On buoyancy in disperse two-phase flow and its impact on the well-posedness of two-fluid models

The Maxey–Riley–Gatignol equation for the flow around a sphere at low particle Reynolds number tells us that the fluid–particle interaction force decomposes into a contribution from the local flow disturbance caused by the particle’s boundary – consisting of the drag, virtual-mass and history forces, and their Faxén corrections – and another contribution from the stress of the background flow, termed generalized-buoyancy force. There is also a consensus that, for general disperse two-phase flow, the interfacial force density, coupling the average fluid phase and dispersed-phase momentum balances, decomposes in a like manner. However, there has been a long-standing controversy about the physical closure separating the generalized buoyancy from the interfacial force density, especially whether or not pseudostresses, such as the Reynolds stress, should be attributed to the background flow. Furthermore, most existing propositions for this closure involve small-particle approximations. Here, we show that all existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments designed to determine the roles of pseudostresses and small-particle approximations. We then derive the unique consistent closure. It requires no approximation and implies that all stresses and pseudostresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. Remarkably, it exhibits a low-pass filter property, attenuating buoyancy at short wavelengths, that prevents it from causing Hadamard instabilities, constituting a first-principles-based solution to the long-standing ill-posedness problem of two-fluid models. When employing the derived closure, even simplistic two-fluid models are linearly well-posed.