Tim Reis

Lattice Boltzmann model for simulating immiscible two-phase flows

The lattice Boltzmann equation is often promoted as a numerical simulation tool that is particularly suitable for predicting the flow of complex fluids. This paper develops a two-dimensional 9-velocity (D2Q9) lattice Boltzmann model for immiscible binary fluids with variable viscosities and density ratio using a single relaxation time for each fluid. In the macroscopic limit, this model is shown to recover the Navier–Stokes equations for two-phase flows. This is achieved by constructing a two-phase component of the collision operator that induces the appropriate surface tension term in the macroscopic equations. A theoretical expression for surface tension is determined. The validity of this analysis is confirmed by comparing numerical and theoretical predictions of surface tension as a function of density. The model is also shown to predict Laplaces law for surface tension and Poiseuille flow of layered immiscible binary fluids. The spinodal decomposition of two fluids of equal density but different viscosity is then studied. At equilibrium, the system comprises one large low viscosity bubble enclosed by the more viscous fluid in agreement with theoretical arguments of Renardy and Joseph (1993 Fundamentals of Two-Fluid Dynamics (New York: Springer)). Two other simulations, namely the non-equilibrium rod rest and the coalescence of two bubbles, are performed to show that this model can be used to simulate two fluids with a large density ratio.

Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier–Maxwell slip boundary conditions

We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, and nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity and the pressure relative to asymptotic solutions of the compressible Navier–Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier–Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back and other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself.

Moment Method Boundary Conditions for Multiphase Lattice Boltzmann Simulations with Partially-wetted Walls

We propose a lattice Boltzmann approach for simulating contact angle phenomena in multiphase fluid systems. Boundary conditions for partially-wetted walls are introduced using the moment method. The algorithm with our boundary conditions allows for a maximum density ratio of 200000 for neutral wetting. The achievable density ratio decreases as the contact angle departs from 90°, but remains of the order ℴ(102) for all but extreme contact angles. In all simulations an excellent agreement between the simulated and nominal contact angles is observed.

Moment-based boundary conditions for lattice Boltzmann simulations of natural convection in cavities

We study a multiple relaxation time lattice Boltzmann model for natural convection with moment-based boundary conditions. The unknown primary variables of the algorithm at a boundary are found by imposing conditions directly upon hydrodynamic moments, which are then translated into conditions for the discrete velocity distribution functions. The method is formulated so that it is consistent with the second order implementation of the discrete velocity Boltzmann equations for fluid flow and temperature. Natural convection in square cavities is studied for Rayleigh numbers ranging from 103 to 108. An excellent agreement with benchmark data is observed and the flow fields are shown to converge with second order accuracy.

Co‐Extrusion Instabilities Modeled with a Single Fluid

Industrial processes involving co‐extrusion of multiple fluids to produce multi‐layered products are rife with instabilities. We consider a simple indicative instance of co‐extrusion, in which there is only a single fluid involved in the flow, but two different channel branches impose differing flow histories on it. The channels merge and, ideally, a smooth film is extruded with two layers having different stress histories. In experimental studies a wavelike instability is observed with a well defined wavelength in the flow direction and a ‘zig‐zag’ like structure, indicating that the extra flow caused by the instability is three‐dimensional. Suggested mechanisms for instabilities in co‐extrusion include a jump in viscosity and/or first normal stress difference across a flat interface, and a coupling of normal stresses with streamline curvature in the region where the two streams merge. Using a numerical linear stability tool we investigate this instability (using a single mode fluid model throughout) and explore which of the known mechanisms is the most likely culprit here.