Question

could you please help me with answering all parts of this question. like and comment are rewarded.

[7] A3. (a) Draw the streamlines and vortex lines of a Rankine vortex. Indicate which field lines are streamlines and which field lines are vortex lines, and label the vortex core. Explain which region of the flow has zero vorticity, and which region of the flow has non-zero vorticity. (b) A fluid with constant density po, pressure p, and velocity v, satisfies the Euler equation po(@xv + (v.7)v) = -Vp-povo where 0 is the gravitational potential. Use the Euler equation and the identity (v.) = (xv) xv + V (50%) to show that Bernoulli's H field v 2+ P +0 po is constant along streamlines in steady flow. [10] (c) A gap between a pair of concentric cylinders is filled with a viscous fluid of constant density. The inner cylinder has radius R, and the outer cylinder has radius R2. The inner cylinder is held at rest whilst the outer cylinder is forced to slowly rotate about the z-axis with constant angular velocity 1. The fluid velocity has the form v = vo(r) 9 in cylindrical polar coordinates, (1,4, z), where Во Wp = Ar + and A, and B, are constants. (i) State the appropriate boundary conditions on vp at each cylinder. Determine A, and B, in terms of R1, R2, and N. (ii) The kinematic viscosity of the fluid is 1.0 x 10-6m’s-! Estimate the Reynolds number of the flow when R1 = 1.0cm, R2 = 1.2 cm, and N = 0.2 rads-1 [13

Answer 1

a)

forced vorted 7a Re EX Potential (free) velocity distance Core radius

One of the most features of the Rankine vortex is its vorticity field. actually consistent with
the definition of the vortex velocity field (1.1) and therefore the the appliance of the curl operator
in cylindrical coordinates (A 4), it's evident that the vortex presents vertical vorticity
component only. Furthermore the vorticity field modulus is constant within the inner a part of
the vortex, it's positive and it's a function of the utmost flow velocity and therefore the vortex
characteristic distance only. within the outer region of the vortex, the flow has no vorticity
at all.

It is worth to notice that the Rankine vortex is characterized by endless velocity
field, but with a discontinuity in vorticity at the characteristic distance.

V xv = k 1 arve) kl ve ave + 24. if 0<r<R 0 if R<r где ar

b) Let u(t,x) represent the speed vector field of the fluid. Let x(t) denote the position of a particle moving with the fluid, then the speed x˙(t) of the particle at a time t are going to be adequate to the speed of the fluid flow at the purpose (t,x(t)), namely
u(t,x(t))=x˙(t)
Now suppose that u(t,x)⋅∇H(t,x)=0. He want to point out that this suggests that H is constant along the trail of a particle moving with he fluid. Notice that for any path x(t) we've

d Ht, x(t) dt = aH (t, x(t)) + X(t) VH(t, x(t)) at

Assuming then that ∂tH=0, and assuming that the trail x(t) is that of a particle moving with he fluid, the equations written above imply

d - H (t, x(t)) = u(t, x(t)) · JH (t, x(t)) = 0 dt

so the quantity H is constant along a flow line, as desired!

c)

Re: fv.d хKхбочx to'x (1-2-) xo'2. 1-охов =k P= Constant density of viscous fund 2-xxlo 3 x 2 16о о Re 160xE у : х. = (, ).л - ( 1.2 - 1). О2 x(0-2. оочxio-m/set