where is the mass density, the pressure, the internal energy per unit mass, the viscous energy dissipation rate per unit volume, and the heat flux density. We also have
where is the entropy per unit mass. Moreover, the entropy flux density at a given point in the fluid is (Hazeltine and Waelbroeck 2004)
where the first term on the right-hand side is due to direct entropy convection by the fluid, and the second is the entropy flux density associated with heat conduction.
Derive an entropy conservation equation of the form
where is the net amount of entropy contained in some fixed volume , the entropy flux out of , and the net rate of entropy creation within . Give expressions for , , and . Demonstrate that the entropy creation rate per unit volume is
Finally, show that , in accordance with the second law of thermodynamics.
where is the fluid velocity, the pressure, the potential energy per unit mass, and the (uniform) kinematic viscosity. The incompressibility constraint requires that
Finally, the quantity
is generally referred to as the fluid vorticity.
Derive the following vorticity evolution equation from the Navier-Stokes equation:
Demonstrate that the equations of incompressible fluid flow (see Exercise 2) can be satisfied by writing
Here, , and . Furthermore, the quantity is termed a stream function, because . In other words, the fluid flow is everywhere parallel to contours of .
as well as
Demonstrate that the previous equations can be satisfied by writing
Here, is a spatial constant. This type of flow is known as potential flow, because the velocity field is derived from a scalar potential.
is a constant of the motion. In other words, . This result is known as Bernoulli's theorem.
is a constant of the motion. In other words, . This result is a more general form of Bernoulli's theorem.