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