Advanced Fluid Dynamics Flashcards
Finding components of curl and divergence
- Divergence (.) is already its own component
- Curl (x) need to take its i-th component
Properties of Tensors
Decomposition of rank-2 tensors
Decomposition of a symmetric tensor
Transformation rule of a rank-2 tensor
State the difference for a rank-3 tensor
With R orthogonal
For rank 3, X gains a third indice, and a third copy of R (still two indicies)
Tensor components
Symmetry of ๐ and ๐ฟ
๐ - anti-symmetric
๐ฟ - symmetric
Polar coordinates
r^2 = x^2 + y^2
x = r cos(theta)
y = r sin(theta)
Tensor quotient rule
The Material derivative
Difference between Eulerian and Lagrangian descriptions of a fluid flow
Eulerian
* specifies the fluid velocity is a function of time t and position x - i.e. u(t,x)
* the velocity is measured at fixed points in space
* the material derivative of a point x^i is u^i
Lagrangian
* specifies fluid particles trajectories as a function of time t and the initial position X - i.e. x(X,t) s.t. x(X,0) = X
* the material derivative of a point X^i is 0
Euler identity
Equation relating Eulerian and Lagrangian flow
Reynolds transport theorem
Conservation of mass
Conservation of mass in differential form
+ incompressibility condition
Fluid is incompressible when density ฯ is constant
Linear velocity at surface (boundary)
radius x velocity comp.
Navier-Stokes equations
Bernoulliโs theorem
Vorticity equation
Describe the body forces
- Forces that act on each particle of fluid across V
- Typical example being gravity
Describe the stress forces
- Stresses, indiciated by ฯ, are forces acting on the surface of the fluid
- Depend on the material and surface orientation
Explain the stress tensor
- ฯ_ij, the stress tensor
- Acts across a surface
- Abides by the tensor quotient rule ฯi = ฯij . nห_j
