What is the fundamental assumption in incompressible flow analysis?
The continuity equation ∇⋅𝑢=0, meaning the fluid density remains constant and there is no volumetric expansion or compression.
How does the Navier-Stokes equation simplify for unidirectional flows?
The nonlinear convective term 𝑢⋅∇𝑢 vanishes, making the equations linear.
What are some common types of unidirectional flows?
Pipe flow, Taylor-Couette flow, flow between plates, and rotating cylinder flow.
Why is the velocity component only in the primary flow direction for straight channels and pipes?
Because of translational symmetry along the primary flow axis, leading to no dependence on that coordinate.
What are the four types of pressure-driven pipe flows?
Steady pressure gradient with stationary boundaries, steady gradient with moving boundaries, time-dependent gradient with stationary boundaries, and time-dependent gradient with moving boundaries.
What is the vorticity equation, and why is it important?
t describes the evolution of vorticity due to viscous diffusion:
∂𝜔/∂𝑡=𝜈∇^2𝜔
It shows how vorticity diffuses over time due to viscosity.
What is the velocity profile for steady, pressure-driven pipe flow?
Parabolic, given by:
𝑢(𝑟)=𝑅^2/4𝜇𝐿 Δ𝑃(1−(𝑟/𝑅)2)
How does the volumetric flow rate Q depend on pipe radius?
Q∝R^4
according to the Hagen-Poiseuille equation:
𝑄=(𝜋𝑅^4/8𝜇𝐿)(Δ𝑃)
What is Darcy’s Law and when is it useful?
Q=(−k/μ)ΔP, useful for flow through porous media.
What is the Reynolds number and its significance in pipe flow?
Re= ρUR/μ
Re<2000 → laminar
Re>4000 → turbulent
2000<Re<4000 → transitional
What happens to the flow profile when the Reynolds number increases?
It transitions from parabolic (laminar) to a more uniform (turbulent) profile.
What is Stokes’ First Problem?
A suddenly moving plate in a viscous fluid generates a velocity field evolving due to vorticity diffusion
How does the velocity profile evolve over time?
The velocity follows:
𝑢(𝑦,𝑡)=𝑈erf(𝑦/sqrt(4𝜈𝑡))
What is the characteristic length scale for vorticity diffusion in this problem?
sqrt(νt), which grows over time.
How does shear stress at the plate scale with time?
τ∼μU/sqrt(νt)
How does vorticity behave away from the plate?
It exponentially decays, given by:
𝜔=(𝑈_0/sqrt(𝜋𝜈𝑡)) 𝑒^−(𝑦^2/4𝜈𝑡)
What is Stokes’ Second Problem?
A plate oscillating sinusoidally in a viscous fluid.
What is the characteristic penetration depth of the oscillatory motion?
δ=sqrt(v/ω)
What is the velocity field in an oscillating flat plate problem?
u(y,t)=U_0e^(−y/δ) cos(ωt−y/δ).
How does vorticity behave in this problem?
Confined to a boundary layer of thickness
𝑂(sqrt(𝜈/𝜔))
What is pulsatile flow?
Flow driven by a periodic pressure gradient, important in applications like blood flow.
What is the governing equation for pulsatile pipe flow?
The unsteady Navier-Stokes equation with a time-dependent pressure gradient.
What is the Womersley number, and what does it represent?
Wo= R/(sqrt(ν/ω)), comparing inertial to viscous effects.
What happens when 𝑊𝑜≪1?
Flow is quasi-steady, meaning it follows the pressure gradient closely.
