Fluid Dynamics Flashcards
Notation for pressure and density
pressure; p = p(x, y, z, t)
density; ρ = ρ(x, y, z, t)
SCALAR FUNCTIONS
Steady flow
The flow is independent of time s.t.
u = u(x)
(where u and x are vectors)
Opposite of steady flow?
Unsteady flow
The flow is dependent of time s.t.
u = u(x, t)
(where u and x are vectors)
Define a stagnation point
A stagnation point is a location, x = x, say, at which u(x) = 0
(where u, x and x* are vectors)
What force does pressure (p) give rise to? And in what direction?
Pressure p gives rise to a force -∇p directed from regions of high pressure to regions of
low pressure.
By laws of vector calculus, what is the relationship between -∇p and the level surfaces?
The vector -∇p is perpendicular to the level surfaces, p = constant.
(NB p = constant is the definition of the level surfaces being referred to in this case)
What law of Newtonian dynamics is the Continuity Equation a result of?
The continuity equation results from one of the most fundamental laws of Newtonian dynamics,
that of CONSERVATION OF MASS.
Relationship between mass, volume and density?
M = V x ρ
Define the flux of a quantity through a surface.
The flux of a quantity through a surface is the rate of flow of the quantity through the surface, or
the amount of quantity that flows through the surface per unit time.
Another term for The Continuity Equation.
Mass Conservation equation
Another term for the Divergence Theorem
Gauss’ Theorem
Define the material derivative for scalar function Φ.
DΦ/Dt = ∂Φ/∂t + u · ∇Φ
Where u is a vector (velocity vector)
State the continuity equation in its material derivative form.
Dρ/Dt + ρ∇ · u = 0
Where u is the velocity VECTOR
What does Dρ/Dt mean in context?
Dρ/Dt is the rate of change of density following the fluid motion.
Define an incompressible fluid.
Incompressible fluids do not expand or contract. Therefore Dρ/Dt = 0 which implies ∇ · u = 0 (the incompressible form of the continuity equation)
Name for the suffix ‘i’
The free suffix.
NB ‘i’ is arbitrary and we could equally have used ‘j’ or another other suffix.
What is a repeated suffix referred to as?
As a dummy suffix.
It is summed over. A suffix that is free of summation is called a free suffix or a live suffix.
The trace of a matrix , A, in suffix notation
Tr(A) = A11 + A22 + A33 = Ajj
Is the Kronecker delta function symmetric?
Yes
δ_ij = δ_ji
Simplify δ_ij a_j
δ_ij a_j = a_i
What is the name for this; ε_ijk
The alternating tensor
Define the cross product using suffix notation.
(a x b)_i = ε_ijk a_j b_k
(NB a & b are vectors on the LHS, and this definition is for the ith component of the cross product of vectors a and b.)
What expression is helpful for remembering the relationship between the alternating tensors and the Kronecker delta function?
middle middle outside outside - middle outside outside middle
ε_ijk ε_ilm = δ_jl δ_km - δ_jm δ_kl
Newton’s second law
F = ma
F & a are vectors
F is force, m is mass and a is acceleration.
Relationship between acceleration and velocity
Du(x,t)/Dt = ∂u/∂t + u · ∇u = a(x,t)
In the Navier-Stokes equation, what does μ represent?
μ represents the coefficient of viscosity, which is a property of the fluid.
What does the viscous TERM in the Navier-Stokes equation represent in context?
The viscous term µ∇²u (u is the velocity vector) gives the resistance of fluid elements to being sheared.
Unless stated otherwise, what should be assumed about a fluid (to complete 5 equations for 5 unknowns)?
Take the density ρ to be a constant, i.e. ∂ρ/∂t = 0 and ∇ρ = 0.
The equation set then simplifies to that for constant density incompressible flow.
i.e. use ∇ · u = 0 as the continuity equation.
Give the two main categories of solutions to the Navier-Stokes equations.
Solutions for which the nonlinear term u · ∇u vanishes and those for which it doesn’t.
What types of flow are an example of when the non-linear term in the Navier-Stokes equation vansishes?
Couette flow
Poiseuille flow
Define streamwise
In the same direction as the velocity vector u.
No slip boundary conditions
Fluid at a boundary moves at the same velocity as the boundary
Regularity ‘boundary conditions’
We eliminate unphysical terms by an appropriate choice of constants
Free surface boundary conditions
The absence or neglect of forces on a surface (e.g. water in air) gives rise to a no stress boundary condition
µ ∂u/∂y = 0
Relationship between angular and linear velocity.
u = Ω x r
Where u is the linear velocity vector and Ω is the angular velocity vector and r is the position vector.
r = (x, y, z) in cartesian coords
r = (R, 0, z) in cylindrical coords
What is the trace of the symmetric matrix, e_ij (or if considering the trace e_ii), equal to?
The divergence of the flow.
∇ · u
Thus it it 0 for the incompressible flows.
If the trace is positive the fluid is expanding.
Define streamlines
dx/u = dy/v = dz/w
dx1/u1 = dx2/u2 = dx3/u3
Define vorticity
ω = ∇ x u
Relationship with velocity and the streamlines.
Velocity is tangent to the streamlines.
Can the vorticity be written in terms of the symmetric or anti-symmetric component of the Deformation matrix?
Anti-symmetric
In context, what do the symmetric and anti-symmetric parts of the Deformation matrix represent?
The symmetric part represents local straining motion
The anti-symmetric part represents local solid body motion
