Fluid flow (In the following question, we write the cartesian variables as x = x1, y = 2, and z = X3. We also write the basis vectors as i = e1, j = e2, and k = e3.) In fluid mechanics, fluids which are incompressible and inviscid are referred to as ideal Let u = u(x, y, z, t) = ulei + uze2 + uges be the velocity of an ideal fluid at an arbitrary point in space and time. Its motion is governed by the Euler equation, du at - J = n( A . n) + dA (1) P Here P is the fluid's pressure, p its constant density, F is the external force on the fluid at any given point, and 3 (u . V)u= Wi axi i=1 Suppose the system is simplified in three ways: - The flow is steady (u = u(x, y, z) is not changing with time) u is a conservative vector field (this occurs when the fluid is irrotational, though we won't elaborate on that here.) - The external force is also conservative, with f = -VV, for some scalar potential V = V(21, 22, 23) a Show that in this case, (u . V)u = , VIlull. (Hint, you can assume that Clairaut's theorem applies) b) Hence show that the Euler equation simplifies to 2 1 14 1 1 2 + V + - = constant c) If V is a constant, what can you say about the relationship between a fluid's speed and its pressure