# Why does the force on bubble on a flow differ from that of a sphere in a flow?

Background:

A tiny sphere of radius $$a$$ in a slow moving laminar fluid flow with viscosity $$\mu$$ is acted on by a force given by the well known Stokes equation: $$F_s = 6 \pi \mu a U,$$ where $$U$$ is the relative velocity of the fluid and the sphere.

This equation is a consequence of the Stokes equation $$\mu \nabla^2 \textbf{u} = \nabla p$$ and the continuity equation $$\nabla \cdot \textbf{u} = 0$$, solved subject to the condition that the velocity $$\bf{u}$$ vanishes on the surface of the sphere.

Meanwhile, the force on a bubble of radius $$a$$ is only $$F_b = 4\pi \mu a U$$ In this case, obviously the boundary conditions of the fluid on the surface of the bubble are different.

Questions:

Why is it that $$F_b = \frac{2}{3} F_s$$?

What is the boundary condition satisfied by a fluid on the surface of a bubble, and why is it different than that for a sphere?

If the bubble doesn't change shape due to flow, why does a bubble experience less acceleration than an otherwise equivalent sphere in a flowing fluid? Where does the lost momentum go?

• the first expression is Stokes' formula for sphere drag in very low-Re flow, before instability arises, and it's know as one of the few exact solution of Navier-Stokes equations. Could you provide a reference for the expression of the force acting on a bubble, because it's the first time I see it and I'd like to have a look before giving you a (hopefully) meaningful answer Oct 21, 2022 at 22:46
• One reference is "Bird, R. B., Armstrong, R. C., and Hassager, O. 1987. Dynamics of Polymeric Liquids. New York: John Wiley & Sons." I guess it is also in "Bubbles, Drops, and Particles" by Clift and Weber, 1978 Oct 22, 2022 at 20:23

A sphere is solid. The velocity of the boundary layer at the surface of a sphere is $$0$$.

A bubble is gas. It is not rigid. There is no requirement that there be a boundary layer with a velocity of $$0$$.

--- Update

As @ChetMiller said, there is no shear stress. Instead, there is surface tension. This tries to minimize the surface area.

It is counteracted by pressure. In quiet conditions, water will compress the bubble until volume is given by $$V = nRT/p$$ and the shape is a sphere.

If $$p$$ varies over the surface, then I expect you have to figure out the shape from the forces on the surface.

• Thank you for your answer! Howver, this is just restating info already in the question. What is the boundary condition and why ? Oct 20, 2022 at 5:43
• The boundary condition on a bubble is negligible (zero) shear stress, Oct 20, 2022 at 11:17
• Why is there no shear stress on a bubble ? Bubbles can elongate as they flow. How does this happen if there is no shear ? Oct 24, 2022 at 9:00