This paper investigates the influence of the use of the cubic equation of state (EOS) in the isothermal cavitation of compressible fluids. To do so, a thermodynamic consistent cavitation model that was recently proposed has been used. This model is derived under the Thermodynamics of Irreversible Processes and considers the irreversible dissipative character of the phase change transformation. Numerical simulations carried out using linear and cubic EOS are presented and compared. Neglecting surface tension effects, the results obtained demonstrate that there is no significant difference between the responses of these two types of EOS for water up to saturation pressures up to about 200 kPa. Hysteresis loops observed in the simulations with both types of EOS are virtually the same. It suggests that linear EOSs can provide good approximations for metastable behaviors (intrinsically present in cubic EOS) as well as for the Gibbs free energy difference (the thermodynamic force associated with irreversible phase change transformation), rendering a great simplification in the analysis.
This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.
The stability of steady, fully developed flow in a long cylindrical pipe for a shear-thinning fluid (which approximates a class of viscoplastic materials) is studied using linear stability analysis. The eigenvalues of the frequency of the perturbation of the steady-state solution are obtained using the shooting method. The eigenvalues are negative in the Reynolds number range studied and asymptotically tend to zero as the Reynolds number increases. This shows the pipe flow is stable in the Reynolds number range studied. A qualitatively similar trend is shown by the eigenvalues of a Navier–Stokes fluid of equivalent viscosity. However, the eigenvalues are much lesser than those of the shear-thinning fluid, and this shows that the flow of the Navier–Stokes fluid can be expected to be stable over a much larger Reynolds number range than the shear-thinning fluid.