1. Math
  2. Advanced Math
  3. 4the velocity field in a fluid is given by v...

Question: 4the velocity field in a fluid is given by v...

Question details

4The velocity field in a fluid is given by v uit j wk, and the density by p, where u, v, w and ρ are functions of x, y, z and t (a) The divergence of w is defined by (AV) where Δ)/ is an infinitesimal control volume and -is the substantial Dt derivative. By considering the mass of the moving control volume. Δm= ρΔν, derive the equation of continuity (ie. of mass conservation), and show that it can be written as p a (b) Let xX yxand z-x (i) The i component of the curl of a vector u Vxuca be written using index notation as ах; where ε * is the Lev-Civita tensor Show that l.e for any, φ (ii) The vorticity ω is defined by Show that [ You may assume that εk,ekim δ, δ,m-dim δ. ] ) Te momentum equation may be written as @w. θη 18p. 180.. where force is the pressure. ơ, is the stress tensor, and F is the body

4(continued) Show that the momentum equation can be rewritten as F (2) and show that [ Again you may assume that Ek,Ekim-δ0m-0m5-] Hence, by taking the curl of (2), show that if the density ρ is constant, the body force is conservative (ie ▽ x F = 0) and the flow is incompressible (i e ▽·v= 0). [ You may assume that V.(Vx for any u. J

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution