What Exactly is Stokes Flow?

By Eliza Jones, Monash University

In fluid dynamics, we are trying to see how a flow will move and change in time, either by analysing the flow by hand or numerically analysing the flow. This gives us information about what happens inside the flow and how will the flow interact with objects inside it. Stokes flow is a particular type of flow, often referred to as creeping or sticky flow as it often has a very small velocity. Stokes flow is characterised by having a large viscosity, or a low Reynolds number so Re<<0, where the Reynolds number is a ratio of the advective inertial forces to the viscous forces. There are many physical examples of Stokes flow, in my project we explored magma which is known for having a large viscosity, other examples are very small objects moving in a fluid, such as microorganisms swimming or fluid flow through small cracks such as ground water.

Stokes flow has four properties which define it, they are instantaneity, reversibility, linearity and uniqueness. Instantaneity of the flow means there are no time derivatives, this mean we assume the flow responds instantly to boundary motion and forces. Linearity means that the flow is linearly forced whether it is by a boundary motion or body force. This means we can say that the force is proportional to the velocity rather than the acceleration of the object. If the flow is reversible, we can say if the velocity of the flow is reversed then it is reversed everywhere in the fluid. The same goes for a boundary motion, which if reversed then each point retraces its history. The last property is uniqueness, this means that each Stokes flow is unique but can be represented in many ways. This is summarised in a theorem which states, there exists at most one Stokes flow in a volume V for which u is specified on the boundary.

References

Acheson D, (1990). ‘Elementary Fluid Dynamics’, Oxford, Clarendon Press

Eliza Jones was a recipient of a 2018/19 AMSI Vacation Research Scholarship.