Boundary layers in a dilute particle suspension

Abstract

The general problem of a boundary-layer flow carrying a dilute, mono-disperse suspension of small particles (together with gravitational effects) is considered. The problem is modelled using the 'dusty-gas' equations, which are a coupled equation set linking the fluid motion to that of the particle motion (both of which are modelled as continua). A number of qualitatively distinct potential scenarios are predicted. These include a variety of boundary-layer breakdowns, and the formation of shock transitions in the distribution of the particulate phase (together with the development of particlefree zones). Numerical results predicting these differing behaviours are confirmed through local asymptotic analyses of the governing equations. Although we consider a general class of boundary layer, our results are compared and contrasted with previous studies of specific cases, most notably the constant freestream fluid velocity case (akin to the 'clean' Blasius boundary layer). In the case of a boundary-layer flow driven by a linearly retarding free stream (the analogue of the classical 'Howarth' boundary-layer problem), the effects of the particle phase are surprisingly seen to (slightly) delay the separation of the boundary layer.

Analytical solution of an Eulerian dilute particle-laden flow problem

“…The Eulerian governing equations [1][2][3], in the non-conservative form, of a steady two-dimensional dispersed dilute solid (particle) phase immersed in a uniform gas flow in the Cartesian coordinates, x and y , are where is the solid phase volume fraction; u and v are the velocity components of the solid phase in x and y directions, respectively; u f and v f are the uniform velocity components of the gas phase in x and y directions, respectively; g is the acceleration due to gravity, which is assumed to act, without loss of generality, along the negative y direction; and is the reciprocal of the particle relaxation time. The momentum Eqs.…”

“…Assuming the flow is within the Stokes regime; which is applicable for low Reynolds number, i.e. Re ≪ 1 ; is constant and given by [1,2] where is the dynamic viscosity of the gas, p is the particle material density and d is the particle diameter.…”

“…For the system of hyperbolic PDEs given by Eqs. (2) and (3), the corresponding system of ODEs representing the characteristic equation and compatibility conditions is where s is a dummy variable along the direction of the characteristic curves projected onto the xy plane [7,8].…”

“…The most obvious advantage of the Eulerian approach over the Lagranian one is the reduction of the computational costs required to solve the equations of motion for the huge number of particles encountered in realistic multiphase flow problems. Eulerian models of dilute particle-laden flow, with very low solid volume fraction, are often formulated with the one way coupling assumption [1][2][3]. For such particle concentrations, inter-particle interactions are rare and consequently, the solid phase momentum equations do not contain the stress tensor term which arises from pressure and viscous forces; thus avoiding closure sub-models used for particle flows with relatively higher concentrations.…”

Effects of interphase coupling in one-dimensional particle-laden flow

“…The Lagrangian approach requires much more computational resources to calculate the trajectories and velocities of the massive number of particles encountered in realistic flow situations [10]. This led to extensive research effort in the development of Eulerian models [7,[11][12][13][14][15][16] and their application to different flow problems [7,9,[17][18][19][20][21][22].…”