Self-Consistent Effective Binary Interaction Approximation For Strongly Coupled Multifluid Dynamics

An improved self-consistent effective binary diffusion approximation for multicomponent diffusion was recently described [1]. Here we develop an analogous self-consistent effective binary interaction (SCEBI) approximation for simplifying multifluid dynamical descriptions in which each fluid is strongly coupled to the other fluids by pairwise frictional forces. The net drag force on each fluid is the summation of the drag forces due to each of the other fluids. This summation is approximated by a single term proportional to the velocity of the fluid in question relative to an appropriately weighted average velocity. This approximation permits an explicit numerical solution for the fluid velocities even when the drag terms are evaluated at the advanced time level to avoid explicit stability restrictions on the time step. Introduction and Summary An improved self-consistent effective binary diffusion (SCEBD) approximation for multicomponent diffusion was recently described [1]. This approximation is used to obtain an explicit expression for the diffusion velocities without solving the full StefanMaxwell equations. Our purpose here is to develop an analogous approximation within the context of multifluid dynamical models in which each fluid satisfies its own momentum equation and is strongly coupled to the other fluids by pairwise frictional interaction forces. Such models are used, for example, in plasma physics [2] and multiphase flow [3]. In models of this type, the net drag force on each fluid is obtained by summing up the individual pairwise drag forces due to each of the other fluids, which are presumed to be proportional to the relative velocities of the fluid pairs. We shall approximate this summation by a single term proportional to the velocity of the fluid in question relative to an appropriately weighted average velocity. This approximation will be referred to as the self-consistent effective binary interaction (SCEBI) approximation. It is self-consistent in the sense that it introduces no net force into the total momentum of the multifluid mixture, which would of course be undesirable. 1>This work was performed under the auspices ofthe U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48. J. Non-Equilib. Thermodyn. · 19.98 ·Vol. 23 ·No.2 © Copyright 1998 Walter de Gruyter· Berlin· New York Brought to you by | Portland State University (Portland State University) Authenticated | 172.16.1.226 Download Date | 5/29/12 8:26 PM 136 J.D. Ramshaw The motivation for the SCEBI approximation is primarily computationaL When the friction coefficients are large, the multifluid dynamical equations become stiff. If these equations were solved using an explicit numerical scheme, the stiffness would require the use of very small time steps to satisfy the explicit stability restriction. This restriction may be avoided by using an implicit numerical scheme in which the drag forces are evaluated at the advanced time leveL This produces a linear system of equations for the advanced-time fluid velocities. These equations bear a strong formal resemblance to the Stefan-Maxwell equations, and are accordingly susceptible to an SCEBI approximation which is closely analogous to the SCEBD approximation. The SCEBI approximation permits an explicit solution for the advanced-time fluid velocities, thereby avoiding the solution of the original linear implicit system on each time step. Of course, this approximation· is sensible only when there are at least three interacting fluids, and it is worthwhile only when the number of fluids is large enough that it becomes inconvenient or unp~easant to solve the linear implicit system directly. The SCEBI approximation is only needed when the friction coefficients are large. However, if all of these coefficients are uniformly very large everywhere, then the multifluid description can be reduced to a diffusional description [ 4] to which the original SCEBD approximation could have been applied. The rationale for the SCEBI approximation is that it extends the previm,1s SCEBD approximation to a wider class of multifluid flow problems in which large friction coefficients present unacceptable explicit stability restrictions but the diffusional limit cannot be invoked. Situations of this type include problems in which (a) some but not all of the friction coefficients are large, (b) the friction coefficients are large enough to impose unacceptable explicit stability conditions, but not large enough to reach the diffusional limit, and (c) some or all of the friction coefficients are large locally (or temporarily), but not everywhere. 2. The multifluid momentum equations We consider multifluid momentum equations of the general form o(piuJ __:_..:.........:._ = s. + "a .. (u.u.) at ' L:'1 1 ' J (1) where ui and Pi are respectively the velocity and partial mass density of fluid i, the ai. are friction coefficients, and Si represents convective, viscous, and source terms, the for:n of which is irrelevant for present purposes. Newton's third law implies that ex .. = a .. The lj Jl" termj = i does not contribute to the sum in equation (1), so.we are free to define aii = 0 for later convenience. It is also convenient to define a. = I:. a .. If equation (1) is summed ' J lj" over i, the friction terms cancel in pairs and we obtain o(pu)/ ot = I:i Si, where p = I:i Pi is the total mass density and pu = I:i piui is the total momentum per unit volume. This cancellation occurs because the friction terms represent a conservative momentum exchange between the fluids with no net creation or destruction of total momentum. Equations (1) are not ~losed by themselves, and must be solved simultaneously with the continuity, energy, and state equations for each fluid and/or the mixture as a whole. This can rarely be done analytically, so it is usually necessary to resort to numerical J. Non-Equilib. Thermodyn. ·1998 ·Vol. 23 ·No.2