Analytical solution of an Eulerian dilute particle-laden flow problem

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Dilute particle-laden flows are encountered in various natural processes and manmade applications. To reduce the computational resources used to simulate such flows, the particle phase could be formulated using the Eulerian approach, resulting in a continuum hydrodynamic model. The aim of this paper is to present the analytical solution of a steady two-dimensional flow problem using that model. The dispersed solid particles, considered in this problem, are immersed in a uniform fluid flow field. The solid phase is coupled to the fluid using a linear Stokes-drag, which is valid for low slip velocities. The general solution of the solid phase velocity field is obtained by solving its quasi-linear hyperbolic momentum equations using the method of characteristics. For the case of a uniform inlet particle velocity, the solid phase velocity field is obtained in terms of Lambert W function. Subsequently, this velocity field is substituted in the solid phase continuity equation; transforming it to a semi-linear hyperbolic partial differential equation, which is solved to obtain the solid phase volume fraction field.