PDEs (Term 2) Flashcards
What is the rate of change of N, the number of atoms equal to?
dN/dt = -λN
What is c(x,t), Q(x,t) and u(x,t)?
Concentration, heat density and displacement of a string.
State the diffusion equation.
dc(x,t)/dt = D*d^2c(x,t)/dx^2, where D is the diffusion constant.
State the wave equation.
d^2u/dt^2 = c^2 * d^2u/dx^2, where c is the wave velocity.
How do you work out the time which has surpassed using the diffusion equation?
Set Dτ = Δx^2, where Δx is the separation of particles after time τ. Rearrange for τ.
How do you work out the 3D and 1D concentration of particles in volume?
c(3D) = N/V, where N is the number of particles and V is the volume. c(1D) = c(3D)*A, where A is the cross-sectional area.
How do you normalise the concentration?
Integral between 0 and L of c(x,t) dx = N (tube with fixed ends)
What is the flux of particles equal to?
moving to the right - # moving to the left.
State Fick’s Law.
j ∝ -dc/dx (partial)
j = -D dc/dc (partial)
What is the continuity equation?
-The change in # of molecules in Δx per unit time = (j(x,t) - j(x+Δx, t))Δt.
-Same for change in time but with concentration:
= -(c(x,t) - c(x,t+Δt))Δx.
-Set these equal to eachother
-Divide both sides by ΔxΔt
-Take limit Δx, Δt -> 0
-Yields form equal to definition of derivative: df/dx = lim(Δx->0) f(x+Δx)-f(x)/Δx
-Use this to give -dj(x,t)/dx = dc(x,t)/dt
How do you derive the diffusion equation using the continuity equation? What can you change c for in the Diffusion equation?
Use j = -D dc/dx to give the Diffusion equation.
Can change c for Q (heat density)
What is the equation relating temperature and heat capacity, and how can we use this to get the heat equation?
- CQ = T, where C is the heat capacity and Q is the heat.
- Multiply both sides of the heat density diffusion equation by C
- Substitute in T for CQ
- Change D for K, the thermal diffusivity
