<P> Since T depends on x' and t only through the combination x = x ′ + v t (\ displaystyle x = x'+ vt), we have: ∂ T ∂ x ′ = 1 v ⋅ ∂ T ∂ t (\ displaystyle (\ frac (\ partial T) (\ partial x')) = (\ frac (1) (v)) \ cdot (\ frac (\ partial T) (\ partial t))) </P> <P> Thus: ∂ T ∂ t = κ ∇ 2 T = κ ∂ 2 T ∂ 2 z + κ v 2 ∂ 2 T ∂ 2 t (\ displaystyle (\ frac (\ partial T) (\ partial t)) = \ kappa \ nabla ^ (2) T = \ kappa (\ frac (\ partial ^ (2) T) (\ partial ^ (2) z)) + (\ frac (\ kappa) (v ^ (2))) (\ frac (\ partial ^ (2) T) (\ partial ^ (2) t))) </P> <P> We now use the assumption that v (\ displaystyle v) is large compared to other scales in the problem; we therefore neglect the last term in the equation, and get a 1 - dimensional diffusion equation: ∂ T ∂ t = κ ∂ 2 T ∂ 2 z (\ displaystyle (\ frac (\ partial T) (\ partial t)) = \ kappa (\ frac (\ partial ^ (2) T) (\ partial ^ (2) z))) with the initial conditions T (t = 0) = T 1 ⋅ Θ (− z) (\ displaystyle T (t = 0) = T_ (1) \ cdot \ Theta (- z)). </P> <P> The solution for z ≤ 0 (\ displaystyle z \ leq 0) is given by the error function erf (\ displaystyle \ operatorname (erf)): </P>

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