Post Midterm Flashcards
How did we derive the Navier-Cauchy equilibrium equations?
Combine Hooke’s law + Cauchy’s strain tensor
Stick it into Cauchy’s equilibrium eqn
How to estimate the relative size of terms (ex. Deformation of legs of a chair?)
Find |Δuij|
Navier-Cauchy eqn: type of PDE
Elliptic
Navier-Cauchy eqn: solid wall BC
u = 0 (rigid wall that doesn’t move)
Navier-Cauchy eqn: open boundary BC
Specify stress tensor
- typically t=sigma•n = 0 (no force)
Navier-Cauchy eqn: where is the boundary?
- who knows? It can move
- we will consider it at rest + small displacements (BCs imposed are slightly deformed boundaries)
- avoids complications
Saint-Venant’s principle
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
Solving problems with uniform settling
- 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)
Alternate def of Laplacian of a vector
Nabla^2 (u) = nabla(nabla•u) - nabla x (nabla x u)
Strain tensor in cylindrical coordinates
1/2 (nabla u + (nabla u)^T)
- assuming symmetric + solid at rest
- hasn’t changed from xyz I don’t think
Newton’s second law (integral form)
Triple integral (rho * u_tt dV) = triple integral (fi dV) + double integral (sigma_ij nj dS)
Newton’s second law (small displacements)
rho * u_tt = f + nabla*sigma^T
Total momentum of idealized particle (mass dM)
dP = v(x, t)dM = vrhodV
- P, v, x, are vectors
P = triple integral (rho*v)dV
Pathlines
- particle trajectory as a function of initial position
- x(X, t) where x is current position, X is initial position, t is time
Lagrangian description
Describes a fluid/particle based on initial position X and time t
- follows a fluid parcel (lazy surfer riding a wave)
Lagrangian description: variables
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
Lagrangian description: requirement
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
Eulerian description
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)
Eulerian description: variables
rho(x, t) = density at position x and time t
v(x, t) = velocity at position x and time t
Relationship between Eulerian and Lagrangian fields
Scalar: rho_L(X, t) = rho(x(X, t), t)
Vector: v_L(x, t) = v(x(X, t), t)
How to solve: given v(x, t), find particle trajectories
Solve for x(X, t) from
1) dx/dt = v(x, t) [1-3 eqns]
2) x = X at t=0
Def: pathlines
x(X, t) for fixed X
- must solve dx/dt = v(x(t), t)
- trajectories
Def: streamlines
Trajectories for a velocity frozen in time, t0 fixed
Must solve dx/dt = v(x(t), t0)
- snapshot of velocity field
Def: streaklines
Finding al, the positions that pass through a given location
- paintbrush lines (paint all fluid parcels)
- not many uses
What abt pathlines, streamlines, streaklines do we know if the velocity is steady?
They’re all the same!
Physical meaning of d(rho_L)/dt
Rate of change of density of a moving fluid parcel initially at X
Physical meaning of d(rho(x, t))/dt
Rate of change of density at fixed point x(X, t)
- we care more abt the lagrangian description!
Def: material derivative
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)
Alternate way to write v•nabla(f)
|v| ( v/|v| • nabla(f))
- directional derivative
Material derivative for vector fields
DFi/Dt = dFi/dt + vj(dFi/dxj)
Assumptions for conservation laws
Bounded, simply connected, smooth orientable body
Other nice things lol
Material volume
D(t) volume in conservation laws
Reynolds Transport Theorem
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))
RTT physical interpretation of terms
- 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
Leibniz’s theorem
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
Principle of conservation of mass
d/dt (triple integral (D(t)) rho(x, t) dV) = 0
Continuity equation
- local form of conservation of mass
- D(rho)/Dt + rho*nabla•v = 0
- d(rho)/dt + nabla•(rho*v) = 0 (flux form)
Continuity equation physical interpretation + alternate form
d(rho)/dt = -v•nabla(rho) - rho*nabla•v
- term 1: local changes in rho
- term 2: advection
- term 3: velocity divergence
Def: incompressible
- a flow that conserves volume
- conservation of mass -> conservation of volume
- nabla•v = 0
- alt: d(rho)/dp = 0
Is there a perfectly incompressible fluid?
Nope
It’s a useful approximation though
Incompressibility and conservation of mass
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)
Principle of conservation of linear momentum
d/dt(triple integral (D(t)) rho(x,t)*v(x,t) dV) = F
- F = sum of forces
- hard to solve
Conservation of linear momentum (local form)
Triple integral(D(t)) rho*Dvi/Dt dV = F - F is a sum of body and contact forces
Conservation of linear momentum (global eqn after RTT)
Triple integral(D(t), (rho*Dvi/Dt - fi - d(sigma_ij)/dxj) dV) = 0
Momentum equation
rho*Dvi/Dt = fi + d(sigma_ij)/dxj
Application: Big Bang
Eqn: Dv/Dt = 0
Sol: v=x/t
Hubble’s law
v = H0*x
Mass density (Newtonian cosmology)
rho(x, t) = rho(t)
Velocity field (Newtonian cosmology)
v(x, t) = H(t)*x
Gravity (Newtonian cosmology)
g(x, t) = -4*pi/3 * G * rho(t) * x
Deriving the cosmological equations
1) sub rho, v into the continuity equation
2) sub rho, v, g into the momentum equation
3) define H = 1/a * da/dt
Def: ideal fluid
1) stress tensor of sigma_ij(x, t) = -p(x, 5)*delta_ij
2) rho is constant
Is an ideal fluid compressible or incompressible?
Incompressible
Euler equations for an ideal fluid
Dv/Dt = f/rho - 1/rho * nabla(p) Nabla•v = 0
What is the ideal fluid model missing? (3)
1) shear stresses (viscosity or stickiness)
2) compressibility
3) changes in density (can be included)
Def: inviscid
Fluid with no viscosity
Characteristics of the Euler equations (4)
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
Def: prognostic
Includes a time derivative (dv/dt = …)
Euler eqns: how do we solve for pressure?
Compute the divergence of the momentum eqn!
Poisson eqn for pressure
Nabla^2(p) = nabla•f - rho*nabla•((v•nabla)v)
Physical meaning of the poisson eqn for pressure
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)
Boundary conditions for an ideal fluid
No normal flow: v•n = 0 on boundary
Velocity doesn’t pass through
Solid body rotation: velocity
v = (-omegay, omegax)
- omega has units of 1/s
Newton’s second law: ideal fluid version (variable density)
rhoDv/Dt = -nabla(p) - ge_z
- e_z is unit vector in z direction
Def: solid body rotation
All solutions rotate at a frequency of omega, like a solid body
What theorem is important for steady incompressible flow?
Bernoulli theorem
Def: vorticity
w = nabla x v
- w = (u, v, w) in general
Def: irrotational
w = 0 everywhere (vorticity is 0)
- no rotation
Euler eqns: incompressible flow
dv/dt + (w x v) = -nabla(p/rho + pi + 1/2 * |v|^2)
Euler eqns: steady incompressible flow
- eqn for incompressible flow
w x v = -nabla(H)
v•(w x v) = 0 = (v•nabla)H
Bernoulli function
H = p/rho + pi + 1/2 * |v|^2
Physical interpretation of (v•nabla)H = 0
- If v!=0 then H is constant along streamlines
- if H is constant in a direction, directional derivative is 0 in that direction
Steady incompressible flow: irrotational case
w x v = 0 = -nabla(H)
- H is constant everywhere
Vorticity in 2D
w = (0, 0, dv/dx - du/dy) Define w (not vector) as the z component
Physical interpretation of vorticity
The local rotation of fluid
- w*(average angular velocity) of two small line elements moving within the fluid
Vorticity: meaning of sign
Positive: counter-clockwise
Negative: clockwise
Vorticity: irrotational vortex
Same procedure, just use polar coordinates!
Vorticity eqn for an ideal fluid: derivation
Compute the curl of the momentum eqn (incompressible flow)
Vorticity eqn for an ideal fluid
dw/dt + (v•nabla)w = (w•nabla)v
Dw/Dt = (w•nabla)v
Rate of change of w moving with fluid = stretching / tilting term
Physical interpretation of vorticity eqn
How vorticity evolves in time
- describes evolution of a needle
Vorticity eqn: 2D Euler eqns
Dw/Dt = 0
- every fluid particle conserves it’s vorticity
Vorticity: shear flow
v = (u(y), 0, 0) w = -du/dy
Vorticity: solid body rotation
w = 2*Omega
Velocity: irrotational vortex
v = Gamma0/2pi * 1/r * e_phi
Def: stream function
2D incompressible motion
u = -dpsi/dy
v = dpsi/dx
- psi is constant along streamlines
Vorticity: stream function
w = nabla^2(psi)
Vorticity eqn: stream function
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
Def: Jacobian
J(A, B) = dA/dx * dB/dy - dA/dy * dB/dx
Difference between centripetal force and centrifugal force
- centripetal: towards centre
- centrifugal: away from centre
Why introduce the stream function?
Velocity can be written in terms of psi, we reduce the number of variables
- ensures nabla•v = 0 always
Def: needle
A material line element
Def: tilting
First two terms of 3D vorticity eqn output
Tilts (different directions)
Def: stretching
- same direction of output of 3D vorticity eqn (ex. wz*dw/dz)
- vorticity size is amplified
Def: vortex lines
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
Properties of vortex lines
Can’t cross each other (since nabla•w = 0)
- can be squished and stretched
Steady Euler’s flow: properties of Bernoulli function (with vorticity)
w x v = -nabla(H)
- v• -> H constant on stream lines
- w• -> H constant on vortex lines
Def: circulation
- 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)
Circulation: physical meaning
How the fluid is flowing along the contour C
Helmholtz decomposition for a 2D vector field
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
Helmholtz decomposition: conditions
1) nabla•v = -nabla^2( phi) -> incompressible nabla^2( phi)=0
2) dv/dx - du/dy = nabla^2 (psi) -> irrotational nabla^2(psi)=0
Potential flow
Helmholtz decomposition + irrotational
- yields Laplace’s eqn
Laplace’s eqn
nabla^2(psi) = 0
And
nabla^2(phi) = 0
Def: stagnation points
|v|^2 = 0
From Bernoulli function
Solving Laplace’s eqn in four steps
1) convert coordinate system if needed
2) separate variables
3) find general solution
4) impose BCs
5) also use Bernoulli function to find pressure
Def: sound
Small-amplitude compression waves
Def: Euler eqns
Continuity eqn + momentum eqn
Euler eqns: compressible flows
Dv/Dt = g - 1/rho * nabla(p) (momentum)
d(rho)/dt + nabla•(rho*v) = 0 (mass)
Deriving the wave eqn for small oscillations
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!
Def: speed of sound
c0 = sqrt(K0/rho0)
- K0 = bulk modulus evaluated at rho0
Adiabatic index of an isothermal gas
1
Solving the wave eqn
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)
Dispersion relation
Output of wave eqn: w(k)
Def: phase velocity
c = w/k
w and k are from plane wave solution
- speed at which crests propagate (crest: kx-wt=constant)
Def: group velocity
cg = dw/dk
- w, k from plane wave solution
- speed at which energy propagates
Longitudinal wave
Velocity of the fluid propagates in the same direction of the wave
Wave eqn assumptions
Assumed the advection was small compared to the temporal acceleration
- can verify using the solution
Jean’s instability: assumptions
Delta(p) = c0^2 * Delta(rho) nabla•g = -4*pi*G*Delta(rho)
Momentum eqn: self gravitating fluid
dv/dt + (v•nabla)v = -nabla(p)/rho - nabla(pi)
g = -nabla(pi)
Dispersion relation: condition for a wave to exist
Require w(k) > 0
Wavelength formula
lambda = 2pi/k
- k from plane wave solution
Steady compressible flow: assumptions
Assume barotropic fluid
Def: Mach number
Ma = |v|/c
c is sound speed
Purpose for defining Mach number
If Ma «_space;1 then flow is effectively incompressible
Isentropic ideal gas
w = cp*T
Cp gamma R eqn
Def: Newtonian fluid
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
Newton’s law of viscosity
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
Kinematic viscosity
Nu = n/rho
- nu units: m^2/s
- usually constant
Diffusion eqn
dvx/dt = nu*d^2(vx)/dy^2
Shear stress in linear shear?
Constant
- to show: viscosity + steady planar flow
Viscous friction
- drag depends on velocity (can take a long time to slow down)
- can integrate F=ma to find stopping distance
Drag force
D = nAU/d
- A = area
- d = fluid layer thickness
- U = velocity
Def: isotopic fluid
At rest, there is no difference in each direction
sigma_ij = -p*delta_ij
Stress tensor: viscous fluid
sigma_ij = -p*delta_ij + sigma_ij’
- first term: ideal fluid behaviour
- second term: deviation stress tensor from ideal fluid model
Strain rate tensor
e_ij = 1/2 * (dvi/dxj + dvj/dxi)
Note: ekk = nabla•v
Strain rate tensor: physical meaning
Measures the rate of deformation in a fluid and is very similar to uij in solids
Stokes’ hypothesis
sigma_ij’ is linear isotopic function of eij
sigma_ij = -pdelta_ij + mu(dvi/dxj + dvj/dxi) + lambdanabla•v*delta_ij
Stokes’ hypothesis: what are mu and lambda?
Not lamé coefficients, similar though
Incompressible fluid: mu = n (dynamic viscosity)
Viscosity: contact force
nabla•sigma^T = -nabla(p) + munabla^2(v) + (mu + lambda)nabla(nabla•v)
Note: incompressible is the same but missing the last term
Deriving the Navier-Stokes eqns
1) incompressible, homogenous fluid (Euler eqns) (include viscous force)
2) divide by rho
Viscosity: mu/n
Dynamic viscosity
No-slip flow
v•t = 0 on boundary delta(D)
- t is unit tangent vector
Plane-Poisseulle flow
u(y) = P/2*nu * (h^2 - y^2)
- u is x component of velocity
Adjective acceleration
(v•nabla)v
- allows fluid to continue along
Viscosity (in NS eqns)
nu*nabla^2( v)
- tends to slow the fluid down
Def: Reynold’s number
Re = UL/nu = Tdiff/Tflow
- U = velocity scale of motion
- L = length scale of motion
Reynolds number: physical meaning
Measures relative importance of inertia vs viscosity
- can predict whether flow is laminar or turbulent
Re»_space; 1 -> viscosity relatively small (drop it)
Re «_space;1 -> viscosity relatively large
Diffusion time scale
Tdiff = L^2/nu
Flow time scale
Tflow = L/U
Reynolds number in practice
1) Assume every variable is of the form x = L*x$, x$ is non dimensional
2) plug into NS eqns
3) calculate Reynolds number
The wave eqn with viscosity: differences
- 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!
Def: inertial frame of reference
Newton’s first law says an object will continue at a fixed velocity unless acted on by forces
Earth coordinate system
X: East
Y: North
Z: up
Centrifugal force
- from rotation
Fc = -m*Omega x (Omega x x)
X is distance from axis of rotation
Coriolis force
- from rotation + velocity
Fco = -2m*Omega x v
Def: fictitious force
Aka inertial force or pseudo-force
Extra terms on LHS of F=ma
Arises from being in a non-inertial reference frame
Steady rotation: coordinate transformations
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’
Velocity in steady rotating frame
Velocity from fixed frame + rotation
- dude just calculate it out
v’ = dx’/dt, v=dx/dt
Steady rotation: coordinate transformations (matrix form)
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)
Net effect of centrifugal force on earth
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
Effective gravity
Aka geopotential
Centrifugal force + gravity combined
- always orthogonal to the surface
Coriolis acceleration: what can we neglect?
- terms with 2Omegacos(theta) except in tropics (then neglect the sin(theta) terms)
Eqns for an inviscid fluid in a rotating frame
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
Rossby number
R0 = U/2Omegasin(theta)*L
Rossby number: physical interpretation
Relative strength of inertia to coriolis
Ro «_space;1: coriolis is relatively large
Ro»_space; 1: coriolis is relatively small
Geostrophic flow
- steady flow, no motion
- yields geostrophic balance
2*Omega x v = -1/p * nabla(p) + g
Taylor-Proudman theorem
- compute curl of geostrophic momentum eqn
(2*Omega•nabla)v = 0
Taylor columns
Strong rotation makes fluid act like a rigid body throughout the depth
dv/dz = 0 invariant with height
Def: magnetohydrodynamics
Study of magnetic properties of electrically conducting fluids
What’s the point of learning MHD
- study stars and planets
- tsunami detection
- plasma physics
Electric field
E vector
- moves charges
E = Es + Ei
- Es electrostatic field
- Ei electric field induced by changing magnetic field
Magnetic field
B vector
- moves magnets
Current density
J vector
- proportional to Coulomb force
MHD: velocity
u vector
* new notation*
MHD: total charge density
Rho
Electromagnetic forces
f = qE + qu x B
- qE = Coulomb / electrostatic force
- qu x b = Lorentz force
Helmholtz decomposition: E
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
Lorentz force in formula sheet: which term can we usually neglect?
The first one rho*E
Ampère-Maxwell eqn: simplified form
1) compute divergence
2) assume drho/dt small and e0*dE/dt small
Nabla x B = mu*J
- requires nabla•J = 0 (solenoidal condition)
Helmholtz decomposition: B
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
Approximations for MHD
1) qE «_space;qu x B
2) drho/dt ~ 0 + drop Gauss’ law (for ME6)
3) e0*dE/dt «_space;J
4) ignore electrostatic solutions
Deriving the induction eqn
1) sub ohm’s law into ampère’s law
2) compute curl of 1) result
3) use vector IDs and other eqns lol
Baroclinic term
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)
MHD: governing eqns for incompressible, charged fluid (plasma)
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
Stress tensor for plasma
Same as Newtonian fluid normal except two additional terms: magnetic pressure and magnetic stress
- magnetic pressure has a factor of delta_ij on it
Deriving the MHD vorticity eqn
1) assume incompressible + constant density
2) sub into governing eqns
3) apply incompressible + rho constant + solenoidal etc
4) compute curl of momentum eqn
MHD vorticity eqn: terms meanings
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)
MHD: Lorentz torque
(B•nabla)J
2D MHD plasma model
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)
Waves in MHD: background flow
Psi = 0 A = -alpha*y u = (0, 0) B = (alpha, 0) w = 0 J = 0
