Derive two-dimensional heat equation

Setting. A thin homogeneous rectangular plate (thickness h, density ρ, specific heat σ, thermal conductivity K) has its two faces insulated, so heat flows only in the xy-plane. Let u(x, y, t) be the temperature.

Law of conduction. Heat flows from hot to cold at a rate proportional to the temperature gradient: the heat crossing an area A per unit time is −K A ∂u/∂n.

Element. Take a small rectangle with sides δx and δy.

   y+δy  ┌──────────────┐
         │              │
         │   element    │
   y     └──────────────┘
        x             x+δx

Heat flow in the x-direction. Area of the faces = h δy.

• Gain at x: −K h δy (∂u/∂x)ₓ
• Loss at x + δx: −K h δy (∂u/∂x)_{x+δx}

Net gain in x-direction per unit time = K h δy [ (∂u/∂x)_{x+δx} − (∂u/∂x)ₓ ] ≈ K h δy δx ∂²u/∂x²

Heat flow in the y-direction. Area of faces = h δx. Similarly

Net gain = K h δx [ (∂u/∂y)_{y+δy} − (∂u/∂y)_y ] ≈ K h δx δy ∂²u/∂y²

Total net gain = K h δx δy (∂²u/∂x² + ∂²u/∂y²)

Heat absorbed. The element has mass ρ h δx δy, and its temperature changes at rate ∂u/∂t:

heat absorbed per unit time = σ ρ h δx δy ∂u/∂t

Equate:

σρ h δx δy ∂u/∂t = K h δx δy (u_xx + u_yy)

Cancel h δx δy and put c² = K/(σρ), the diffusivity:

∂u/∂t = c² (∂²u/∂x² + ∂²u/∂y²)

This is the two-dimensional heat equation. In steady state (∂u/∂t = 0) it reduces to Laplace's equation u_xx + u_yy = 0.