12. Navier-Stokes Applications CH EN 374: Fluid Mechanics

Problem Consider a steady, two-dimensional, incompressible velocity field: 𝑣 = 𝑎𝑥+𝑏 𝑖 + −𝑎𝑦+𝑐𝑥 𝑗 Calculate the pressure as a function of x and y.

Problem Consider a steady, two-dimensional, incompressible velocity field: 𝑣 = 𝑎𝑥+𝑏 𝑖 + −𝑎𝑦+𝑐𝑥 𝑗 Calculate the pressure as a function of x and y. 𝜌 𝜕 𝑣 𝑥 𝜕𝑡 + 𝑣 𝑥 𝜕 𝑣 𝑥 𝜕𝑥 + 𝑣 𝑦 𝜕 𝑣 𝑥 𝜕𝑦 + 𝑣 𝑧 𝜕 𝑣 𝑥 𝜕𝑧 =− 𝜕𝑃 𝜕𝑥 +𝜌 𝑔 𝑥 +𝜇 𝜕 2 𝑣 𝑥 𝜕 𝑥 2 + 𝜕 2 𝑣 𝑥 𝜕 𝑦 2 + 𝜕 2 𝑣 𝑥 𝜕 𝑧 2 𝜌 𝜕 𝑣 𝑦 𝜕𝑡 + 𝑣 𝑥 𝜕 𝑣 𝑦 𝜕𝑥 + 𝑣 𝑦 𝜕 𝑣 𝑦 𝜕𝑦 + 𝑣 𝑧 𝜕 𝑣 𝑦 𝜕𝑧 =− 𝜕𝑃 𝜕𝑦 +𝜌 𝑔 𝑦 +𝜇 𝜕 2 𝑣 𝑦 𝜕 𝑥 2 + 𝜕 2 𝑣 𝑦 𝜕 𝑦 2 + 𝜕 2 𝑣 𝑦 𝜕 𝑧 2

Laminar Pipe Flow I’ve shown you pictures of the velocity profile of laminar pipe flow. Now let’s find the profile ourselves.

1 𝑟 𝜕(𝑟 𝑣 𝑟 ) 𝜕𝑟 + 1 𝑟 𝜕 𝑣 𝜃 𝜕𝑟 + 𝜕 𝑣 𝑧 𝜕𝑧 =0 Continuity Incompressible continuity equation (incompressible): 1 𝑟 𝜕(𝑟 𝑣 𝑟 ) 𝜕𝑟 + 1 𝑟 𝜕 𝑣 𝜃 𝜕𝑟 + 𝜕 𝑣 𝑧 𝜕𝑧 =0 For laminar flow, we know all flow is in the z direction 𝑣 𝑟 =0 𝑣 𝜃 =0 So what does the continuity equation tell us about 𝜕 𝑣 𝑧 𝜕𝑧 ?

Simplifying NS We did this Friday: 𝜌 𝜕 𝑣 𝑧 𝜕𝑡 + 𝑣 𝑟 𝜕 𝑣 𝑧 𝜕𝑟 + 𝑣 𝜃 𝑟 𝜕 𝑣 𝑧 𝜕𝜃 + 𝑣 𝑧 𝜕 𝑣 𝑧 𝜕𝑧 =− 𝜕𝑃 𝜕𝑧 +𝜌 𝑔 𝑧 +𝜇 1 𝑟 𝜕 𝜕𝑟 (𝑟 𝜕 𝑣 𝑧 𝜕𝑟 )+ 1 𝑟 2 𝜕 2 𝑣 𝑧 𝜕𝜃 2 + 𝜕 𝑣 𝑧 2 𝜕 𝑧 2 PS: What about pressure and gravity in the other directions?

Boundary Conditions At what 𝑟 values do we know what 𝑣 𝑧 is? What else do we know?

Solve

What’s the velocity profile good for? Shear stress

What’s the velocity profile good for? Average velocity 𝑣 𝑎𝑣𝑔 = 𝑉 𝐴 = 1 𝐴 𝐴 𝑣𝑑𝐴