Lecture 2

Question 1

What is the goal of dimensional analysis in fluid mechanics?

Answer 1

Dimensional analysis helps to simplify problems by identifying key variables and their relationships based on dimensions, reducing the complexity of equations and revealing fundamental scaling laws.

Question 2

What are the dimensions of dynamic viscosity (μ)?

Answer 2

[μ]=M/LT

Where 𝑀 is mass, L is length, and 𝑇 is time.

Question 3

How is shear stress related to viscosity?

Answer 3

τ=μ du/dy
​
which represents the force per unit area exerted due to viscosity.

Question 4

What is the significance of the Buckingham Pi theorem?

Answer 4

It states that a problem with 𝑛 variables and 𝑟 fundamental dimensions has 𝑛−𝑟 independent dimensionless groups, which helps in reducing experimental and theoretical complexity.

Question 5

How can dimensional analysis be used to estimate the time for an object to fall under gravity?

Answer 5

By balancing length 𝐿, gravity 𝑔, and time 𝑇, the characteristic time scale is:
𝑇∼(𝐿/𝑔)^(1/2)
which gives an estimate of how long it takes an object to hit the ground.

Question 6

What is the Reynolds number and why is it important?

Answer 6

Re=ρUL/μ
It determines the flow regime: low 𝑅𝑒 indicates laminar flow, while high 𝑅𝑒 suggests turbulent flow.

Question 7

What is the relationship between flow rate and pressure drop in laminar pipe flow?

Answer 7

sing dimensional arguments:
𝑄∝(𝑅^4Δ𝑃)/𝜇𝐿​
which follows from the Hagen-Poiseuille equation.

Question 8

What does the term “natural time scale” refer to in fluid mechanics?

Answer 8

It is the characteristic time of a system, obtained via dimensional analysis, that determines the timescale of a flow process.

Question 9

How can the time scale of oscillations in a spring-mass system be determined dimensionally?

Answer 9

By balancing force terms in:

𝑚𝑥¨+𝑘𝑥=0
the characteristic time is:
𝜏∼sqrt(𝑚/𝑘)

Question 10

What assumption allows for the thin-film approximation?

Answer 10

The assumption that the film height ℎ is much smaller than the lateral dimensions, allowing simplifications in governing equations.

Question 11

What is the governing equation for the height profile ℎ(𝑟,𝑡) of a spreading axisymmetric drop?

Answer 11

∂h/∂t=λ1/r * ∂/∂r(h^nr∂h/∂r)

Question 12

How does the spreading radius of a drop scale with time?

Answer 12

Using mass conservation and the governing PDE, the radius increases as:

𝑅(𝑡)∼𝑡^1/8

Question 13

How is the pressure field determined in the thin-film approximation?

Answer 13

By integrating the vertical momentum equation, the pressure field is:

𝑝(𝑟,𝑧,𝑡)=𝑝_0+𝜌𝑔[ℎ(𝑟,𝑡)−𝑧]

Question 14

What is the velocity profile for a thin film spreading under gravity?

Answer 14

The velocity in the radial direction is:

𝑢_𝑟(𝑟,𝑧,𝑡)=(𝜌𝑔/2𝜇)(∂ℎ/∂𝑟) 𝑧[𝑧−2ℎ(𝑟,𝑡)]
which satisfies no-slip at the base and zero shear at the free surface.

Question 15

What condition must the vertical velocity satisfy at the free surface?

Answer 15

Since fluid particles remain on the free surface, the kinematic boundary condition gives:

𝑢_𝑧 from (𝑧=ℎ) =𝑢𝑟 from (𝑧=ℎ) (∂ℎ/∂𝑟)+(∂ℎ/∂𝑡)

Question 16

How does the falling speed of a cylindrical plug depend on the gap width 𝜖?

Answer 16

By balancing forces (gravity, pressure gradient, and viscous resistance), the velocity scales as:

𝑈∝𝜖^𝛽

where 𝛽≠2

Question 17

How can the viscosity 𝜇μ of a fluid be estimated using the falling plug method?

Answer 17

Using known geometry and weight of the plug, 𝜇 can be extracted from measurements of 𝑈 with an error of 𝑂(𝜖)

Question 18

What equation describes vorticity diffusion?

Answer 18

By taking the curl of the momentum equation, vorticity 𝜔 satisfies: ∂𝜔/∂𝑡=𝜈∇2𝜔

Question 19

What is the physical significance of vorticity in fluid mechanics?

Answer 19

Vorticity represents the local rotation of fluid elements