Fluids and Waves

Question 1

Define Lagrangian coordinates

Answer 1

A coordinate system with a reference point given by the position of the particle at t = 0.

Question 2

Define Eulerian coordinates

Answer 2

A coordinate system with a reference point given by the position of the particle at the current time.

Question 3

Define a convective derivative

Answer 3

The rate of change with X held constant.

Question 4

Define the velocity of fluid

Answer 4

u = Dx/DT

Question 5

Define steady flow

Answer 5

Flow where the velocity of fluid is independent of t.

Question 6

Define a streamline

Answer 6

A snapshot of the flow at a fixed time t, with curves parallel to the velocity field.

Question 7

Define a stagnation point

Answer 7

A point where the velocity is 0.

Question 8

Define particle paths

Answer 8

A description of the flow given by following the path of a particle.

Question 9

Give Euler’s identity

Answer 9

DJ/Dt = J∇ · u, where J is the Jacobian relating the Lagrangian and Eulerian coordinate systems.

Question 10

Define incompressible flow

Answer 10

Flow is incompressible if infinitesimal volumes are preserved, that is DJ/Dt = 0.

Question 11

Give Reynold’s Transport Theorem

Answer 11

d/dt of the triple integral of f over a volume V is given by the triple integral over V of ∂f/∂t + ∇ · (fu).

Question 12

Give the density equation

Answer 12

Dρ/Dt + ρ∇·u = 0

Question 13

Give the momentum equation

Answer 13

ρ(Du/Dt) = −∇p + ρg.

Question 14

Give Bernoulli’s equation for steady flow

Answer 14

p/ρ + (1/2)|u|^2 + χ is constant along streamlines for steady flow.

Question 15

Define vorticity

Answer 15

Vorticity is the curl of the velocity field.

Question 16

Give the vorticity equation

Answer 16

∂ω/∂t + (u·∇)ω = (ω·∇)u

Question 17

Define circulation

Answer 17

The circulation around a closed curve C is given by the path integral along C of u with respect to x.

Question 18

Give Kelvin’s Circulation Theorem

Answer 18

The derivative of circulation with respect to time is 0.

Question 19

Define irrotational flow

Answer 19

Flow where the vorticity is 0.

Question 20

Give Bernoulli’s equation for steady, irrotational flow

Answer 20

p/ρ + (1/2)|u|^2 + χ is constant everywhere if the flow is steady and irrotational.

Question 21

Define a velocity potential

Answer 21

For irrotational flow, there is a potential ϕ such that u = ∇ϕ, this potential is known as the velocity potential.

Question 22

Give Bernoulli’s equation for irrotational flow

Answer 22

∂ϕ/∂t + p/ρ + (½)|∇ϕ|^2 + χ = F(t), where F is independent of position.

Question 23

Give the potential of a line source

Answer 23

At a line source, the potential is given by φ = Qlog(r)/2π.

Question 24

Define the streamfunction

Answer 24

The function ψ(x, y, t) satisfying u = ∇ × (ψk).

Question 25

Define flux

Answer 25

The flux between two streamlines is given by the path integral along C of u.nds, where C is a smooth path joining the streamlines and n is the unit normal.

Question 26

Give the streamfunction of a line vortex

Answer 26

At a line vortex, the streamfunction is given by ψ = −Γlog(r)/2π.

Question 27

Define the complex potential

Answer 27

Provided that both φ and ψ are continuously differentiable, the complex potential is given by w(z) = φ(x, y) + iψ(x, y).

Question 28

Give the complex potential of a line source

Answer 28

The complex potential of a line source of strength Q at (a, b) is given by w(z) = Qlog(z − c)/2π, where c = a + bi.

Question 29

Give the complex potential of a line vortex

Answer 29

The complex potential of a line vortex of strength Γ at (a, b) is given by w(z) = -iΓlog(z − c)/2π, where c = a + bi.

Question 30

Give Milne-Thomson's circle Theorem

Answer 30

Suppose a velocity potential w(z) = f(z) is given with the property that any singularities of f(z) occur in |z| > a. Then the potential w(z) = f(z) + [f(a^2/z*)]* has the same singularities as f(z) in |z| > a and the circle |z| = a as a streamline.

Question 31

Give Blasius' Theorem

Answer 31

Suppose fluid flows steadily past an obstacle B with simple closed boundary ∂B. If gravity is neglected, the net force (Fx, Fy) exerted on B by the fluid is given by Fx − iFy = iρ/2 time the path integral along ∂B of (dw/dz)^2.

Question 32

Give the Kutta-Joukowski Lift Theorem

Answer 32

Consider steady uniform flow at speed U past an obstacle B, where there is a circulation Γ about B but there are no singularities in the flow. Then the obstacle experiences a drag force D parallel to the flow and lift force L perpendicular to the flow given by D = 0, L = −ρUΓ.

Question 33

Give the Joukowski transformation

Answer 33

The map z = G(ζ) = ζ + a^2/ζ.

Question 34

Define the Kutta condition

Answer 34

The circulation must be chosen such that the velocity at the trailing edge is finite.

Question 35

State Helmholtz's Principle

Answer 35

A vortex moves with the velocity field due to everything except itself.

Question 36

Give the water wave boundary condition far away from the surface

Answer 36

∇ϕ → 0 as y → −∞.

Question 37

Give the dynamic boundary condition of water waves

Answer 37

p = Patm at y = η, where Patm denotes the atmospheric pressure above the fluid.

Question 38

Define the amplitude of a water wave

Answer 38

Given a water wave η(x, t) = Acos(kx − ωt − β), the amplitude is given by A.

Question 38

Give the kinematic boundary condition of water waves

Answer 38

The velocity of the fluid normal to the boundary must equal the velocity of the boundary normal to itself.

Question 39

Define the frequency of a water wave

Answer 39

Given a water wave η(x, t) = Acos(kx − ωt − β), the frequency is given by ω.

Question 40

Define the wave number of a water wave

Answer 40

Given a water wave η(x, t) = Acos(kx − ωt − β), the wavenumber is given by k.

Question 41

Define the wave length of a water wave

Answer 41

Given a water wave η(x, t) = Acos(kx − ωt − β), the wavelength is given by 2π/k.

Question 42

Define the wave speed of a water wave

Answer 42

Given a water wave η(x, t) = Acos(kx − ωt − β), the wave speed is given by c = ω/k.

Question 42

Define a dispersive wave

Answer 42

A wave where the velocity is dependent on the frequency of the wave.