POTENTIAL FLOWS OF VISCOUS AND VISCOELASTIC FLUIDS

Potential flows of incompressible fluids admit a pressure (Bernoulli) equation when the divergence of the stress is a gradient as in invisci d fluids, viscous fluids, linear viscoelastic fluids and second-order fluids. We show that in potential flow without boundary layers the equ ation 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 when the flow is steady. But, if the potential flow is viewed as an approximation to the actual flow field, the unsteady d rag on bubbles in a viscous (and possibly in a viscoelastic) fluid may be approximated by evaluating the dissipation integral of the approxi mating potential flow because the neglected dissipation in the vortici ty layer at the traction-free boundary of the bubble gets smaller as t he Reynolds number is increased. Using the potential flow approximatio n, the actual drag D on a spherical gas bubble of radius a rising with velocity U(t) in a linear viscoelastic liquid of density rho and shea r modules G(s) is estimated to be D = 2/3pia3 rhoU +12pia integral-t/- infinity G(t-tau) U(tau) dtau and, in a second-order fluid, D = pia(2/ 3a2rho + 12alpha1) U + 12piamuU, where alpha1 < 0 is the coefficient o f the first normal stress and mu is the viscosity of the fluid. Becaus e alpha1 is negative, we see from this formula that the unsteady norma l stresses oppose inertia; that is, oppose the acceleration reaction. When U(t) is slowly varying, the two formulae coincide. For steady flo w, we obtain the approximate drag D = 12piamuU for both viscous and vi scoelastic fluids. In the case where the dynamic contribution of the i nterior flow of the bubble cannot be ignored as in the case of liquid bubbles, the dissipation method gives an estimation of the rate of tot al kinetic energy of the flows instead of the drag. When the dynamic e ffect of the interior flow is negligible but the density is important, this formula for the rate of total kinetic energy leads to D = (rho(a lpha) - rho) V(B) g . e(x) - rho(a) V(B) U where rho(a) is the density of the fluid (or air) inside the bubble and V(B) is the volume of the bubble. Classical theorems of vorticity for potential flow of ideal f luids hold equally for second-order fluid. The drag and lift on two-di mensional bodies of arbitrary cross-section in a potential flow of sec ond-order and linear viscoelastic fluids are the same as in potential flow of an inviscid fluid but the moment M in a linear viscoelastic fl uid is given by M = M(I) + 2 integral-t/-infinity [G(t-tau) GAMMA(tau) ]dtau, where M(I) is the inviscid moment and GAMMA(t) is the circulati on, and M = M(I) + 2muGAMMA + 2alpha1 partial derivative GAMMA/partial derivative t in a second-order fluid. When GAMMA(t) is slowly varying , the two formulae for M coincide. For a steady flow, the reduce to M = M(I) + 2muGAMMA, which is also the expression for M in both steady a nd unsteady potential flow of a viscous fluid. Moreover, when there is no stream, the moment reduces to the actual moment M = 2muGAMMA on a rotating rod. Potential flows of models of a viscoelastic fluid like M axwell's are studied. These models do not admit potential flows unless the curl of the divergence of the extra stress vanishes. This leads t o an over-determined system of equations for the components of the str ess. Special potential flow solutions like uniform flow and simple ext ension satisfy these extra conditions automatically but other special solutions like the potential vortex can satisfy the equations for some models and not for others.