Fluid flow with three upstream configurations in freezing tubes

Abstract

The accumulation of frozen liquid around a central passageway of melt as it flows through a freezing region can make calculations very challenging. To both illustrate and to quantify some of these challenges from freezing, a model equation is developed. It simplifies the solution of Holmes (2007, https://gfd.whoi.edu/wp-content/uploads/sites/18/2018/03/MHolmesGFDReport_30151.pdf) for low Reynolds number single component liquid flow through a long tube that has a wall kept at subfreezing temperature. This model equation is used in conjunction with three different upstream configurations, each with parameters expressing their behavior. Analytical and numerical results give the parameters that have criteria for: the freezing of a compressible upstream reservoir that includes oscillatory behavior; the freezing of flow fed through a constriction with a large upstream pressure, just like a dripping water faucet during winter; the evolution of flow in multiple tubes connected by an upstream manifold, where some tubes end up with full flow and others freeze shut. Numerical runs with 1,000 tubes give a formula for the spacing between actively flowing (non-frozen) tubes over wide ranges of the two upstream parameters (flow rate and manifold resistance). Results have implications in various areas in earth science. Some are: oscillatory and freezing shut criteria for flow of magma from a compressible region, a criterion for wintertime ice accumulation at natural springs, and the spacing between volcanos.