Consider time-dependent (unsteady) 1D flow with in only the x direction, given by u. (y, t)...

 

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Consider time-dependent (unsteady) 1D flow with in only the x direction, given by u. (y, t) a function of y and t only. Assume that u.(y, t) satisfies the linearized Navier-Stokes equation with no pressure gradient: and satisfies as well as the boundary conditions lim, 0% (y.t) = uo and limyo (y.t) = 0 for any t> 0. (b) Show that 8² = V vz(y, t), əy² (a) By applying to both sides of Eq. (1), show that the z component of the vorticity w(y, t) = V x v(y, t) is given by satisfies Eq. (2), where (y,t)= (c) Show that W₂ ду w.(y, t) = V. 2² Əy² Vz =140 y 2√ut w-(y.t). avt (1) 00 v. (y, t) = w. (y', t)dy = u₁¬ satisfies Eq. (1), as well as the boundary conditions above. 2 Consider time-dependent (unsteady) 1D flow with in only the x direction, given by u. (y, t) a function of y and t only. Assume that u.(y, t) satisfies the linearized Navier-Stokes equation with no pressure gradient: and satisfies as well as the boundary conditions lim, 0% (y.t) = uo and limyo (y.t) = 0 for any t> 0. (b) Show that 8² = V vz(y, t), əy² (a) By applying to both sides of Eq. (1), show that the z component of the vorticity w(y, t) = V x v(y, t) is given by satisfies Eq. (2), where (y,t)= (c) Show that W₂ ду w.(y, t) = V. 2² Əy² Vz =140 y 2√ut w-(y.t). avt (1) 00 v. (y, t) = w. (y', t)dy = u₁¬ satisfies Eq. (1), as well as the boundary conditions above. 2

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a To show that the vorticity component wz is given by wz vzy and satisfies the partial differential ... View the full answer