Problem 4.20: Consider the steady laminar flow of a Newtonian fluid with constant density...

Problem 4.20: Consider the steady laminar flow of a Newtonian fluid with constant density in a long annular region between two coaxial cylinders of radii and (see Fig. P4.20). The differential equation for this case is given by

where is the velocity along the cylinders (i.e., the component of velocity), is the viscosity, L is the length of the region along the cylinders in which the flow is fully developed, and P1 and P2 are the pressures at respectively (P1 and P2 represent the combined effect of static pressure and gravitational force). The boundary conditions are

Solve the problem using (a) two linear elements and (b) one quadratic element, and compare the finite element solutions with the exact solution at the nodes:

where Determine the shear stress at the walls using (i) the velocity field and (ii) the equilibrium equations, and compare with the exact values. (Note that the steady laminar flow of a viscous fluid through a long cylinder or a circular tube can be obtained as a limiting case of