Solve the Graetz Problem 11.9 but use a boundary condition of T = 1 at the outlet. Solve with Pe =..

Solve the Graetz Problem 11.9 but use a
boundary condition of T = 1 at the outlet. Solve with Pe = 30 on a short
domain, L = 6; solve on a long domain, L = 30. This problem illustrates the
importance of having the correct boundary conditions (see Chang and Finlayson,
1980).

Problem 11.9

Consider flow into a 3D channel that is one
unit high, eight units across, and ten units long for a Reynolds number of 1.0.
Solve for the flow coming in the 1 × 8 cross section. Use “P1 + P1”
discretization for flow. First, use a uniform velocity in, to check for any
errors in your representation. Next, apply linear coupling (see Chapter 10) to
have a fully developed flow coming in. Then solve the convective diffusion
equation with the concentration = 1 on the right-hand side of the inlet and
zero on the left-hand side of the inlet. Use linear discretization for
concentration. Compute the variance at the outlet for Pe = 100, 200, 500, 1000.
Discuss the accuracy of your calculations and indicate steps that could be
taken to improve the accuracy. Also solve the problem when the velocity profile
is the fully developed velocity in a channel, which is equivalent to having
slip on the side walls. To achieve this, just take the velocity in the x
direction as 6∗z∗(1−z). Although the problem with slip
could be solved in two dimensions, solve it in the 3D geometry with the same
mesh used with no-slip.