potential analogies

Potential Problem Analogies PhL 8.5.10 This is very incomplete. Use split-screen to compare two worlds. 1. Electrostatics: potential V potential difference ΔV drives electric current dQ/dt via ohms law current I = dQ/dt = ΔV /R surface of metal is place where V = constant electric field E = -V = electric flux density, constant = 1 current density J = κE constant = κ = electric conductivity ∂nV = En = 4πσ at a metal boundary surface, σ = charge density = electric flux source 2V ~ ρ = 3D charge density ∂nV ~ σ = charge density source Dirichlet BC: specify V on boundary Neumann BC: charge density flow on boundary positive and negative charge charge-free boundary (such as cap of bowl) has ∂nV = 0 2. Heat flow: temperature u temperature difference Δu drives heat current dQ/dt via the conduction heat transfer law heat current I = dQ/dt = Δu/R R = thermal resistor resistance surface against heat reservoir is place where u = constant heat flow field F = - k u(x) constant = k = thermal conductivity heat current density J = F constant = 1 ∂nu = heat current source at a boundary = (-1/k) Fn = like charge density σ but has no name ( heat flow: ∂tQ = -k ∂nu ∂tQ is like σ, heat flow through unit area of surface. ) 2u = 3D heat source density = q(x) ∂tu - (k/c)2u = f(x,t)/c = 3D heat source // Stak v I p 328 static temperature distribution: - (k/c)2u = f(x,t)/c f ~ ρ -k2u = f(x,t) dimensions: f = energy/vol/time. k = energy/time/degree TBC Dirichlet BC: specify u on boundary Neumann BC: specify heat flow ∂nu on boundary positive and negative heat flow source (source and sink) insulated boundary means ∂nu = 0 since no heat flow cross boundary. (Neumann) 3. Fluid flow velocity field v is like E v = φ φ is the velocity potential hard boundary wall sets vperp = 0 = ∂nφ = Neumann with σ = 0 ∂nφ ~ source at a boundary also at wall we have vparallel = 0 like we have Eparallel= 0 at a metal wall. (so this is true where the wall has no source or sink, ie, it is not a leaky wall)