Thermal diffusion of Gaussian

Thermal diffusion of Gaussian refers to the specific solution

\(\Psi=\Psi(x,t)=\alpha(t) \exp(-\beta(t) x^2)=\alpha \epsilon\)

of the equation of thermal conductivity

\(\dot \Psi=\Psi''\)

where dot differentiates with respect to time \(t\) and prime differentiates with respect the spatial variables \(x\). Parameters \(\alpha\) and \(\beta\) are supposed to be functions of time. Evolution of these parameters can be determined from the thermal diffusion equation.

Calculation of \(\alpha\) and \(\beta\)

Derivative of \(\Psi\) with respect to \(j\)th coordinates appears as follows:

\(\Psi'_{,j}=-2 \alpha \beta x_j \epsilon\)

and the second derivative

\(\Psi''_{,j,j}=-2 \alpha \beta x_{j,j} \epsilon+ 4\alpha\beta^2 x_j x_j \epsilon =-6 \alpha \beta \epsilon+ 4\alpha\beta^2 x^2 \epsilon\)

where summation with respect to repeating index \(j\) is assumed. For 3-dimensional case, \(j\) takes values 1,2,3; so, \(x_{j,j}=3\).

The time derivative of solution is expressed as follows:

\(\dot \Psi = \dot \alpha\epsilon - \alpha \dot \beta \epsilon x^2\)

Then, the thermal diffusion equation gives

\(\dot\alpha=-6\alpha \beta~\), and \(~ \dot \beta=-4\beta^2\)

The last equation has solution \( ~\displaystyle \beta=\frac{1}{4t}\)

Then, for \(\alpha\),

\(\displaystyle \frac{\dot\alpha}{\alpha}=-6 \frac{1}{4t}= -\frac{3}{2} \frac{1}{t}\)

\(\displaystyle (\ln(\alpha))^{\bullet} = -\frac{3}{2} \frac{1}{t}\)

\(\displaystyle \ln(\alpha) = -\frac{3}{2} \ln(t)=\ln\big( t^{-3/2} \big)\)

\(\alpha=t^{-3/2}\)

Constants of integration

In the deduction above, the constants of integration are omitted. They can be recovered using the translational invariance of the diffusion equation. For example, after such a recovery, the temperature at coordinate zero can be expressed as follows:

\(y=y(t)=a + b\, (t\!-\!c)^{-3/2}\)

where \(a\) refers to the background constant temperature, \(c\) indicate the place of singularity, and \(b\) refers to the thermal capacity and thermal conductivity, that can easy included into the initial equation by scaling of the coordinate \(x\) and/or time \(t\).

Such a representation of temperature can be used to fit experimental data of evolution of temperature at the center of localised (punctual) warming or cooling.

Recovery of dimensions

The scaling of coordinate and/or scaling of time allows to write the temperature distribution as follows:

\(\displaystyle \Phi=\Phi(X,T)= \frac{W r^{1/2}T^{-3/2} }{\pi^{3/2} s^{3/2} } \exp\left( - \frac{r/s}{4T} X^2 \right) \)

where
\(X\) are dimensional coordinates, proportional to \(x\)
\(T\) is dimensional time, proportional to \(t\)
\(r\) is thermal capacity of the material
\(s\) is its termal conductivity
\(W\) is total energy of pulse used to hit the material. In the case, if the half–space is occupied by the material, this \(W\) is doubled energy of the initial heating pulse.

This \(\Phi\) satisfies the physical, dimensional equation of thermal conductivity

\(r\, \dot \Phi = k\, \Phi ''\)

Measurements of temperature at \(X=0\) as function of time and approximating it with power function \(\displaystyle f(T)= \frac{W }{\pi^{3/2}} q^{1/2} T^{-3/2} \) gives the fitting parameter \(q=r/s^3\).

In such way, with the single pulse, ratio of the thermal capacity of the material to the cube of the thermal conductivity can be measured.

References

http://web.csulb.edu/~kmerry/FourierAnalysis/Fourier_09.pdf Kent G. Merryfield. 9. The Gaussian and the Heat Equation on the Line.

Keywords

Diffusion