Viscous impinging streams

In all the above derivations in this section, the influence of viscosity is neglected so that analytical solutions for velocity and pressure profiles can be obtained. When the viscosity of fluid is taken into account, it is difficult to obtain any analytical solution. Kuts and Dolgushev [35] solved numerically the flow field in the impingement of two axial round jets of a viscous impressible liquid ejected at the same velocity from conduits with the same diameter and located very close to each other. The mathematical formulation incorporated the complete Navier-Stokes equations transformed into stream and velocity functions in cylindrical coordinates r and z, with the assumption that the velocity profiles at the entrance and the exit of the conduit were parabolic. The continuity equation is given by Eq. (1.22) and the equations for motion in dimensionless form are  

R is the radius of the conduits n the velocity at the outlet of the conduit, i.e., the impinging velocity. The velocity components in terms of stream function were given by Eq. (1.26) while the conditions of irrotational flow were determined by Eq. (1.21). 

Elperin et al. [3, 36] investigated experimentally the hydrodynamics of co-axial gas-solid suspension impinging streams. The dimensions of the device they used are  

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The principle by which this is done is to pass the water-contaminated oil through a dense inorganic fibre bed or filter mat. Water droplets are intercepted by, and impinge on, the fibres. The oil on the fibres is thinned by displacement and the effect of viscous drag, until ultimately the oil film ruptures and allows the water droplets to attach themselves completely to the fibre, with the oil film dispersed and passed on through the mat. Other water droplets are now collected by the fibres in a similar matter, and these will join with others, forming streams along the fibres. 

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