Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius R_i and outer radius R_o (Fig. P9–101). Ignore the effects of gravity. A constant negative pressure gradient ∂P/∂x is applied in the x-direction,

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, θ, x) and (u_r, u_θ, u). Derive an expression for the velocity field in the annular space in the pipe.

FIGURE P9–101

Transcribed Image Text:

(ap/dx) (P₂P₁)/(x₂-x₁),