On the constitutive relations for dispersed particles in nonuniform flows. I: Dispersion in a simple shear flow

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Abstract
The work presented follows on from a previous study in which a kinetic approach was used to generate the continuum equations for a dilute dispersed phase of noncolliding particles in a nonuniform turbulent flow. Here this approach is used to derive forms for the Reynolds stresses and the fluctuating interphase momentum transfer for particles at equilibrium in a bounded simple shear flow; the combination of the two also gives the total stress and long‐term particle diffusion coefficients. These forms are based on equilibrium solutions of a kinetic equation describing the transport of the particle phase space probability density <W(v,x,t)≳ for a particle with velocity v and position x at time t. The crucial constituent of the equation is the form of the phase space diffusion current given here by −(∂/∂v⋅μ+∂/∂x⋅λ+γ)<W(v,x,t)≳, where μ, λ, and γ are tensors whose components depend upon gradients of the local mean carrier flow, as well as particle and carrier flow time scales. While some of the results quoted here can be obtained by more direct means, this approach identifies those contributions arising explicitly from the mean shear gradient k p of the dispersed phase (the viscous contributions) and those arising explicitly from that of the carrier flow. These forms highlight fundamental deficiencies in the traditional use of the Boussinesq approximation, that have hitherto gone unrecognized. The Reynolds stresses for instance are composed of a homogeneous component (as if the carrier flow is homogeneous) and a deviatoric component (due to the mean shear gradient): The Boussinesq approximation ignores the homogeneous components in the shear stresses and accounts only for the viscous contribution in the deviatoric component. For small particles (approaching fluid point motion) the deviatoric Reynolds shear stresses (in the streamwise direction) are proportional to the relative mean shearing between the two phases. For large particles (approaching infinite response time) the normal Reynolds stress in the streamwise direction is proportional to k p 2, as is also the corresponding particle diffusion coefficient which is also proportional to particle response time.