Asymptotically-consistent analytical solutions for the non-Newtonian Sakiadis boundary layer
Authors: Nastaran Naghshineh, Nathaniel S. Barlow, Mohamed A. Samaha, Steven J. Weinstein
Abstract
The Sakiadis boundary layer induced by a moving wall in a semi-infinite fluid domain is a fundamental laminar flow field relevant to high speed coating processes. This work provides an analytical solution to the boundary layer problem for Ostwald-de Waele power law fluids via a power series expansion, and extends the approach taken for Newtonian fluids (Naghshineh et al., 2023) in which variable substitutions (i.e. gauge functions) are chosen to be consistent with the large distance behavior away from the wall. Contrary to prior literature, the asymptotic behavior dictates that a solution only exists in the range of power law exponents, $\alpha$, lying in the range $0.5 < \alpha \leq 1$. A presumed exact analytical solution is obtained here in the range of approximately $0.74 \leq \alpha < 1$, using a convergent power series with an asymptotically motivated gauge function. For power laws corresponding to $0.5 < \alpha<0.74$, the gauge function becomes ill-defined over the full domain, and an approximate analytical solution is obtained using the method of asymptotic approximants (Barlow et al., 2017). The approximant requires knowledge of two physical constants, which we compute a priori using a numerical shooting method. The values of these constants, and the numerical solution itself, are sensitive to domain length, and this sensitivity increases as $\alpha$ decreases. This demonstrates the utility of power series solutions, which by their nature are independent of domain length.
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