Basis transform solution for Brinkman equation to describe unsteady hydrodynamic interactions

This dissertation elucidates how inherently unsteady hydrodynamic interactions between two closely situated spheres in viscous liquid affect their time-dependent motion. The system represents two spherical Brownian particles for which temporal inertia is always comparable to the viscous forces even though convective inertia is negligible. Therefore, instead of Stokes equation, the linearized unsteady Navier-Stokes is considered. We apply Fourier transform in frequency space to convert this to Brinkman equation. Then a novel mathematical formulation is proposed to investigate vector field solution governed by Brinkman equation. The methodology relies on the expansions in multiple sets of vector basis functions corresponding to each sphere. The key result in the formulation is the mutual transformations between the basis functions of two such sets. This allows the derivation of the matrix relations coupling the unknown amplitudes with the given inhomogeneous boundary conditions. The presented mathematical theory is validated by complementing numerical calculations. Accordingly, the solution is constructed using the outlined method, and the error in the form of departure from the intended boundary condition is evaluated. This error vanishes very quickly with increasing number of basis solutions demonstrating high accuracy and exponential spectral convergence of the numerical scheme. Construction of hydrodynamic force or torque for each basis function in the derived vector field provide the frequency-dependent two-body frictions. On the contrary, inverse Fourier trans-form of these after adding appropriate inertial contributions yield time-dependent mobility response. The friction and mobility values are validated in limiting cases for short- and long-time limits. The scaling laws of these quantities are also explored as functions of the separation distance between two solid bodies revealing important physical insight into the complicated dynamics.