Question

A fluid with constant viscosity and density flows in radial direction from R₁ radius surface to R₂ radius surface in the annular region, which consists of porous horizontal cylinders with coaxial R₂ (outer) and R₁ (inner) radii. (V = V (r)).

Make your simplifications using the continuity equation and motion equations by specifying your required assumptions.

Since the fluid is incompressible and is in laminar flow, obtain the steady pressure profile P(r) in terms of velocity in the outer radius v_r(R₂) and pressure P(R₂) in the outer radius.

Answer 1

Note, derivation contain some images including derivation, as it's not possible to derieve here.

Consider the fluid is flowing through annular region from surface if r1 to r1 as shown in fig

& velocity profile is shown in fig below

In the tube of length D shown in Fig, a pressure d ifference p2 − p3 causes a liquid of viscosity µ to flow steadily from left to right in the annular area between two fixed concentric cylinders. Note that p2 is chosen f or the inlet pressure in order to correspond to the extruder exit pressure from

The inner cylinder is solid, whereas the outer one is hollow; their r adii are r1 and r2, respectively. total

volumetric flow rate Q. Note that cylindrical coordinates are now involved

assumptions and continuity equation. The following assumptions are realistic:
1. There is only one nonzero velocity component, namely that in the direction of
flow, vz. Thus, vr = vθ = 0.
2. Gravity acts vertically downwards, so that gz = 0.
3. The axial velocity is independent of the angular location; that is, ∂vz/∂θ = 0.

To analyze the situation, again start from the continuity equation,

∂ρ/ ∂t + (1/ r )*∂(ρrvr)/ ∂r +(1/r)*∂(ρvθ)/∂θ +∂(ρvz)/∂z = 0

which, for constant density and vr = vθ = 0, reduces to:
∂(vz) /∂z = 0,

verifying that vz is independent of distance from the inlet, and that the velocity
profile vz = vz(r) appears the same for all values of z.

Momentum balances:There are again three momentum balances, one

for each of the r, θ, and z directions

With vr = vθ = 0 (from assumption 1), ∂vz/∂z = 0 from above equation
∂vz/∂θ = 0 (assumption 3), and gz = 0 (assumption 2), this momentum balance
simplifies to:

Shortly, we shall prove that the pressure gradient is uniform between the die

inlet and exit, being given by:

−∂p /∂z = (p2 − p3) /D ,

in which both sides of the equation are positive quantities. Two successive inte- grations of eqn may then be performed, yielding:

vz = − (1 /4)µ −∂p /∂z* r2 + c1 ln r + c2

The two constants may be evaluated by applying the boundary conditions of zero

velocity at the inner and outer walls,

r = r1: vz = 0, r = r2: vz = 0,

Giving

Substitution of these values for the constants of integration into Eqn. We get

yields the final expression for the velocity profile.

ote that the maximum velocity occurs some-
what before the halfway point in progressing from the inner cylinder to the outer
cylinder