Reference no: EM13929198

Problems:

1 Determine the material derivative of the flux of any vector property Q*_i through the spatial area S. Specifically, show that

d/dt ∫_S Q*_in_idS =  ∫_S (Q^.*_i + Qi^*Vk,k - Q^*V_i,k) n_idS

2. Let the property P*_ij..... be the scalar 1 so that the integral in that equation represents the instantaneous volume V. Show that in this case

P^._ij.. = d/dt ∫_v dV = ∫_v v_i,jdV

3 Verify the identity

ε_ijk = 2 (w^._i + w_iv_j,j - w_j v_i,j

and, by using this identity as well as the result of Problem 1, prove that the material derivative of the vorticity flux equals one half the flux of the curl of the acceleration; that is, show that

d/dt ∫_s w_in_idS = 1/2 ∫_s ε_ijk, α_k,jn_idS

4 Making use of the divergence theorem of Gauss together with the identity

∂w_i/ ∂t = 1/2 ε_ijkα_k,j - ε_ijk ε_kmq (w_mv_q),_j

5 Show that the material derivative of the vorticity of the material contained in a volume V is given by

d/dt ∫_v w_idV = ∫_s(1/2 ε_ijkα_k + w_jv_i)n_jdS

6 Given the velocity field

v₁ = ax₁ - bx₂, v₂ = bx₁ + ax₂, v3 = c√(x²₁ + x²₂)

7 For a certain contiuum at rest, the stress is given by

σ_ij = - p_oδ_ij

where p_o is a constant. Use the continuity equation to show that for this case the stress power may be expressed as

σ_ijD_ij =  p₀ρ^./ρ

8 Consider the motion x_i = (1 + t/k)X_i where k is a constant. From the conservation of mass and the initial condition ρ = ρ_o at t = 0, determine ρ as a function of ρ_o, t, and k.

9. Using the identity

ε_ijkα_k,j = 2(w^._i + w_iv_j,j - w_jv_i,j)

as well as the continuity equation, show that

d/dt (w_i/ρ) = (ε_ijkα_k,j + 2 w_jv_i,j )_{/ 2}ρ