Viscous and Viscoelastic Potential Flow

Potential flows of incompressible fluids admit a pressure (Bernoulli) equation when the divergence of the stress is a gradient as in inviscid fluids, viscous fluids, linear viscoelastic fluids and second-order fluids. We show that the equation balancing drag and acceleration is the same for all these fluids independent of the viscosity or any viscoelastic parameter and that the drag is zero in steady flow. The unsteady drag on bubbles in a viscous (and possibly in a viscoelastic) fluid may be approximated by evaluating the dissipation integral of the approximating potential flow because the neglected dissipation in the vorticity layer at the traction-free boundary of the bubble gets smaller as the Reynolds number is increased. Using the potential flow approximation, the drag D on a spherical gas bubble of radius a rising with velocity U(t) in a linear viscoelastic liquid of density ρ and shear modules G(s) is given by