CL336 Advanced Transport Phenomena

Assignment 3

1. Calculate the curvature for a perturbed planar interface if y = 1 + δheikx, if

y = 1 is the unperturbed interface. Use the fact that δ is a small parameter

and derive expression correct to O(δ).

2. The curvature is given by ∇ · n, where n = ∇F/|∇F | and F = r −

r(θ). Calculate the curvature for a perturbed spherical interface if r = 1 +

δhPn(cos θ), if r = 1 is the unperturbed interface. Use the fact that δ is a

small parameter and derive expression correct to O(δ). Legendre equation

for axisymmetric case is given by

∂2f(θ)

∂θ2+ cot θ

∂f(θ)

∂θ+ l(l + 1)f(θ) = 0

In spherical coordinates

∇ = er∂

∂r+

eθr

∂

∂θ+

eφr sin θ

∂

∂φ

3. The curvature is given by∇·n, where n = −∇F/|∇F | and F = r−r(θ, z).

Calculate the curvature for a perturbed cylindrical interface if r = 1 +

δhei(mθ+kz), if r = 1 is the unperturbed interface. Use the fact that δ is a

small parameter and derive expression correct to O(δ).

General form of∇ in cylindrical co-ordinates

∇f =∂f

∂rer +

1

r

∂f

∂θeθ +

∂f

∂zez (1)

∇ · A =1

r

∂rAr∂r

+1

r

∂Aθ∂θ

+∂Az∂z

(2)

4. Calculate angular momentum for a square element (as shown in figure 1)

and prove that stress tensor is symmetric for planar geometries.

1

Figure 1

5. Fluid flow through a pipe with velocity u at pressure difference ∆P . If the

pipe losses water through the wall with a rate ur = K∆P then find the flow

rate at the exit (Assume local Hagen - Poiseuille equation).

2