Fundamentals Flashcards
What does it mean for a flow to be steady?
The flow is independent of time.
What does a stagnation point x* mean?
u(x*) = 0 } all vectors
How do we define the flux?
The flux of quantity through a surface is the rate of flow of the quantity through the surface per unit time.
What is the mathematical definition of a volume flux?
(vec{nhat} . vec{u})dS = vec{u} . vec{dS}
What is the mathematical definition of the mass flux?
rho vec{u} . vec{dS}
What is the mathematical definition of the total mass flux?
\int_{S} rho vec{u} . vec{dS}
What does mass conservation mean?
The rate of increase in mass in V must equal the rate at which mass flows in through the boundary.
What is the material derivative operator?
D/Dt = delta/deltat + vec{u}.grad
What does it mean for a fluid to be incompressible?
Drho/Dt = 0 (no expansion or contraction) so grad.vec{u} = 0.
If vec{u} is a function of time, how do we calculate streamlines?
Set t to be a constant as streamlines are properties of flow at a given time.
Streamlines are continuously changing except in what case?
The case of steady motion (flow is independent of time)
At which special location do streamlines cross?
Stagnation points (vec{u} = vec{0})
What is a particle path (pathline)?
A trajectory of a ‘fluid element’ of fixed identity over a period of time.
Which term in the Navier-Stokes equation refers to acceleration?
Dvec{u}/Dt (the material derivative)
Which term in the Navier-Stokes equation refers to pressure forces?
-gradP
Which term in the Navier-Stokes equation refers to gravitational forces?
rho vec{g}.
Which term in the Navier-Stokes equation refers to viscous forces?
mu del squared vec{u}
What is the summation convention in suffix notation?
A repeated suffix j in a term implies the term is to be summed from j=1 to j=3.
How is the Kronecker Delta defined?
delta_{ij} = 1 if i=j, 0 otherwise.
How is the Alternating Tensor defined?
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.
How does the alternating tensor relate to the cross product?
(vec{a} x vec{b}){i} = epsilon{ijk}a_{j}b_{j}
What is the relationship between the alternating tensor and the kronecker delta?
epsilon_{ijk}epsilon_{ilm} = delta_{jl}delta_{km} - delta_{jm}delta_{kl}
What is the difference between the gradient and the divergence operators?
The gradient forms a vector, with each term being the respective partial derivatives. The divergence is the sum of these individual terms, a scalar.
