Laminar flow velocity field from a green's function

Abstract

The determination of velocity field of the fluid with laminar flow through randomly distributed spheres using an effective medium treatment (EMT) is presented. In the EMT, the system of fluid and spheres is replaced by a composite sphere -- a representative sphere of radius a enclosed by a fluid shell of radius b -- embedded in an effective medium of different viscosity. This type of fluid flow is described by Navier-Stokes equation which is equivalent to Poisson's equation, so that the velocity field in the fluid shell and in the effective medium as a function of sphere volume packing fraction (gamma [superscript 3]=a[superscript 3]/b[superscript 3]) or density of spheres in the fluid can be determined by using Green's theorem and proper boundary conditions. The approximate fluid velocity in analytic closed form is obtained for low packing fraction ( gamma [superscript 3]<0.1) and for the other range of gamma[superscript 3]>0.1. The comparison of flow fields in the fluid shell obtained in this research with Happel flow fields is shown for varying gamma. The results of fluid velocity are applied to study the capture of magnetic particles by an assemblage of magnetic spheres by using Mathematica program. The capture radius as a function of gamma is obtained and compared with the results from previous study based on Happel's theory. The investigation shows that the flow fields within the fluid shell are very similar to Happel flow fields especially for the dilute range of packing fractions. The general features of the variation of capture radii with gamma for the EMT and Happel flow fields are also similar, but for higher gamma our results are lower than the corresponding Happel model results.