Exact solution for heat transfer across the Sakiadis boundary layer

TL;DR

This paper derives an exact analytical solution for the thermal boundary layer in convective heat transfer over a moving surface, extending previous hydrodynamic solutions to include temperature effects for all Prandtl numbers.

Contribution

It develops an exact solution for the thermal boundary layer based on the Sakiadis flow, incorporating Prandtl number effects and providing a composite solution with high accuracy.

Findings

Exact solution for temperature gradient expressed as an integral of flow solution

Large Prandtl number expansion obtained using Laplace's method

Composite solution accurate to 10^{-10}

Abstract

We consider the problem of convective heat transfer across the laminar boundary-layer induced by an isothermal moving surface in a Newtonian fluid. In previous work (Barlow, Reinberger, and Weinstein, 2024, \textit{Physics of Fluids}, \textbf{36} (031703), 1-3) an exact power series solution was provided for the hydrodynamic flow, often referred to as the Sakiadis boundary layer. Here, we utilize this expression to develop an exact solution for the associated thermal boundary layer as characterized by the Prandtl number ($\Pr$) and local Reynolds number along the surface. To extract the location-dependent heat-transfer coefficient (expressed in dimensionless form as the Nusselt number), the dimensionless temperature gradient at the wall is required; this gradient is solely a function of $\Pr$, and is expressed as an integral of the exact boundary layer flow solution. We find that the exact solution for the temperature gradient is computationally unstable at large $\Pr$, and a large $\Pr$ expansion for the temperature gradient is obtained using Laplace's method. A composite solution is obtained that is accurate to $O(10^{-10})$. Although divergent, the classical power series solution for the Sakiadis boundary layer -- expanded about the wall -- may be used to obtain all higher-order corrections in the asymptotic expansion. We show that this result is connected to the physics of large Prandtl number flows where the thickness of the hydrodynamic boundary layer is much larger than that of the thermal boundary layer. The present model is valid for all Prandtl numbers and attractive for ease of use.