Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius Ri and outer radius Ro Ignore the effects of gravity. A constant negative pressure gradient _P/_x is applied in the x-direction, (∂P/∂x)=(P1-P2)/(x2-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 cylindricaloordinate system here with x instead of z for the axial component, namely, ( r, θ ,x) and (Ur, Uθ ,Ux) Derive an expression for the velocity field in the annular space in the pipe.
