Mathematical Model of Gas Bubble Evolution in a Straight Tube

Abstract

Deep sea divers suffer from decompression sickness (DCS) when their rate of ascent to the surface is too rapid. When the ambient pressure drops, inert gas bubbles may form in blood vessels and tissues. The evolution of a gas bubble in a rigid tube filled with slowly moving fluid, intended to simulate a bubble in a blood vessel, is studied by solving a coupled system of fluid-flow and gas transport equations. The governing equations for the fluid motion are solved using two techniques: an analytical method appropriate for small nondeformable spherical bubbles, and the boundary element method for deformable bubbles of arbitrary size, given an applied steady flow rate. A steady convection-diffusion equation is then solved numerically to determine the concentration of gas. The bubble volume, or equivalently the gas mass inside the bubble for a constant bubble pressure, is adjusted over time according to the mass flux at the bubble surface. Using a quasi-steady approximation, the evolution of a gas bubble in a tube is obtained. Results show that convection increases the gas pressure gradient at the bubble surface, hence increasing the rate of bubble evolution. Comparing with the result for a single gas bubble in an infinite tissue, the rate of evolution in a tube is approximately twice as fast. Surface tension is also shown to have a significant effect. These findings may have important implications for our understanding of the mechanisms of inert gas bubbles in the circulation underlying decompression sickness.