Shallow current of viscous fluid flowing between diverging or converging walls

TL;DR

This paper studies shallow viscous fluid flows in channels with power-law widening or narrowing, deriving similarity solutions and analyzing convergence, revealing no divergence in the shallow case and an exponential relation between channel power and similarity degree.

Contribution

It provides new similarity solutions for shallow viscous flows in channels with power-law geometries, extending previous deep-channel analyses and exploring convergence and asymptotic behaviors.

Findings

No divergence in similarity variable as channel power approaches 1 or 3.

Inverse power dependence of equilibration time on ratio disagreement.

Exponential decay relationship between channel power and similarity degree.

Abstract

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power-law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid has higher viscosity than the ambient fluid, thus avoiding Saffman-Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al [2014] studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow1$ and $n\rightarrow3$, when fluid flows into and out of the channel respectively. No divergence is found in the shallow case presented here. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming inverse dependence on the ratio disagreement between the current front using the method of lines (MOL). The inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta}$, which does not show any divergent behaviour for large $n$. An asymptotic behaviour is found for $\delta$ that approaches $1/2$ as $n\rightarrow\infty$.