10.5.2. Continuity and Navier-Stokes Equations

Try these first

1. 1.

   What does the continuity equation represent?

   Hint

   Think about the principle of mass in a closed system.

2. 2.

   Describe irrotational flow.

   Hint

   What happens to rotation in the fluid?

3. 3.

   What is the main purpose of the continuity equation?

   • Energy conservation
   • Mass conservation
   • Momentum conservation

   Hint

   Recap on what happens to mass in a closed system.

4. 4.

   True or False: The Navier-Stokes equations can only be applied to incompressible flows.

   • True
   • False

   Hint

   Think about whether fluids can be compressed and how that impacts equations.

5. 5.

   Given a velocity field of u = Ax + By and v = Cx - Dy, derive the continuity equation and analyze if it satisfies mass conservation.

   Hint

   Utilize partial derivatives to solve.

6. 6.

   For a flow scenario involving high-pressure gradients across a cylindrical pipe, derive the velocity profile using the Navier-Stokes equations.

   Hint

   Remember to set gravity terms to zero for horizontal flow.