Fundamentals

Question 1

What does it mean for a flow to be steady?

Answer 1

The flow is independent of time.

Question 2

What does a stagnation point x* mean?

Answer 2

u(x*) = 0 } all vectors

Question 3

How do we define the flux?

Answer 3

The flux of quantity through a surface is the rate of flow of the quantity through the surface per unit time.

Question 4

What is the mathematical definition of a volume flux?

Answer 4

(vec{nhat} . vec{u})dS = vec{u} . vec{dS}

Question 5

What is the mathematical definition of the mass flux?

Answer 5

rho vec{u} . vec{dS}

Question 6

What is the mathematical definition of the total mass flux?

Answer 6

\int_{S} rho vec{u} . vec{dS}

Question 7

What does mass conservation mean?

Answer 7

The rate of increase in mass in V must equal the rate at which mass flows in through the boundary.

Question 8

What is the material derivative operator?

Answer 8

D/Dt = delta/deltat + vec{u}.grad

Question 9

What does it mean for a fluid to be incompressible?

Answer 9

Drho/Dt = 0 (no expansion or contraction) so grad.vec{u} = 0.

Question 10

If vec{u} is a function of time, how do we calculate streamlines?

Answer 10

Set t to be a constant as streamlines are properties of flow at a given time.

Question 11

Streamlines are continuously changing except in what case?

Answer 11

The case of steady motion (flow is independent of time)

Question 12

At which special location do streamlines cross?

Answer 12

Stagnation points (vec{u} = vec{0})

Question 13

What is a particle path (pathline)?

Answer 13

A trajectory of a ‘fluid element’ of fixed identity over a period of time.

Question 14

Which term in the Navier-Stokes equation refers to acceleration?

Answer 14

Dvec{u}/Dt (the material derivative)

Question 15

Which term in the Navier-Stokes equation refers to pressure forces?

Answer 15

-gradP

Question 16

Which term in the Navier-Stokes equation refers to gravitational forces?

Answer 16

rho vec{g}.

Question 17

Which term in the Navier-Stokes equation refers to viscous forces?

Answer 17

mu del squared vec{u}

Question 18

What is the summation convention in suffix notation?

Answer 18

A repeated suffix j in a term implies the term is to be summed from j=1 to j=3.

Question 19

How is the Kronecker Delta defined?

Answer 19

delta_{ij} = 1 if i=j, 0 otherwise.

Question 20

How is the Alternating Tensor defined?

Answer 20

epsilon_{ijk} = 1 if i,j,k = 123 or any clockwise permutation. epsilon_{ijk} = -1 if i,j,k = 132 or any clockwise permutation, and epsilon_{ijk} = 0 if any of the suffices are equal.

Question 21

How does the alternating tensor relate to the cross product?

Answer 21

(vec{a} x vec{b}){i} = epsilon{ijk}a_{j}b_{j}

Question 22

What is the relationship between the alternating tensor and the kronecker delta?

Answer 22

epsilon_{ijk}epsilon_{ilm} = delta_{jl}delta_{km} - delta_{jm}delta_{kl}

Question 23

What is the difference between the gradient and the divergence operators?

Answer 23

The gradient forms a vector, with each term being the respective partial derivatives. The divergence is the sum of these individual terms, a scalar.