Representations of solutions of vector partial differential equations in terms of scalar harmonic functions is a popular method to solve boundary-value problems (BVPs) in fluid dynamics and elasticity. The most famous solution forms include those due to Galerkin, Papkovitch and Neuber among others. In this talk I will show a few general solution representations in terms of biharmonic and harmonic scalar functions for studying viscous fluid flow problems. These representations yield general solutions for the BVPs with planar and spherical boundaries leading to reflection theorems for arbitrary incident flow fields. Our solution forms contain the existing representations as special cases. I will also discuss analytic solutions for the motion of a circular disk in a viscous fluid which rely on solutions of dual integral equations involving Bessel functions. General solutions for an axisymmetric flow in and around a compound droplet consisting of two orthogonally intersecting spherical surfaces will be shown as well.