The flow of viscoelastic fluids is an area in which analytical results are difficult to attain,
yet can provide invaluable information.
We develop a weak-coupling expansion that allows for semi-analytical computations of
viscoelastic fluid flows coupled to immersed structures.
We apply the expansion to the transient benchmark problem of a rigid
sphere settling through a viscoelastic fluid using the Oldryod-B model
and recover previous results and observed behavior.
The theory presented here is in contrast to the retarded-motion,
or low Weissenberg number, expansions that have received much attention,
and one advantage is that the weak-coupling expansion offers information
for Weissenberg numbers larger than one.
The expansion's limit of validity is closely related to the diluteness criterion
for a Boger fluid.
We extend the classical settling problem to include an oscillating body-force,
and show how the introduction of a forcing time-scale modifies the body-dynamics.
Erosion by flowing fluids carves striking landforms on Earth and also provides important clues to the past and present environments of other worlds.
In these processes, solid boundaries both influence and are shaped by the surrounding fluid, but the emergence of morphology as a result of this interaction is not well understood.
Here, we study the coevolution of shape and flow in the context of erodible bodies molded from clay and immersed in fast flowing water.
Though commonly viewed as a smoothing process, we find that erosion sculpts pointed and corner-like features that persist as the solid shrinks.
We explain these observations using flow visualization and a fluid mechanical model in which the local shear stress dictates the rate of material removal.
Experiments and simulations show that this interaction ultimately leads to self-similarly receding boundaries and a unique front surface characterized by nearly uniform shear stress.
This tendency toward conformity of stress offers a principle for understanding erosion in more complex geometries and flows, such as those present in nature.
The vertical motion of bodies through a (density) stratified fluid is a problem with important applications
in the ocean and atmosphere.
Here we create a "stripped-down" flow configuration in order to isolate one feature of vertical motion
through stratification - namely the creation of vertical density layers by viscous entrainment.
This flow configuration is created in a table-top experiment by towing a fiber vertically through a
stratified corn syrup solution and lubrication theory is used to model the dynamic flow configuration.
Theoretically we find long-wave instabilites along the fluid interface for layers that
are larger than a critical length-scale.
Here we characterize the brachistochrone path, or path of shortest time, in potential flow past a body,
and we apply our theory to obtain new results related to Darwin's drift volume.
Drift volume is related to a body's added mass and has important practical applications.
Our theory extends previous infinite and semi-infinite results to finite distances and times,
which is an important extension in the context of laboratory experiments.
We also discuss an application to a rigid sphere sedimenting through a sharply stratified fluid in
order to account for an experimentally observed levitation phenomenon.
Further theoretical work related to the potential energy carried by the drift volume is underway.