(a) Consider a perfect fluid with density , pressure P, and velocity v that vary in time and space. Explain why the fluids momentum density

(a) Consider a perfect fluid with density ρ, pressure P, and velocity v that vary in time and space. Explain why the fluid’s momentum density is G = ρv, and explain why its momentum flux (stress tensor) is

(b) Explain why the law of mass conservation for this fluid is

(c) Explain why the derivative operator

describes the rate of change as measured by somebody who moves locally with the fluid (i.e., with velocity v). This is sometimes called the fluid’s advective time derivative or convective time derivative or material derivative.

(d) Show that the fluid’s law of mass conservation (1.37b) can be rewritten as

which says that the divergence of the fluid’s velocity field is minus the fractional rate of change of its density, as measured in the fluid’s local rest frame.

(e) Show that the differential law of momentum conservation (1.36) for the fluid can be written as

This is called the fluid’s Euler equation. Explain why this Euler equation is Newton’s second law of motion, F = ma, written on a per unit mass basis.

T = Pg+pv@v, or, in slot-naming index notation, T = P9j + pvvj. Tij (1.37a)