. By using similarity approach, in addition to single solution, the dual and triple closed-form solutions of the flow field and temperature are obtained in some specific flow situations of two fluids, namely, second-grade and Walters liquid B. Dual solutions for the flow of second-grade fluid are detected only for stronger mass suction with smaller viscoelastic parameter of second-grade fluid and for both weaker and stronger mass suctions (for two ranges) with larger viscoelastic parameter and unique solution in case of mass injection. However, for the flow of Walters liquid B, boundary layer solution is attained for mass suction only, and most importantly, triple solutions for some mass suction range are found. Now for heat flow, an exceptional result is witnessed: if the wall temperature distribution is assumed suitably, then the heat transfer rate is not dependent on the fluid rheology. Also, influences of involved physical parameters on velocity, temperature and other important quantities are described through some graphs.