Lecture 9

Question 1

What is the dimensionless form of the Navier-Stokes equations for incompressible flow, and what assumption simplifies them for Re ≪ 1?

Answer 1

Dimensionless form:
Re(∂𝑢⃗/∂𝑡 +𝑢⃗ ⋅ ∇𝑢⃗ )=−∇𝑝+∇^2𝑢⃗
For Re ≪ 1, inertial terms are negligible, yielding:
0=−∇𝑝+∇^2𝑢⃗
These are the Stokes equations — linear, time-independent (in steady case), and dominated by viscosity.

Question 2

How is the Reynolds number interpreted as a ratio of time scales?

Answer 2

Re= UL/ν= (timeforvorticitytodiffuseacrossL)/(timeforfluidtoadvectacrossL
timeforvorticitytodiffuseacrossL)

So Re ≪ 1 implies that vorticity diffuses faster than advection moves fluid parcels.

Question 3

In unsteady Stokes flow, why do we decouple the time derivative from convection?

Answer 3

Because for Re ≪ 1, nonlinear convection terms are negligible. The time-dependent Stokes equation becomes:
∂𝑢⃗/∂𝑡=−∇𝑝+𝜈∇^2𝑢⃗
This decouples time evolution from advection, making the system linear and more tractable.

Question 4

What is the hydrodynamic force and torque on an object in Stokes flow?

Answer 4

Force:
F^𝐻=∫from∂Ω (𝑛⃗⋅𝜏𝑑𝑆∼𝜇𝑈_0𝐿)
Torque:
𝐿⃗^𝐻=∫from∂Ω 𝑥⃗×(𝑛⃗⋅𝜏) 𝑑𝑆∼𝜇𝑈_0𝐿^2

Question 5

Why can inertia of a particle be neglected in many low Re problems?

Answer 5

Because if fluid and particle density are comparable and Re ≪ 1, the particle reacts almost instantaneously to forces. The equation

𝑚 𝑑𝑉⃗/𝑑𝑡=𝐹⃗^𝐻+𝐹⃗^ext
reduces to a force balance, as inertial (LHS) terms are negligible.

Question 6

What key property allows Stokes equations to be solved via stream functions?

Answer 6

They are linear and divergence-free, which means a stream function 𝜓 can automatically enforce continuity. In 2D:

𝑢=∂𝜓/∂𝑦, 𝑣=−∂𝜓/∂𝑥

This reduces the governing equations to a biharmonic equation:
∇^4𝜓=0

Question 7

For a sphere moving slowly in a fluid, what is the drag force?

Answer 7

F _drag =−6πμaU_0
This is Stokes drag, valid for creeping flow (Re ≪ 1) around a sphere.

Question 8

How does Brownian motion arise and what is the effective diffusion coefficient?

Answer 8

Brownian motion is due to random molecular impacts on a small particle.
Diffusion coefficient:
𝐷_trans=𝑘_𝐵𝑇/6𝜋𝜇𝑎

Question 9

How does adding rigid particles to a Newtonian fluid affect its viscosity?

Answer 9

For neutrally buoyant spheres at low volume fraction 𝜙,
𝜇_eff=𝜇(1+5/2 𝜙)
This is Einstein’s viscosity formula.

Question 10

What does Purcell’s scallop theorem state?

Answer 10

In Stokes flow (linear, time-reversible), reciprocal motion (e.g. opening/closing without asymmetry) produces no net locomotion.
Therefore, a scallop-like swimmer cannot move by simply opening and closing — it needs asymmetric motion or multiple degrees of freedom.

Question 11

For an object with initial velocity 𝑈_0, how far does it glide in a fluid at low Re?

Answer 11

Neglecting inertia, solve
𝑚 𝑑𝑈/𝑑𝑡=−6𝜋𝜇𝑎𝑈⇒𝑈(𝑡)=𝑈_0𝑒^−𝑡/𝜏
Time scale:
𝜏=𝑚/6𝜋𝜇𝑎
Glide distance:
∼Re⋅𝑎≪𝑎
So the object comes to rest almost immediately.