# A Linearization of nonlinear grad(T) in the diffusion equation

#### stockzahn

Homework Helper
Gold Member
Dear all,

I would like to perform numerical simulations of the heat transfer/temperature field in a static bath of superfluid helium. The heat conduction in superfluid helium can be modelled in two regimes depending on the heat flux. For low heat fluxes $\dot{q}$, the temperature gradient depends linearly on the heat flux (Landau regime). With increasing heat flux, a second term becomes more important (Goerter-Mellink regime), in which the temperature gradient depends non-linearly on the transferred heat flux (the exponent $m$ experimentally was determined to be $3.4$):

$$grad(T) = \underbrace{- f_{L} \dot{q}}_{Landau} + \underbrace{- f_{GM} \dot{q}^m}_{Goerter-Mellink}$$

Assuming the contribution of the Landau-term negligible (high $\dot{q}$), after re-arranging the correlation can be plugged in into the diffusion equation ($f_{GM}^{-1} = const.$):

$$\rho c \frac{\partial T}{\partial t} = -\sqrt[m]{f_{GM}^{-1}}\; div\;\sqrt[m]{grad(T)}$$
$$\frac{\partial T}{\partial t} = \underbrace{\frac{-\sqrt[m]{f_{GM}^{-1}}}{\rho c}}_{a_{GM}}\; div\;\sqrt[m]{grad(T)}$$

Exemplarily for the x-direction:

$$\frac{\partial T}{\partial t} = a_{GM}\frac{\partial\left(\frac{\partial T}{\partial x}\right)^{\frac{1}{m}}}{\partial x}$$

Now I would like to discretize the above equation and I obtain

$$\frac{T_i^{n+1}-T_i^{n}}{\Delta t} =a_{GM}\frac{\left(\frac{T_{i+1}-T_{i}}{\Delta x}\right)^{\frac{1}{m}} - \left(\frac{T_{i}-T_{i-1}}{\Delta x}\right)^{\frac{1}{m}}}{\Delta x}$$

$$\frac{T_i^{n+1}-T_i^{n}}{\Delta t} =a_{GM}\frac{\sqrt[m]{T_{i+1}-T_{i}} - \sqrt[m]{T_{i}-T_{i-1}}}{\Delta x^{\frac{m+1}{m}}}$$

$$T_i^{n+1}-T_i^{n} =\underbrace{\frac{a_{GM}\Delta t}{\Delta x^{\frac{m+1}{m}}}}_{M}\left[\sqrt[m]{T_{i+1}-T_{i}} - \sqrt[m]{T_{i}-T_{i-1}}\right]$$

Using the Euler Upwind Scheme:

$$T_i^{n+1} -M\;\sqrt[m]{T_{i+1}^{n+1}-T_{i}^{n+1}} +M\; \sqrt[m]{T_{i}^{n+1}-T_{i-1}^{n+1}}=T_i^{n}$$

This last equation cannot be transformed into an LES for the cells of the numerical mesh. I hope that I'm correct by stating that I need to linearize the equation at a certain point and/or maybe solve it iteratively. Unfortunately I'm not sure, if what I try is even possible at all. I'd appreciate, if someone could guide, help or explain to me if and how this nonlinear correlation can be discretized.

Thanks,
stockzahn

Last edited:
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#### Chestermiller

Mentor
Why not use an automatic stiff integration package (based on the method of lines) such as the stiff package in the IMSL library or the DASSL package available online (probably the double precision versions).?

"Linearization of nonlinear grad(T) in the diffusion equation"

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