Hele-Shaw flow

About: Hele-Shaw flow is a research topic. Over the lifetime, 5451 publications have been published within this topic receiving 151320 citations.

Miscible displacement between two parallel plates : BGK lattice gas simulations

Abstract: We study the displacement of miscible fluids between two parallel plates, for different values of the Peclet number Pe and of the viscosity ratio M. The full Navier–Stokes problem is addressed. As an alternative to the conventional finite difference methods, we use the BGK lattice gas method, which is well suited to miscible fluids and allows us to incorporate molecular diffusion at the microscopic scale of the lattice. This numerical experiment leads to a symmetric concentration profile about the middle of the gap between the plates; its shape is determined as a function of the Peclet number and the viscosity ratio. At Pe of the order of 1, mixing involves diffusion and advection in the flow direction. At large Pe, the fluids do not mix and an interface between them can be defined. Moreover, above M∼10, the interface becomes a well-defined finger, the reduced width of which tends to λ∞=0.56 at large values of M. Assuming that miscible fluids at high Pe are similar to immiscible fluids at high capillary numbers, we find the analytical shape of that finger, using an extrapolation of the Reinelt–Saffman calculations for a Stokes immiscible flow. Surprisingly, the result is that our finger can be deduced from the famous Saffman–Taylor one, obtained in a potential flow, by a stretching in the flow direction by a factor of 2.12.

The effect of surface tension on the shape of a Hele–Shaw cell bubble

Abstract: Numerical and asymptotic solutions are found for the steady motion of a symmetrical bubble through a parallel‐sided channel in a Hele–Shaw cell containing a viscous liquid. The degeneracy of the Taylor–Saffman zero surface‐tension solution is shown to be removed by the effect of surface tension. An apparent contradiction between numerics and perturbation arises here as it does for the finger. This contradiction is resolved analytically for small bubbles and is shown to be the result of exponentially small terms. Numerical results suggest that this is true for bubbles of arbitrary size. The limit of infinite surface tension is also analyzed.

Saffman–Taylor instability in yield-stress fluids

Abstract: When a fluid is pushed by a less viscous one the well-known Saffman–Taylor instability phenomenon arises, which takes the form of fingering. Since this phenomenon is important in a wide variety of applications involving strongly non-Newtonian fluids – in other words, fluids that exhibit yield stress – we undertake a full theoretical examination of Saffman–Taylor instability in this type of fluid, in both longitudinal and radial flows in Hele-Shaw cells. In particular, we establish the detailed form of Darcy's law for yield-stress fluids. Basically the dispersion equation for both flows is similar to equations obtained for ordinary viscous fluids but the viscous terms in the dimensionless numbers conditioning the instability contain the yield stress. As a consequence the wavelength of maximum growth can be extremely small even at vanishing velocities. Additionally an approximate analysis shows that the fingers which are left behind at the beginning of destabilization should tend to stop completely. Fingering of yield-stress fluids therefore has some peculiar characteristics which nevertheless are not sufficient to explain the fractal pattern observed with colloidal systems.