Potential Flows of Non-Newtonian Fluids

The theory of potential flow in ideal fluid is extended to Newtonian fluid as well as non-Newtonian fluid. 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. Potential flows of models of a viscoelastic fluid like Maxwell's are studied. These models do not admit potential flows unless the curl of the divergence of the extra-stress vanishes. This leads to an over-determined system of equations for the components of the stress. Special potential flow solutions like uniform flow and simple extension satisfy these extra conditions automatically but other special solutions like the potential vortex can satisfy the equations for some models and not for others. Classical theorems of vorticity for potential flow of ideal fluids hold equally for second-order fluids. Several applications based on this potential flow theory for second -order fluid are studied. Especially, the Blasius' integral formulas for the forces and the moment on two-dimensional bodies of arbitrary cross section in a potential flow of second-order and linear viscoelastic fluids are derived. If potential flows are viewed as approximations to actual flow fields, then the unsteady drag on a rising air bubble in viscous (and possibly in viscoelastic) fluids 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 becomes smaller as the Reynolds number is increased. For example, using the potential flow approximation, the dissipation formula gives the drag of the value pi a(2a^2{rho } / 3 + 12alpha_{1 })dU(t)/dt + 12pialpha mu U(t) for a rising spherical air bubble with radius a and rising velocity U(t) in a second-order fluid where rho is the fluid density, alpha_{1} is a coefficient related to the first normal stress difference and mu is the viscosity of the fluid. Because alpha_{1} is negative, we see from this formula that the unsteady normal stresses oppose inertia. A similar result is found in a linear viscoelastic liquid by using the potential flow approximation.