EPR in ducts with non-circular cross sections

In engineering, not only cylindrical pipes are used, but also pipelines with elliptic, square, or triangular cross sections [380], The area D and the cross section shape are taken constant along the longitudinal flow direction Ox. It is known for this problem that the normal flow velocity components Uy and U- are equal to zero, and the only non-zero component is the longitudinal velocity U = U(y, z) that depends on two coordinates (y, z)) in D. Therefore, the complete Navier—Stokes equations are reduced to 

This formulation generalizes problem (3.6). Two boundary conditions are to be imposed, one non-slip condition on the boundary dD and the maximum condition dD/dl = 0 on the geometric center of the area D (median intersection point). Isolines of a constant velocity value (isotaches) have been plotted on the bottom of Fig. 3.6. They are parallel lines in the flow between two plates, and are circular ones in a round tube. The isotaches reflect the influence of corners where the obstructed roughness is especially deep. 

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