Fluid Flow With Three Upstream Configurations in Freezing Tubes.

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%5f30151.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. Plain Language Summary: The dynamics of liquid flow in an upstream region are considered in conjunction with flow through a freezing region. This is because when liquid flows into a freezing region, the resistance change that arises from the accumulation of solid modifies the upstream pressure and flow rate in both upstream and freezing regions. This study shows examples of three different upstream situations with dynamic interaction between upstream and freezing regions. The interaction leads to complicated results such as oscillations, intense flow channelization in subfreezing surroundings, and complete freezing shut in some portions of the downstream region. Through the use of three examples, the fundamental nature of the interaction between upstream and freezing flows helps to begin to explain the complicated nature of freezing flows in Earth Science. Key Points: A simple model is developed for liquid flow into a freezing tubeThree upstream conditions are considered: compressible, throttled, and multiple tubes fed by a manifoldTimes and locations of freezing are quantified [ABSTRACT FROM AUTHOR]