Apply the conditions of steady state and solenoidal vector fields \(\mathbf{y}\) to the Hopf fde for cylindrical coordinates in the flow domain \(\mathcal{D}\) obtained in the previous Problem (10.2).

10.3.1 Specialize the result to constant \(\mathbf{G}\) and \(\frac{\partial P_{0}}{\partial z}\).

10.3.2 Show that the scalar product of two modes in \(\mathcal{B}_{e}\) is reduced to an integral w.r.t. the radial coordinate.

Problem (10.2)

Transform the Hopf fde (9.40) to cylindrical coordinates in \(\mathcal{D}\). The explicit form of the pressure gradient term as functional of the velocity is not required. 

Transcribed Image Text:

1 82 80 [y; t]+illaly, g; t, x]+ [y; 1, x] Re Ox30x8 Sya t] dxya(x) lim 820 () 0[y; 1] = [ dxya (x) {i x x x 5y3(x)ya (x) \ +i(- Fr Ja(x). -- axa [) 0 [y: 1}]} (9.40)