How are fluids represented in Eulerian and Lagrangian description?
Eulerian: grid-based
Lagrangian: particle-based
What is the Nabla operator?
partial derivatives
What does the gradient of a scalar field tell us?
vector field which points in the direction of greatest increase
What is the gradient of a vector field?
Tensor field (Jacobian Matrix at each point)
Which information does the deformation tensor encode?
How a point is moving with its neighbors
Explain divergence in your own words!
Scalar value which represents how much a infinitesmal volume is expanding/contracting
What is the curl of a 2D/3D vector field?
rotation of the field
Can we compute "curl" of a 2D scalar field?
yes, compute to cogradient
What does the Laplace operator do?
measures how far a quantity is from the mean around it
What does a material derivative compute?
Describes the change of a physical quantity for a material element
moving with space-time dependent velocity
Compute all differential operators for a given analytical scalar and/or vector field!
Gradient: 𝛻𝜙(𝑥, 𝑦)
Divergenz: 𝛻 ⋅ 𝑢(𝑥, 𝑦)
Curl: 𝛻 × 𝑢(𝑥, 𝑦, 𝑧)
Laplacian: 𝛻 ⋅ 𝛻𝑝(𝑥, 𝑦, 𝑧)
What is the difference between a PDE and an ODE?
An ordinary differential equation (ODE) contains only derivatives with
respect to a single variable
A partial differential equation (PDE) contains derivatives with respect to
multiple variables
What does a conservation law describe?
How can we discretize the state of a dynamical system described by a PDE?
finite differences, initial value problem
Name the advection equation, diffusion equation and wave equation!
Advection:
Diffusion:
Wave:
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