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

A small gas bubble grows in a large container of a Newtonian fluid. As the bubble grows, it pushes liquid away radially. We can ignore the inertia in the liquid, i.e. Re is small. By solving for the flow in the liquid and using the boundary condition at the bubble/liquid interface, we can determine how the bubble growth rate, i.e. the bubble radius R as a function of time, depends on the pressure in the bubble and liquid parameters.

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

b) Substitute your result into the r-component of the Stokes equation to show that the pressure in the liquid is constant everywhere! 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 condition at the interface becomes the normal stress ?rr must be continuous there. The normal stress on the bubble side is ?rr = -pb, where pb is the pressure inside the bubble.

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samet mamat answered 1 year ago

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