Potential flows of viscous and viscoelastic fluids

Extract

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 in potential flow without boundary layers 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 when the flow is steady. But, if the potential flow is viewed as an approximation to the actual flow field, 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 actual 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 estimated to be \[D = \frac{2}{3}\pi a^3 \rho {\dot U} + 12\pi a \int_{-\infty}^t G(t - \tau)U(\tau){\rm d}\tau\] and, in a second-order fluid, \[D = \pi a\left(\frac{2}{3}a^2 \rho + 12\alpha _1\right ) {\dot U} + 12\pi a\mu U,\] where α₁, < 0 is the coefficient of the first normal stress and μ is the viscosity of the fluid. Because α₁ is negative, we see from this formula that the unsteady normal stresses oppose inertia; that is, oppose the acceleration reaction. When U(t) is slowly varying, the two formulae coincide. For steady flow, we obtain the approximate drag D = 12πaμU for both viscous and viscoelastic fluids. In the case where the dynamic contribution of the interior flow of the bubble cannot be ignored as in the case of liquid bubbles, the dissipation method gives an estimation of the rate of total kinetic energy of the flows instead of the drag. When the dynamic effect of the interior flow is negligible but the density is important, this formula for the rate of total kinetic energy leads to D = (ρ_a – ρ) V_Bg · e_x – ρ_aV_B U where ρ_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 fluids hold equally for second-order fluid. The drag and lift on two-dimensional bodies of arbitrary cross-section in a potential flow of second-order and linear viscoelastic fluids are the same as in potential flow of an inviscid fluid but the moment M in a linear viscoelastic fluid is given by \[M = M_I + 2 \int_{-\infty}^t [G(t - \tau)\Gamma (\tau)]{\rm d}\tau,\] where M_I is the inviscid moment and Γ(t) is the circulation, and \[M = M_I + 2 \mu \Gamma + 2\alpha _1 \partial \Gamma /\partial t\] in a second-order fluid. When Γ(t) is slowly varying, the two formulae for M coincide. For steady flow, they reduce to \[M = M_I + 2 \mu \Gamma ,\] which is also the expression for M in both steady and unsteady potential flow of a viscous fluid. Moreover, when there is no stream, this moment reduces to the actual moment M = 2μΓ on a rotating rod.

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.