Chapter 9, Problem 98P

Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely lons round pipe annulus of inner radius $R_i$ and outer radius $R_0$(Fig. 9-98). Ignore the effects of gravity. A constant negative pressure gradient $\partial P/\partial x$ is applied in the x-direction, $(\partial P/dx)=\left(P_2-P_1\right)/\left(x_2-x_1\right)$ , where $x_1$and $x_2$ are two arbitrary locations along

FIGURE P9-98
the x-axis, and $P_1$ and $P_2$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, namenly, $\left(r,\theta ,x\right)$ and $\left(u_r,u_{\theta },u\right)$ . Derive an expression for the velocity field in the annular space in the pipe.