Lecture 8

Question 1

In squeeze flow for a disk, how does pressure scale with gap height ℎ_0?

Answer 1

Δp∼μVa^2/h_0^3. Pressure increases sharply as gap height decreases due to higher resistance to flow.

Question 2

What is the time evolution of the gap height ℎ_0(𝑡) for a disk under constant force?

Answer 2

h_0(t)∼(2Ft/βμa^4)^−1/2. This shows a power law decay unlike the exponential decay for a sphere.

Question 3

What velocity profile do we expect in the disk squeeze problem?

Answer 3

Parabolic in the gap:
𝑢(𝑟,𝑧,𝑡)=1/2𝜇 ∂𝑝/∂𝑟(𝑧−ℎ), where ℎ=ℎ(r,𝑡).

Question 4

In a spreading gravity current over a planar surface, what drives the flow?

Answer 4

A horizontal gradient in hydrostatic pressure due to the slope of the free surface.

Question 5

What are the assumptions for using the lubrication approximation in spreading films?

Answer 5

Thin film ℎ≪𝐿, negligible inertia, flat interface (no tangential stress), and flow mainly in the 𝑥-direction.

Question 6

hat PDE governs the film thickness ℎ(𝑥,𝑡) in spreading gravity currents?

Answer 6

∂h/∂t=ρg/3μ ∂/∂x(h^3 ∂h/∂x) This is a nonlinear second-order diffusion equation.

Question 7

What is the physical interpretation of the PDE for spreading films?

Answer 7

Thicker regions flow faster due to higher pressure gradients; this leads to flattening of the film over time.

Question 8

How is the front location 𝐿(𝑡) and film height ℎ(𝑥,𝑡) expected to scale with time?

Answer 8

L(t)∼t^1/5
h(t)∼t^−1/5
Derived via dimensional analysis and similarity solution constraints.

Question 9

What similarity form is used to solve the nonlinear PDE for spreading films?

Answer 9

h(x,t)=t^−β H(η), η=x/t^β
with 𝛽=1/5, to satisfy mass conservation and reduce PDE to ODE.

Question 10

What condition ensures the similarity solution for spreading films is valid?

Answer 10

The integral constraint
∫0 to 𝐿(𝑡) ℎ(𝑥,𝑡)𝑑𝑥=𝐴_0(finite total fluid volume) must be time-independent.