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A small gas bubble grows in a large container of a Newtonian fluid. As the bubble...

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Mechanical Engineering

A small gas bubble grows in a large container of a Newtonianfluid. As the bubble grows, it pushes liquid away radially. We canignore the inertia in the liquid, i.e. Re is small. By solving forthe flow in the liquid and using the boundary condition at thebubble/liquid interface, we can determine how the bubble growthrate, i.e. the bubble radius R as a function of time, depends onthe pressure in the bubble and liquid parameters.

a) Use the continuity equation to find the form of the radialvelocity profile in the liquid.

b) Substitute your result into the r-component of the Stokesequation to show that the pressure in the liquid is constanteverywhere! Call that pressure p?.

c) write the boundary condition at the bubble/liquid interface,i.e. at r = R. Hint: ignore surface tension so that the conditionat the interface becomes the normal stress ?rr must be continuousthere. The normal stress on the bubble side is ?rr = -pb, where pbis the pressure inside the bubble.

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