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AMATH 361 > Post Midterm > Flashcards

Post Midterm Flashcards

1
Q

How did we derive the Navier-Cauchy equilibrium equations?

A

Combine Hooke’s law + Cauchy’s strain tensor

Stick it into Cauchy’s equilibrium eqn

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2
Q

How to estimate the relative size of terms (ex. Deformation of legs of a chair?)

A

Find |Δuij|

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3
Q

Navier-Cauchy eqn: type of PDE

A

Elliptic

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4
Q

Navier-Cauchy eqn: solid wall BC

A

u = 0 (rigid wall that doesn’t move)

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5
Q

Navier-Cauchy eqn: open boundary BC

A

Specify stress tensor

- typically t=sigma•n = 0 (no force)

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6
Q

Navier-Cauchy eqn: where is the boundary?

A
  • who knows? It can move
  • we will consider it at rest + small displacements (BCs imposed are slightly deformed boundaries)
  • avoids complications
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7
Q

Saint-Venant’s principle

A

Deformation from localized external F distribution with vanishing total F and total moment of force will not reach much beyond the linear size of the region of F application

  • to illustrate: deformation of coaster on table doesn’t go far beyond edge of coaster
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8
Q

Solving problems with uniform settling

A
  • uniform settling: only vertical displacement
  • step 1: find the BCs (either for u or sigma)
  • step 2: find nonzero components of uij
  • step 3: sub into Hooke’s law for sigma_ij (in terms of nonzero uij)
  • step 4: sub into cauchy’s equilibrium eqn
  • step 5: solve resulting DEs
  • step 6: keep going until you find ui (displacement vectors)
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9
Q

Alternate def of Laplacian of a vector

A

Nabla^2 (u) = nabla(nabla•u) - nabla x (nabla x u)

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10
Q

Strain tensor in cylindrical coordinates

A

1/2 (nabla u + (nabla u)^T)

  • assuming symmetric + solid at rest
  • hasn’t changed from xyz I don’t think
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11
Q

Newton’s second law (integral form)

A

Triple integral (rho * u_tt dV) = triple integral (fi dV) + double integral (sigma_ij nj dS)

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12
Q

Newton’s second law (small displacements)

A

rho * u_tt = f + nabla*sigma^T

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13
Q

Total momentum of idealized particle (mass dM)

A

dP = v(x, t)dM = vrhodV
- P, v, x, are vectors

P = triple integral (rho*v)dV

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14
Q

Pathlines

A
  • particle trajectory as a function of initial position

- x(X, t) where x is current position, X is initial position, t is time

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15
Q

Lagrangian description

A

Describes a fluid/particle based on initial position X and time t

  • follows a fluid parcel (lazy surfer riding a wave)
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16
Q

Lagrangian description: variables

A

x(X, t) = position of particle at time t original at X
rho_L(X, t) = density of particle at time t originally at X
v_L(X, t) = velocity of particle at time t originally at X

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17
Q

Lagrangian description: requirement

A

v_L(X, t) = dx(X, t)/dt

  • 3 ODEs: as long as v_L is continuous, we know that there’s a unique solution
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18
Q

Eulerian description

A

Describes the flow with x, t, current position and time
- used more often

  • observes at a fixed position (coral reef watching the waves go by)
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19
Q

Eulerian description: variables

A

rho(x, t) = density at position x and time t

v(x, t) = velocity at position x and time t

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20
Q

Relationship between Eulerian and Lagrangian fields

A

Scalar: rho_L(X, t) = rho(x(X, t), t)
Vector: v_L(x, t) = v(x(X, t), t)

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21
Q

How to solve: given v(x, t), find particle trajectories

A

Solve for x(X, t) from

1) dx/dt = v(x, t) [1-3 eqns]
2) x = X at t=0

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22
Q

Def: pathlines

A

x(X, t) for fixed X

  • must solve dx/dt = v(x(t), t)
  • trajectories
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23
Q

Def: streamlines

A

Trajectories for a velocity frozen in time, t0 fixed
Must solve dx/dt = v(x(t), t0)
- snapshot of velocity field

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24
Q

Def: streaklines

A

Finding al, the positions that pass through a given location

  • paintbrush lines (paint all fluid parcels)
  • not many uses
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25
Q

What abt pathlines, streamlines, streaklines do we know if the velocity is steady?

A

They’re all the same!

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26
Q

Physical meaning of d(rho_L)/dt

A

Rate of change of density of a moving fluid parcel initially at X

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27
Q

Physical meaning of d(rho(x, t))/dt

A

Rate of change of density at fixed point x(X, t)

  • we care more abt the lagrangian description!
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28
Q

Def: material derivative

A

D/Dt = d/dt + v•nabla

  • differential operator for Eulerian fields
  • first term: what you get by sitting at a given location (time passing)
  • second term: what you get from surfing with velocity (moving in space)
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29
Q

Alternate way to write v•nabla(f)

A

|v| ( v/|v| • nabla(f))

- directional derivative

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30
Q

Material derivative for vector fields

A

DFi/Dt = dFi/dt + vj(dFi/dxj)

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31
Q

Assumptions for conservation laws

A

Bounded, simply connected, smooth orientable body

Other nice things lol

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32
Q

Material volume

A

D(t) volume in conservation laws

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33
Q

Reynolds Transport Theorem

A

d/dt (triple integral (D(t)) (f(x, t) dV)) = triple integral (D(t)) (Df/Dt + f nabla•v dV)
= triple integral (df/dt dV) + closed surface integral (fv•n dS)
- closed surface integral over boundary delta(D(t))

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34
Q

RTT physical interpretation of terms

A
  • term 1: rate of change of total f
  • term 2: sum of changes in f in volume
  • term 3: total changes in f through boundary
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35
Q

Leibniz’s theorem

A

d/dt(int(g(t), h(t)) f(x,t) dx) = int(g(t), h(t)) df/dt dx + dg/dtf(g,t) - dh/dtf(h,t) - (dx/dt*f(x,t))|(g, h)
- last term is evaluated at

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36
Q

Principle of conservation of mass

A

d/dt (triple integral (D(t)) rho(x, t) dV) = 0

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37
Q

Continuity equation

A
  • local form of conservation of mass
  • D(rho)/Dt + rho*nabla•v = 0
  • d(rho)/dt + nabla•(rho*v) = 0 (flux form)
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38
Q

Continuity equation physical interpretation + alternate form

A

d(rho)/dt = -v•nabla(rho) - rho*nabla•v

  • term 1: local changes in rho
  • term 2: advection
  • term 3: velocity divergence
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39
Q

Def: incompressible

A
  • a flow that conserves volume
  • conservation of mass -> conservation of volume
  • nabla•v = 0
  • alt: d(rho)/dp = 0
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40
Q

Is there a perfectly incompressible fluid?

A

Nope

It’s a useful approximation though

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41
Q

Incompressibility and conservation of mass

A

Rewrite continuity eqn: nabla•v = -1/rho *D(rho)/Dt

If changes in density are very small compared to the mean density, then the RHS in the continuity eqn is very small -> nabla•v ~ 0

  • cannot plug back in and say 1/rho*D(rho)/Dt = 0 (can’t get two eqns from one)
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42
Q

Principle of conservation of linear momentum

A

d/dt(triple integral (D(t)) rho(x,t)*v(x,t) dV) = F

  • F = sum of forces
  • hard to solve
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43
Q

Conservation of linear momentum (local form)

A 
Triple integral(D(t)) rho*Dvi/Dt dV = F
- F is a sum of body and contact forces
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44
Q

Conservation of linear momentum (global eqn after RTT)

A

Triple integral(D(t), (rho*Dvi/Dt - fi - d(sigma_ij)/dxj) dV) = 0

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45
Q

Momentum equation

A

rho*Dvi/Dt = fi + d(sigma_ij)/dxj

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46
Q

Application: Big Bang

A

Eqn: Dv/Dt = 0
Sol: v=x/t

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47
Q

Hubble’s law

A

v = H0*x

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48
Q

Mass density (Newtonian cosmology)

A

rho(x, t) = rho(t)

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49
Q

Velocity field (Newtonian cosmology)

A

v(x, t) = H(t)*x

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50
Q

Gravity (Newtonian cosmology)

A

g(x, t) = -4*pi/3 * G * rho(t) * x

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51
Q

Deriving the cosmological equations

A

1) sub rho, v into the continuity equation
2) sub rho, v, g into the momentum equation
3) define H = 1/a * da/dt

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52
Q

Def: ideal fluid

A

1) stress tensor of sigma_ij(x, t) = -p(x, 5)*delta_ij

2) rho is constant

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53
Q

Is an ideal fluid compressible or incompressible?

A

Incompressible

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54
Q

Euler equations for an ideal fluid

A 
Dv/Dt = f/rho - 1/rho * nabla(p)
Nabla•v = 0
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55
Q

What is the ideal fluid model missing? (3)

A

1) shear stresses (viscosity or stickiness)
2) compressibility
3) changes in density (can be included)

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56
Q

Def: inviscid

A

Fluid with no viscosity

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57
Q

Characteristics of the Euler equations (4)

A

1) four eqns, four unknowns (closed system)
2) can generalize to allow for variable density, but then we need a 5th eqn for a closed system
3) 3 eqns for momentum are prognostic
4) fourth variable is pressure but fourth eqn is nabla•v=0 -> no p

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58
Q

Def: prognostic

A

Includes a time derivative (dv/dt = …)

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59
Q

Euler eqns: how do we solve for pressure?

A

Compute the divergence of the momentum eqn!

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60
Q

Poisson eqn for pressure

A

Nabla^2(p) = nabla•f - rho*nabla•((v•nabla)v)

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61
Q

Physical meaning of the poisson eqn for pressure

A

A diagnostic eqn that ensures the motion is incompressible (pressure adjusts to ensure it is)
- need BCs to solve (usually flat earth gravity is picked)

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62
Q

Boundary conditions for an ideal fluid

A

No normal flow: v•n = 0 on boundary

Velocity doesn’t pass through

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63
Q

Solid body rotation: velocity

A

v = (-omegay, omegax)

- omega has units of 1/s

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64
Q

Newton’s second law: ideal fluid version (variable density)

A

rhoDv/Dt = -nabla(p) - ge_z

- e_z is unit vector in z direction

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65
Q

Def: solid body rotation

A

All solutions rotate at a frequency of omega, like a solid body

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66
Q

What theorem is important for steady incompressible flow?

A

Bernoulli theorem

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67
Q

Def: vorticity

A

w = nabla x v

- w = (u, v, w) in general

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68
Q

Def: irrotational

A

w = 0 everywhere (vorticity is 0)

- no rotation

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69
Q

Euler eqns: incompressible flow

A

dv/dt + (w x v) = -nabla(p/rho + pi + 1/2 * |v|^2)

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70
Q

Euler eqns: steady incompressible flow

A
  • eqn for incompressible flow
    w x v = -nabla(H)
    v•(w x v) = 0 = (v•nabla)H
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71
Q

Bernoulli function

A

H = p/rho + pi + 1/2 * |v|^2

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72
Q

Physical interpretation of (v•nabla)H = 0

A
  • If v!=0 then H is constant along streamlines

- if H is constant in a direction, directional derivative is 0 in that direction

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73
Q

Steady incompressible flow: irrotational case

A

w x v = 0 = -nabla(H)

- H is constant everywhere

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74
Q

Vorticity in 2D

A 
w = (0, 0, dv/dx - du/dy)
Define w (not vector) as the z component
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75
Q

Physical interpretation of vorticity

A

The local rotation of fluid

- w*(average angular velocity) of two small line elements moving within the fluid

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76
Q

Vorticity: meaning of sign

A

Positive: counter-clockwise
Negative: clockwise

77
Q

Vorticity: irrotational vortex

A

Same procedure, just use polar coordinates!

78
Q

Vorticity eqn for an ideal fluid: derivation

A

Compute the curl of the momentum eqn (incompressible flow)

79
Q

Vorticity eqn for an ideal fluid

A

dw/dt + (v•nabla)w = (w•nabla)v
Dw/Dt = (w•nabla)v

Rate of change of w moving with fluid = stretching / tilting term

80
Q

Physical interpretation of vorticity eqn

A

How vorticity evolves in time

- describes evolution of a needle

81
Q

Vorticity eqn: 2D Euler eqns

A

Dw/Dt = 0

- every fluid particle conserves it’s vorticity

82
Q

Vorticity: shear flow

A 
v = (u(y), 0, 0)
w = -du/dy
83
Q

Vorticity: solid body rotation

A

w = 2*Omega

84
Q

Velocity: irrotational vortex

A

v = Gamma0/2pi * 1/r * e_phi

85
Q

Def: stream function

A

2D incompressible motion
u = -dpsi/dy
v = dpsi/dx

  • psi is constant along streamlines
86
Q

Vorticity: stream function

A

w = nabla^2(psi)

87
Q

Vorticity eqn: stream function

A

d(nabla^2(psi))/dt + J(psi, nabla^2(psi)) = 0

  • J is jacobian
  • one eqn and one unknown -> use to determine velocity, pressure, with BCs
88
Q

Def: Jacobian

A

J(A, B) = dA/dx * dB/dy - dA/dy * dB/dx

89
Q

Difference between centripetal force and centrifugal force

A
  • centripetal: towards centre

- centrifugal: away from centre

90
Q

Why introduce the stream function?

A

Velocity can be written in terms of psi, we reduce the number of variables
- ensures nabla•v = 0 always

91
Q

Def: needle

A

A material line element

92
Q

Def: tilting

A

First two terms of 3D vorticity eqn output

Tilts (different directions)

93
Q

Def: stretching

A
  • same direction of output of 3D vorticity eqn (ex. wz*dw/dz)
  • vorticity size is amplified
94
Q

Def: vortex lines

A

Field lines of the vorticity field

  • solution to dx/ds = w(x, t0)
  • s has units of time x length
  • x is the vortex lines here
95
Q

Properties of vortex lines

A

Can’t cross each other (since nabla•w = 0)

- can be squished and stretched

96
Q

Steady Euler’s flow: properties of Bernoulli function (with vorticity)

A

w x v = -nabla(H)

  • v• -> H constant on stream lines
  • w• -> H constant on vortex lines
97
Q

Def: circulation

A
  • global property
    Gamma(c, t) = integral (c, v(x, t) • dl)
  • c is closed contour

Gamma(c, t) = surface integral (S, nabla x v • dS)

98
Q

Circulation: physical meaning

A

How the fluid is flowing along the contour C

99
Q

Helmholtz decomposition for a 2D vector field

A

v = e_z x nabla(psi) =? nabla(phi)
(u, v) = (-dpsi/dy, dpsi/dx) = (dphi/dx, dphi/dy)

  • psi is stream function
  • phi is potential function
100
Q

Helmholtz decomposition: conditions

A

1) nabla•v = -nabla^2( phi) -> incompressible nabla^2( phi)=0
2) dv/dx - du/dy = nabla^2 (psi) -> irrotational nabla^2(psi)=0

101
Q

Potential flow

A

Helmholtz decomposition + irrotational

- yields Laplace’s eqn

102
Q

Laplace’s eqn

A

nabla^2(psi) = 0
And
nabla^2(phi) = 0

103
Q

Def: stagnation points

A

|v|^2 = 0

From Bernoulli function

104
Q

Solving Laplace’s eqn in four steps

A

1) convert coordinate system if needed
2) separate variables
3) find general solution
4) impose BCs
5) also use Bernoulli function to find pressure

105
Q

Def: sound

A

Small-amplitude compression waves

106
Q

Def: Euler eqns

A

Continuity eqn + momentum eqn

107
Q

Euler eqns: compressible flows

A

Dv/Dt = g - 1/rho * nabla(p) (momentum)

d(rho)/dt + nabla•(rho*v) = 0 (mass)

108
Q

Deriving the wave eqn for small oscillations

A

1) assume a steady state (v = 0, density constant, etc) + in hydrostatic equilibrium
2) find the steady state solution (p0, rho0, v=0, …)
3) perturb the steady state solution slightly (ex. p=p0 + Delta(p))
4) sub perturbed solutions into the Euler eqns
5) linearize (should end up with the two Euler eqns for perturbations now)
6) combine the two eqns together somehow (take derivative and sub in or smth)
7) make any last assumptions (ex. Barotropic lol)
8) tada!

109
Q

Def: speed of sound

A

c0 = sqrt(K0/rho0)

- K0 = bulk modulus evaluated at rho0

110
Q

Adiabatic index of an isothermal gas

A

1

111
Q

Solving the wave eqn

A

1) assume plane wave solution Delta(rho) = rho_1*sin(kx-wt)
2) solve for rho_1, k, w using BCs and subbing into PDE
3) assume similar form for v and p and other variables
4) use the derivation eqns to find them (sub in rho)

112
Q

Dispersion relation

A

Output of wave eqn: w(k)

113
Q

Def: phase velocity

A

c = w/k
w and k are from plane wave solution

  • speed at which crests propagate (crest: kx-wt=constant)
114
Q

Def: group velocity

A

cg = dw/dk
- w, k from plane wave solution

  • speed at which energy propagates
115
Q

Longitudinal wave

A

Velocity of the fluid propagates in the same direction of the wave

116
Q

Wave eqn assumptions

A

Assumed the advection was small compared to the temporal acceleration
- can verify using the solution

117
Q

Jean’s instability: assumptions

A 
Delta(p) = c0^2 * Delta(rho)
nabla•g = -4*pi*G*Delta(rho)
118
Q

Momentum eqn: self gravitating fluid

A

dv/dt + (v•nabla)v = -nabla(p)/rho - nabla(pi)

g = -nabla(pi)

119
Q

Dispersion relation: condition for a wave to exist

A

Require w(k) > 0

120
Q

Wavelength formula

A

lambda = 2pi/k

  • k from plane wave solution
121
Q

Steady compressible flow: assumptions

A

Assume barotropic fluid

122
Q

Def: Mach number

A

Ma = |v|/c

c is sound speed

123
Q

Purpose for defining Mach number

A

If Ma &laquo_space;1 then flow is effectively incompressible

124
Q

Isentropic ideal gas

A

w = cp*T

Cp gamma R eqn

125
Q

Def: Newtonian fluid

A

Fluid with viscosity

  • shear stress sigma_xy directly proportional to the velocity gradient dvx/dy (velocity only has x component)
  • proportionality constant is mu
  • satisfy Stokes’ hypothesis
126
Q

Newton’s law of viscosity

A

sigma_xy = n*dvx(y)/dy
n is shear viscosity, dynamic viscosity, or “viscosity” lol

  • only valid for one specific case
  • n units: Ns/m^2
127
Q

Kinematic viscosity

A

Nu = n/rho

  • nu units: m^2/s
  • usually constant
128
Q

Diffusion eqn

A

dvx/dt = nu*d^2(vx)/dy^2

129
Q

Shear stress in linear shear?

A

Constant

- to show: viscosity + steady planar flow

130
Q

Viscous friction

A
  • drag depends on velocity (can take a long time to slow down)
  • can integrate F=ma to find stopping distance
131
Q

Drag force

A

D = nAU/d

  • A = area
  • d = fluid layer thickness
  • U = velocity
132
Q

Def: isotopic fluid

A

At rest, there is no difference in each direction

sigma_ij = -p*delta_ij

133
Q

Stress tensor: viscous fluid

A

sigma_ij = -p*delta_ij + sigma_ij’

  • first term: ideal fluid behaviour
  • second term: deviation stress tensor from ideal fluid model
134
Q

Strain rate tensor

A

e_ij = 1/2 * (dvi/dxj + dvj/dxi)

Note: ekk = nabla•v

135
Q

Strain rate tensor: physical meaning

A

Measures the rate of deformation in a fluid and is very similar to uij in solids

136
Q

Stokes’ hypothesis

A

sigma_ij’ is linear isotopic function of eij

sigma_ij = -pdelta_ij + mu(dvi/dxj + dvj/dxi) + lambdanabla•v*delta_ij

137
Q

Stokes’ hypothesis: what are mu and lambda?

A

Not lamé coefficients, similar though

Incompressible fluid: mu = n (dynamic viscosity)

138
Q

Viscosity: contact force

A

nabla•sigma^T = -nabla(p) + munabla^2(v) + (mu + lambda)nabla(nabla•v)

Note: incompressible is the same but missing the last term

139
Q

Deriving the Navier-Stokes eqns

A

1) incompressible, homogenous fluid (Euler eqns) (include viscous force)
2) divide by rho

140
Q

Viscosity: mu/n

A

Dynamic viscosity

141
Q

No-slip flow

A

v•t = 0 on boundary delta(D)

- t is unit tangent vector

142
Q

Plane-Poisseulle flow

A

u(y) = P/2*nu * (h^2 - y^2)

  • u is x component of velocity
143
Q

Adjective acceleration

A

(v•nabla)v

- allows fluid to continue along

144
Q

Viscosity (in NS eqns)

A

nu*nabla^2( v)

  • tends to slow the fluid down
145
Q

Def: Reynold’s number

A

Re = UL/nu = Tdiff/Tflow

  • U = velocity scale of motion
  • L = length scale of motion
146
Q

Reynolds number: physical meaning

A

Measures relative importance of inertia vs viscosity
- can predict whether flow is laminar or turbulent

Re&raquo_space; 1 -> viscosity relatively small (drop it)
Re &laquo_space;1 -> viscosity relatively large

147
Q

Diffusion time scale

A

Tdiff = L^2/nu

148
Q

Flow time scale

A

Tflow = L/U

149
Q

Reynolds number in practice

A

1) Assume every variable is of the form x = L*x$, x$ is non dimensional
2) plug into NS eqns
3) calculate Reynolds number

150
Q

The wave eqn with viscosity: differences

A
  • assume solution of the form exp(i(kx-wt))
  • sol is real part
  • use real part for dispersion relation, phase velocity, etc
  • usually multiple cases for k, be sure to consider all of them!
151
Q

Def: inertial frame of reference

A

Newton’s first law says an object will continue at a fixed velocity unless acted on by forces

152
Q

Earth coordinate system

A

X: East
Y: North
Z: up

153
Q

Centrifugal force

A
  • from rotation

Fc = -m*Omega x (Omega x x)
X is distance from axis of rotation

154
Q

Coriolis force

A
  • from rotation + velocity

Fco = -2m*Omega x v

155
Q

Def: fictitious force

A

Aka inertial force or pseudo-force
Extra terms on LHS of F=ma
Arises from being in a non-inertial reference frame

156
Q

Steady rotation: coordinate transformations

A

Prime is in IRF
- Omega = 2pi/rotation period

x = x’cos(Omega*t) + y’sin(Omega*t)
y = -x’sin(Omega*t) + y’cos(Omega*t)
z = z’
157
Q

Velocity in steady rotating frame

A

Velocity from fixed frame + rotation

  • dude just calculate it out
    v’ = dx’/dt, v=dx/dt
158
Q

Steady rotation: coordinate transformations (matrix form)

A 
x = Ax’
v = Av’ - Omega x x
a = Aa’ - Omega x (Omega x x) - (2*Omega x v)
  • Omega = Omega*e_z
  • A is typical rotation matrix with 1 at zz index (cos sin / -sin cos)
159
Q

Net effect of centrifugal force on earth

A

Expand the earth in the tropics and less expand at poles
- oblate spheroid heck yeah

  • can find this by calculating centrifugal force at poles and at equator
160
Q

Effective gravity

A

Aka geopotential
Centrifugal force + gravity combined

  • always orthogonal to the surface
161
Q

Coriolis acceleration: what can we neglect?

A
  • terms with 2Omegacos(theta) except in tropics (then neglect the sin(theta) terms)
162
Q

Eqns for an inviscid fluid in a rotating frame

A

1) NS eqn one, there’s an extra term of 2Omega x v on the LHS and viscosity term is ignored
2) d(rho)/dt + nabla•(rhov) = 0 conservation of mass
3) barotropic

163
Q

Rossby number

A

R0 = U/2Omegasin(theta)*L

164
Q

Rossby number: physical interpretation

A

Relative strength of inertia to coriolis

Ro &laquo_space;1: coriolis is relatively large
Ro&raquo_space; 1: coriolis is relatively small

165
Q

Geostrophic flow

A
  • steady flow, no motion
  • yields geostrophic balance
    2*Omega x v = -1/p * nabla(p) + g
166
Q

Taylor-Proudman theorem

A
  • compute curl of geostrophic momentum eqn

(2*Omega•nabla)v = 0

167
Q

Taylor columns

A

Strong rotation makes fluid act like a rigid body throughout the depth

dv/dz = 0 invariant with height

168
Q

Def: magnetohydrodynamics

A

Study of magnetic properties of electrically conducting fluids

169
Q

What’s the point of learning MHD

A
  • study stars and planets
  • tsunami detection
  • plasma physics
170
Q

Electric field

A

E vector
- moves charges

E = Es + Ei

  • Es electrostatic field
  • Ei electric field induced by changing magnetic field
171
Q

Magnetic field

A

B vector

- moves magnets

172
Q

Current density

A

J vector

- proportional to Coulomb force

173
Q

MHD: velocity

A

u vector

* new notation*

174
Q

MHD: total charge density

A

Rho

175
Q

Electromagnetic forces

A

f = qE + qu x B

  • qE = Coulomb / electrostatic force
  • qu x b = Lorentz force
176
Q

Helmholtz decomposition: E

A 
E = -nabla(V) + nabla x psi
V = potential function (set by Gauss’ law)
Psi = stream vector (set by Faraday’s law)

If rho~0, E = -dA/dt

177
Q

Lorentz force in formula sheet: which term can we usually neglect?

A

The first one rho*E

178
Q

Ampère-Maxwell eqn: simplified form

A

1) compute divergence
2) assume drho/dt small and e0*dE/dt small

Nabla x B = mu*J
- requires nabla•J = 0 (solenoidal condition)

179
Q

Helmholtz decomposition: B

A

B = nabla x A
Nabla • A = 0 -> nabla•B=0

Sub into Faraday’s eqn -> nabla x psi = -dA/dt

nabla^2(A) = -mu*J

180
Q

Approximations for MHD

A

1) qE &laquo_space;qu x B
2) drho/dt ~ 0 + drop Gauss’ law (for ME6)
3) e0*dE/dt &laquo_space;J
4) ignore electrostatic solutions

181
Q

Deriving the induction eqn

A

1) sub ohm’s law into ampère’s law
2) compute curl of 1) result
3) use vector IDs and other eqns lol

182
Q

Baroclinic term

A

Induction eqn and incompressible vorticity eqn are equivalent (with B instead of w) if nabla(rho) x nabla(p) = 0

In the absence of that term, B and w evolve the same way over time
The term is called the baroclinic term

  • since B is indep of u, we can let B evolve linearly (can’t with w bc w depends on u)
183
Q

MHD: governing eqns for incompressible, charged fluid (plasma)

A

1) momentum: Du/Dt = -1/rho * nabla(p) + mu*nabla^2(u) + 1/rho * (J x B)
2) induction: formula sheet
3) incompressible mass eqn: nabla•u = 0

Combine:
J x B = d/dxj (BiBj/mu) + d/dxi (BjBj/2mu)
- first term: magnetic stress
- second term: magnetic pressure

184
Q

Stress tensor for plasma

A

Same as Newtonian fluid normal except two additional terms: magnetic pressure and magnetic stress

  • magnetic pressure has a factor of delta_ij on it
185
Q

Deriving the MHD vorticity eqn

A

1) assume incompressible + constant density
2) sub into governing eqns
3) apply incompressible + rho constant + solenoidal etc
4) compute curl of momentum eqn

186
Q

MHD vorticity eqn: terms meanings

A

1: dw/dt
2: advection (u•nabla)w
3: tilting/stretching (w•nabla)u
4: like advection 1/rho * (B•nabla)J
5: like tilting/stretching -1/rho * (J•nabla)B
6: viscosity nu*nabla^2( w)

187
Q

MHD: Lorentz torque

A

(B•nabla)J

188
Q

2D MHD plasma model

A

1) d(nabla^2 psi)/dt + J(psi, nabla^2 psi) = 1/murho J(A, nabla^2(A)) + nunabla^2(nabla^2(psi))
2) dA/dt + J(psi, A) = lambda*nabla^2(A)

3) u = -nabla x (ezpsi)
4) B = -nabla x (ezA)
5) w = nabla^2(psi)
6) J = nabla^2(A)

189
Q

Waves in MHD: background flow

A 
Psi = 0
A = -alpha*y
u = (0, 0)
B = (alpha, 0)
w = 0
J = 0

AMATH 361 (2 decks)

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