Lecture 5

Question 1

What is the fundamental assumption in incompressible flow analysis?

Answer 1

The continuity equation ∇⋅𝑢=0, meaning the fluid density remains constant and there is no volumetric expansion or compression.

Question 2

How does the Navier-Stokes equation simplify for unidirectional flows?

Answer 2

The nonlinear convective term 𝑢⋅∇𝑢 vanishes, making the equations linear.

Question 3

What are some common types of unidirectional flows?

Answer 3

Pipe flow, Taylor-Couette flow, flow between plates, and rotating cylinder flow.

Question 4

Why is the velocity component only in the primary flow direction for straight channels and pipes?

Answer 4

Because of translational symmetry along the primary flow axis, leading to no dependence on that coordinate.

Question 5

What are the four types of pressure-driven pipe flows?

Answer 5

Steady pressure gradient with stationary boundaries, steady gradient with moving boundaries, time-dependent gradient with stationary boundaries, and time-dependent gradient with moving boundaries.

Question 6

What is the vorticity equation, and why is it important?

Answer 6

t describes the evolution of vorticity due to viscous diffusion:
∂𝜔/∂𝑡=𝜈∇^2𝜔
It shows how vorticity diffuses over time due to viscosity.

Question 7

What is the velocity profile for steady, pressure-driven pipe flow?

Answer 7

Parabolic, given by:
𝑢(𝑟)=𝑅^2/4𝜇𝐿 Δ𝑃(1−(𝑟/𝑅)2)

Question 8

How does the volumetric flow rate Q depend on pipe radius?

Answer 8

Q∝R^4
according to the Hagen-Poiseuille equation:
𝑄=(𝜋𝑅^4/8𝜇𝐿)(Δ𝑃)

Question 9

What is Darcy’s Law and when is it useful?

Answer 9

Q=(−k/μ)ΔP, useful for flow through porous media.

Question 10

What is the Reynolds number and its significance in pipe flow?

Answer 10

Re= ρUR/μ
Re<2000 → laminar
Re>4000 → turbulent
2000<Re<4000 → transitional

Question 11

What happens to the flow profile when the Reynolds number increases?

Answer 11

It transitions from parabolic (laminar) to a more uniform (turbulent) profile.

Question 12

What is Stokes’ First Problem?

Answer 12

A suddenly moving plate in a viscous fluid generates a velocity field evolving due to vorticity diffusion

Question 13

How does the velocity profile evolve over time?

Answer 13

The velocity follows:
𝑢(𝑦,𝑡)=𝑈erf(𝑦/sqrt(4𝜈𝑡))

Question 14

What is the characteristic length scale for vorticity diffusion in this problem?

Answer 14

sqrt(νt), which grows over time.

Question 15

How does shear stress at the plate scale with time?

Answer 15

τ∼μU/sqrt(νt)

Question 16

How does vorticity behave away from the plate?

Answer 16

It exponentially decays, given by:
𝜔=(𝑈_0/sqrt(𝜋𝜈𝑡)) 𝑒^−(𝑦^2/4𝜈𝑡)

Question 17

What is Stokes’ Second Problem?

Answer 17

A plate oscillating sinusoidally in a viscous fluid.

Question 18

What is the characteristic penetration depth of the oscillatory motion?

Answer 18

δ=sqrt(v/ω)

Question 19

What is the velocity field in an oscillating flat plate problem?

Answer 19

u(y,t)=U_0e^(−y/δ) cos(ωt−y/δ).

Question 20

How does vorticity behave in this problem?

Answer 20

Confined to a boundary layer of thickness
𝑂(sqrt(𝜈/𝜔))

Question 21

What is pulsatile flow?

Answer 21

Flow driven by a periodic pressure gradient, important in applications like blood flow.

Question 22

What is the governing equation for pulsatile pipe flow?

Answer 22

The unsteady Navier-Stokes equation with a time-dependent pressure gradient.

Question 23

What is the Womersley number, and what does it represent?

Answer 23

Wo= R/(sqrt(ν/ω)), comparing inertial to viscous effects.

Question 24

What happens when 𝑊𝑜≪1?

Answer 24

Flow is quasi-steady, meaning it follows the pressure gradient closely.

Question 25

What happens when 𝑊𝑜≫1?

Answer 25

Flow behaves like a solid-body oscillation, except for thin boundary layers near the walls.