Viscous flow entrance to a duct with EPRs

5 Viscous flow entrance to a duct with EPRs 

The previous problem made use of the Prandtl s boundary layer approach. Although it is widely applied, it would be worth to examine this approximation on a test problem, where the complete Navier-Stokes equations should be solved. Here is such a test problem that has also its own significance. 

We emphasize again that the porous inserts are of infinite length and interpreted as an area z [0,/t] u [2H - h,2H] with distributed local mass force f = -AU U a l, for which a = 1 expresses the linear force law, and a = 2 is hold for the quadratic law [219], This kind of a flow generalizes the flow problems for smooth or rough 

The problem considered here differs from the canonical problem by the presence of a source term (i.e. the force) on the right-hand side of the complete Navier—Stokes equations (3.29). This force vanishes outside the EPR, for z (h, 1 - h), is opposite to the local flow direction, and is proportional to some power of its velocity (here, we consider the linear or quadratic law). The boundary condition at the entrance x = 0 is evident, U = 1, V = 0 (homogeneous velocity distribution). There are non-slip conditions on the walls z = 0 and z = 1. The further formulation of the problem is somewhat different for linear and quadratic EPRs. 

The above boundary conditions are however insufficient for the unique determination of the two-dimensional flow field U(x,z), V(x,z), an additional information about p(x,z) is necessary. For the original problem with smooth walls, the problem closure was suggested by the transition to a steady-state flow solution at a certain sufficient distance Lx called the entrance region length [566, 605], To this end, the pressure gradient can be taken constant = -/ , and the steady-state solution gives the relation/ = A =  

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