Two Models on the Unsteady Heat and Fluid Flow Induced By Stretching or Shrinking Sheets and Novel Time-Dependent Solutions

Abstract

This study sheds light on unsteady heat and fluid flow problems over stretching and shrinking surfaces, enriching our understanding of these complex phenomena. We derive two mathematical models using a rigorous approach. The first model aligns with the model commonly employed by researchers in this field, but its steady-state solution remains trivial. The second model, introduced in this work, demonstrably captures the physically relevant steady-state solutions of Sakiadis [B. C. Sakiadis, AIChE J., 7, 26 (1961)] and Crane [L. J. Crane, J. Appl. Math. Phys., 21, 645 (1970)] as well as Miklavcic and Wang [M. Miklavcic, C. Y. Wang, Q. Appl. Math., 46, 283 (2006)]. Notably, we introduce new similarity solutions for the temperature field specifically within the first model. We further demonstrate that a uniform wall temperature condition leads to the optimal heat transfer rate. While similarity solutions can be derived for specific cases with the second model, non-similar solutions may be necessary for more general scenarios. We discuss the implications of our analysis for stagnation-point flow and non-Newtonian viscoelastic fluid flow problems, illuminating future research directions in the open literature.