Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long

Chapter 9, Problem 9-101

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Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius Ri and outer radius Ro (Fig. P9101). Ignore the effects of gravity. A constant negative pressure gradient P/x is applied in the x-direction, (P/dx) 5 (P2 2 P1)/(x2 2 x1), where x1 and x2 are two arbitrary locations along the x-axis, and P1 and P2 are the pressures at those two locations. The pressure gradient may be caused by a pump and/or gravity. Note that we adopt a modified cylindrical coordinate system here with x instead of z for the axial component, namely, (r, u, x) and (ur, uu, u). Derive an expression for the velocity field in the annular space in the pipe.