Find an equation for the material surface for t gt 0 and

Question

Problem 1. Consider the material "surface" (i.e., line) x₀² + y₀² = a² at t = 0 in the 2d flow (u,v) = (x, -y)/τ, where τ is a constant.

i. Find an equation for the material surface for t > 0 and show that its "volume" (i.e., area) is conserved. What property of the velocity guarantees that the volume must be conserved?

ii. Calculate the rate-of-strain tensor E_ij for this flow (a 2x2 matrix) and show that it determines the growth of the distance between x_a = (a, a) and x_b, = x_a + (δx₀, δy₀) according to the equation D/Dt (δx_k, δy_k) = 2δx_iE_ij.

iii. Verify your answer by solving for the particle paths that start at x_a and x_b, and by calculating the rate of change of the distance squared between the particles.

Problem 2. For an inviscid, adiabatic, compressible fluid with momentum equation Dv/Dt = -α∇p - ∇Φ, show that the energy density (per unit volume) E = ρ(v²/2 + I + Φ) obeys the conservation law

∂_tE + ∇.[(E + p)v] = 0,

where Φ is a time independent potential (e.g., the gravitational potential). (I is the internal energy per unit mass from the first law of thermodynamics - use the equation for DI/Dt developed in class. α ≡ 1/ρ.) Interpret the equation - why is there a pv term?

Problem 3. i. Show that, for an ideal gas, dQ = c_vdT + pdα and dQ = c_pdT - αdp are equivalent expressions of the first law of thermodynamics.

ii. Starting with the expression θ = T(p₀/p)^K, with K ≡ R/c_p, for the potential temperature of a simple (i.e., dry) ideal gas, show that

dθ = θ/T (dT - α/c_p.dp) , and

dQ = Tdη = cpT/θ dθ

iii. An air parcel moves adiabatically from 700 hPa to 200 hPa. If the parcel has a temperature of 280K at 700hPa, what is its temperature on. arrival at 200hPa?