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The Local Interaction Model

  Shan et al. [60, 61, 62] consider a fluid with S different components on a regular lattice with b links in D-dimensional space. The direction of the b links are given by the vectors tex2html_wrap_inline13399 , i = 1, ... b and each link has length unity. The distribution functions of the S components on link tex2html_wrap_inline13399 at site tex2html_wrap_inline12911 at time t is given by tex2html_wrap_inline14405 , i = 0, ..., b, a = 1, ..., S and is described by the Boltzmann equations

  equation2791

The collision term tex2html_wrap_inline14417 has the form

  equation2805

where each component has a single relaxation time tex2html_wrap_inline14425 . Here tex2html_wrap_inline14427 is the equilibrium distribution of the ath component on link tex2html_wrap_inline13399 at site tex2html_wrap_inline12911 at time t and is given by tex2html_wrap_inline14437 where tex2html_wrap_inline14439 and tex2html_wrap_inline14441 is the `equilibrium' velocity of the ath component which has still to be defined,

  equation2839

and tex2html_wrap_inline14157 is a positive constant. If we set tex2html_wrap_inline14441 to a common value tex2html_wrap_inline14459 for each component,

equation2863

then the equilibrium distribution in equation (4.88) is simply that of S ideal gases, see equation (4.45). To incorporate inter-component interactions Shan et al. [60] introduce a nonlocal interaction with potential

  equation2878

where tex2html_wrap_inline14479 is a Green's function and tex2html_wrap_inline14481 , the `effective mass', is a function of the density of the ath component, tex2html_wrap_inline14485 . G is restricted to act only on nearest neighbours and so can be simplified to

equation2897

The force acting on the ath component can be found, by summing over all the components and all the neighbouring sites, and is

  equation2911

The rate of change of momentum of the ath component induced at each site by the additional interactions is given by

equation2929

This momentum change can be combined with the momentum change of each component due to the collisions at each site. This gives the equilibrium velocity

  equation2939

Combining equations (4.86) and (4.87) and summing over all directions gives

  equation2952

Substituting the expression for tex2html_wrap_inline14539 given in equation (4.88) we see that

  equation2978

The change in momentum at each site is found by combining equations (4.86) and (4.87), multiplying by tex2html_wrap_inline13399 and summing over all components and all directions,

  equation2992

where tex2html_wrap_inline14557 is given by

  equation3008

The conservation of mass, equation (4.96), and the momentum equation,

  equation3021

can be Taylor expanded and a Chapman-Enskog expansion performed. This gives the `diffusion' equation [62]

  equation3039

and the Navier-Stokes equation [62, 60, 44]

equation3071

where the kinematic shear viscosity is tex2html_wrap_inline14575 , the bulk viscosity is tex2html_wrap_inline14577 , the pressure is tex2html_wrap_inline14579 and tex2html_wrap_inline12875 is defined by

equation3089

and is the average of the fluid velocity before and after the inter-particle force acts. Summing equation (4.100) over a gives the usual continuity equation for the whole fluid

equation3097

as expected. It can also be shown [62] that for a two component system with tex2html_wrap_inline14591 the following diffusion equation is satisfied,

equation3102

where tex2html_wrap_inline14593 is a function of the deviation of the density from its equilibrium value. A binary fluid can be simulated using this model when S = 2. If the effective mass tex2html_wrap_inline14597 and the strength of the inter-component interaction tex2html_wrap_inline14599 are suitably chosen the two components are observed to separate [60]. One component forms circular drops inside the other component and the density difference across the boundary of the drops is seen to obey Laplace's Law. When S = 1 a single component fluid is modelled. This fluid is seen to phase separate into a liquid and gaseous phase with surface tension between the two phases, again for a suitable choice of tex2html_wrap_inline14603 and tex2html_wrap_inline14599 [61]. Full details of the properties of the binary fluid model and the liquid-gas model are given in references [60, 62] and [60, 61] respectively.


next up previous contents
Next: The Free Energy Model Up: Binary-Fluid and Liquid-Gas Lattice Previous: Miscible Binary Fluid

James Buick
Tue Mar 17 17:29:36 GMT 1998