Boundary Layers
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Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid
We determine the globally minimum time 𝑇 needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients.
Fluid Dynamics
Fluid Mechanics Boundary LayersA university course in fluid dynamics for undergraduates in Maths and Physics. (Some version of this is taught at Warwick as MA3D1 and at Brown as ENGN0810.) The HTML and PDF lecture notes may be found at this link. Here is the learning material, including recordings of videos made in the unusual year of 2020:
Fluid mechanical kinematic optimization
Optimization Boundary Layers Fluid MechanicsIn 1696, Johann Bernoulli posed a challenge called the brachistochrone problem, which kickstarted the field of calculus of variations. Here is simplest fluid mechanical version of the brachistochcrone problem.
Work-minimizing kinematics for small displacement of an infinitely long cylinder
We consider the time-dependent speed of an infinitely long cylinder that minimizes the net work done on the surrounding fluid to travel a given distance perpendicular to its axis in a fixed amount of time. The flow that develops is two-dimensional. An analytical solution is possible using calculus of variations for the case that the distance travelled and the viscous boundary layer thickness that develops are much smaller than the circle radius.