Example Start-Up Flow in a Pipe

A classic problem in fluid mechanics is the transient development of flow in a pipe. In this idealized problem, the fluid is initially at rest with no applied pressure gradient. At time 

Step 1 To solve this problem in FEMLAB, choose Chemical Engineering Module/ Momentum balance/lncompressible Navier-Stokes/Transient analysis in the Axial Symmetry (2D) option. 

Step 2 Construct a domain with a radius of 0.0025 m and a length of 0.025 m. At this point, you do not know what the length should be, but it turns out that it does not matter. 

Step 3 You solve for properties appropriate to water (p = 1000 kg/m p, = 0.001 Pa s) in a small pipe with a diameter of 0.005 m. The applied pressure drop is taken as 100 Pa/ m. At steady state, the average velocity should then be 0.078 m/s, using Eq. (9.9). Under Subdomain Settings, put in the parameters. Then click the Tnit tab and be sure that the velocity is initially zero. 

Step 4 Set the boundary conditions as you did for entry flow slip/symmetry at the centerline and no-slip on the solid wall. However, at the inlet choose the Normal flow/ pressure option and put in the value 2.5 Pa. At the exit choose the Normal flow/pressure option and put in zero. 

---