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

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4 The velocity field in a fluid is given by v-ui vj+ wk, and the density by p where u, v, w and p are functions of x, y, z and t (a) The divergence of v is defined by (AV) where Δ)/ is an infinitesimal control yolume and is the substantial derivative By considering the mass of the moving control yolume. Δm = ρΔν, derive the equation of continuity (ie. of mass conservation) and show that it can be written as Dt (i) The i component of the curl of a vector u using index notation as x u, can be written where ε¡jk is the Levi-Civita tensor Show that (оф l.e for any ф (ii) The vorticity w is defined by 0% Show that 0x, [ You may assume that εk,ekin.-6,,9m-6,mo, . J (ii) The momentum equation may be written as @w. θν, 18p 10o,. . t Oxs p@x,.p@x,. where P is the pressure, σ is the stress tensor, and F is the body force

(continued Show that the momentum equation can be rewritten as +-F (2 p@x, ρου and show that [ Again you may assume that Eku Ekim-δ9m-0m5-1 Hence, by taking the curl of (2), show that 0% 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 ▽·(▽ x u) = 0 for any u

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