Reference no: EM132681006

Question 1. Plane Poiseuille Flow of a Convected Jeffrey's Fluid: Consider the steady, fully developed, laminar, pressure driven flow of an incompressible Jeffrey's fluid along x between stationary parallel planes located at y = ±H in a Cartesian system such that

v(x, y, z) = (v_x(y), 0, 0)

The viscous stress in an incompressible Jeffrey's fluid obeys

τ + λ1τ(1) = -μΥ^· - μλ₂Υ^·(1)

with it Υ^· = ( ∇v + ∇v⁺)

and where (...)(1) ≡ D/Dt (...) ∇v^t (...) - (...) ∇v

is the "convected" time derivative. In the above μ, and the λ are material constants.
(a) Determine the non-zero components of τ in this flow. How does this differ from the case of an incompressible Newtonian fluid?
(b) Find the velocity profile vx(y) and determine the non-zero components of τ explicitly.

Question 2. Vorticity Transport From a Thin Rotating Rod. A very long, thin, cylindrical rod of radius R is immersed in a large volume of a Newtonian liquid. The rod and surrounding liquid are initially at rest, but at time t = 0 torque is applied to the rod, causing it to rotate around its axis at a constant angular velocity Ωδ₂ for t > 0. The rod's rotation causes the nearby liquid to begin moving. Intuition suggests the kinematic assumption

v = (0, v_θ (r, t), 0)

in a frame of reference fixed with respect to the liquid and using a cylindrical coordinate system whose z axis coincides with the cylinder. One expects the 9 momentum to be transported away from the rod into the surrounding liquid by the viscous mechanism, with the velocity disturbance gradually reaching further in r as t increases.

(a) Show that for the velocity field assumed, the vorticity w = ∇ x v has only one component, along z

w = (0, 0, w₂ = 1/r ∂/∂r.rvθ)

Take the curl of the NS equations to find the equation governing w₂

∂/∂tw₂ = v 1/r ∂/∂r.r.∂/∂rw₂

where v is the kinematic viscosity. This is a diffusion equation for vorticity.

(b) The equation found in (a) needs auxiliary conditions. That the fluid is at rest initially, and at rest far from the cylinder for t > 0 gives two conditions; express these in terms of wz. For a third, treat the rod as a line-source for vorticity The diffusion equation in (a) has the form of a conservation equation for a scalar quantity in motionless media. The right hand side can be interpreted as

-∇. Q = -∇. (-u∇w_x)

where Q = -vVw₂ is a flux with only a radial component. Consequently one can stipulate a constant total outflow flow boundary condition at r = R. Show that this thinking leads to

lim -r. ∂/∂r.w₂ = α constant

which is a line-source. What value of a is suggested by dimensional analysis ?

(c) Find the solution for w₂ (r = 0, t) using similarity transform (Hint: the transform needed is suggested by V&O Exercise 7.9 ).