Chapter 3 Unidirectional transport - Chemical Engineering ...

3.3. EFFECT OF PRESSURE ON MOMENTUM TRANSPORT 39WallαφrWall6. An ideal vortex is a flow with circular streamlines where the particlemotion is incompressible and irrotational. The velocity profile obeysthe equation in cylindrical coordinates:v θ =Γ2πr(3.149)with v r = v z = 0. At the origin, the above equation indicates that thevelocity becomes infinite. But this is prohibited because viscous forcesbecome important and the flow is rotational in a small region near thecore.Consider an ideal vortex in which the velocity is given by the aboveequation for t < 0, and the core velocity is constrained to be zero att = 0. Find the velocity profile for t > 0. Assume that v θ is the onlynon - zero velocity component.7. A cubic solid of side a is initially held at a temperature T 0 . At timest ≥ 0, its lateral faces are held at temperatures T A , T B , T 0 and T 0 as illustratedin figure 3.11. The top and bottom faces are insulated so thatno heat is transferred through them. The cube has heat conductivityC, density ρ and thermal conductivity K.(a) Solve for the steady state temperature in the cube.(b) Show how the transient problem may be set up in a form to whichseparation of variables can be applied.