An example of mixing between two viscous fluids at low Reynolds number. The two fluids are physically identical, and differentiated by a tracer.
A sinusoidally varying Poiseuille flow applied in the upper half of the domain on the left boundary drives mixing between a cavity and the main flow. Over time the fluid in the cavity is replaced by lobes of the main fluid which are injected as the driving flow slows down.
These flows are described by the Reynolds number Re (which is low in this case) and the product of the Reynolds number and a Strouhal number, Sr, which defines the frequency with which the driving Pouseille flow oscillates. For more details, see Horner et al (2002).
These flows are simple systems which can represent a number of mixing processes in geological fluid mechanics, such as mixing two different kinds of magma. Similar flows have also been suggested as strategies for mixing (or preventing mixing) groundwater in underground aquifers, where the driving flow is provided by injecting and withdrawing water from pairs of wells.
Adaptive refinement is used to resolve the interface and the velocity field. The simulation slows down over time as the interface stretches rapidly both in the cavity and within the main flow. A high degree of refinement is needed to resolve the interface at long times. In this example the lack of resolution results in the formation of droplets from the stretched and thinned lobes both in the cavity and in the channel.
The movie which is generated shows the interface between the two fluids and the logarithm of the dimensionless velocity magnitude (Figure 13), which shows the weak counter-rotating flows which form in the cavity in response to the driving Poiseuille flow.