What is the dimensionless form of the Navier-Stokes equations for incompressible flow, and what assumption simplifies them for Re ≪ 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.
How is the Reynolds number interpreted as a ratio of time scales?
Re= UL/ν= (timeforvorticitytodiffuseacrossL)/(timeforfluidtoadvectacrossL
timeforvorticitytodiffuseacrossL)
So Re ≪ 1 implies that vorticity diffuses faster than advection moves fluid parcels.
In unsteady Stokes flow, why do we decouple the time derivative from convection?
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.
What is the hydrodynamic force and torque on an object in Stokes flow?
Force:
F^𝐻=∫from∂Ω (𝑛⃗⋅𝜏𝑑𝑆∼𝜇𝑈_0𝐿)
Torque:
𝐿⃗^𝐻=∫from∂Ω 𝑥⃗×(𝑛⃗⋅𝜏) 𝑑𝑆∼𝜇𝑈_0𝐿^2
Why can inertia of a particle be neglected in many low Re problems?
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.
What key property allows Stokes equations to be solved via stream functions?
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
For a sphere moving slowly in a fluid, what is the drag force?
F _drag =−6πμaU_0
This is Stokes drag, valid for creeping flow (Re ≪ 1) around a sphere.
How does Brownian motion arise and what is the effective diffusion coefficient?
Brownian motion is due to random molecular impacts on a small particle.
Diffusion coefficient:
𝐷_trans=𝑘_𝐵𝑇/6𝜋𝜇𝑎
How does adding rigid particles to a Newtonian fluid affect its viscosity?
For neutrally buoyant spheres at low volume fraction 𝜙,
𝜇_eff=𝜇(1+5/2 𝜙)
This is Einstein’s viscosity formula.
What does Purcell’s scallop theorem state?
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.
For an object with initial velocity 𝑈_0, how far does it glide in a fluid at low Re?
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.
