**Appendix 2**

### **2.2. The temperature model**

At the thermal equilibrium, Eq. (7) are:

$$\frac{1}{r^2}\frac{\partial}{\partial r}\Big|r^2\frac{\partial T\_i}{\partial r}\Big| + \beta\_i^2 \langle T\_i(\mathbf{r}) - T\_i \rangle = 0 \text{ or } \frac{\partial^2 T\_i}{\partial r^2} + \frac{2}{r}\frac{\partial T\_i}{\partial r} - \beta\_i^2 \ T\_i + \beta\_i^2 \langle T\_i(\mathbf{r}) \rangle = 0 \tag{A2.1}$$

Using substitution *Ri*  = *rTi* , these equations become:

$$\frac{d^2R\_i}{dr^2} - \beta\_i^2 \ R\_i + \beta\_i^2 \ \text{r } T\_e^i(\mathbf{r}) \ = 0 \tag{A2.2}$$

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with solutions

$$\begin{aligned} R\_{\mathbf{i}}(\mathbf{r}) &= \mathbf{c}\_{\mathbf{i}} \cosh(\boldsymbol{\beta}\_{\mathbf{i}} \cdot \mathbf{r}) + \mathbf{c}\_{\mathbf{z}} \sinh(\boldsymbol{\beta}\_{\mathbf{i}} \cdot \mathbf{r}) + \mathbf{v}\_{\mathbf{i}} \cosh(\boldsymbol{\beta}\_{\mathbf{i}} \cdot \mathbf{r}) + \mathbf{v}\_{\mathbf{z}} \sinh(\boldsymbol{\beta}\_{\mathbf{i}} \cdot \mathbf{r}) \\\\ R\_{\mathbf{z}}(\mathbf{r}) &= \mathbf{c}\_{\mathbf{z}} \cosh(\boldsymbol{\beta}\_{\mathbf{z}} \cdot \mathbf{r}) + \mathbf{c}\_{\mathbf{z}} \sinh(\boldsymbol{\beta}\_{\mathbf{z}} \cdot \mathbf{r}) + \mathbf{v}\_{\mathbf{z}} \cosh(\boldsymbol{\beta}\_{\mathbf{z}} \cdot \mathbf{r}) + \mathbf{v}\_{\mathbf{z}} \sinh(\boldsymbol{\beta}\_{\mathbf{z}} \cdot \mathbf{r}) \end{aligned}$$

where

$$\begin{aligned} \mathbf{v}\_{1} &= \frac{1}{\beta\_{1}} \mathbf{f} \left[ \mathbf{a}\_{1} + \mathbf{b}\_{1} \,\, \Phi\_{1}(\mathbf{r}) \right] \sinh(\beta\_{1} \mathbf{r}) \,\mathrm{d}\mathbf{r} \text{ and } \mathbf{v}\_{2} = -\frac{1}{\beta\_{1}} \mathbf{f} \,\, \mathbf{r} \left[ \mathbf{a}\_{1} + \mathbf{b}\_{1} \,\, \Phi\_{1}(\mathbf{r}) \right] \cosh(\beta\_{1} \mathbf{r}) \mathrm{d}\mathbf{r} \\\\ \mathbf{v}\_{3} &= \frac{1}{\beta\_{2}} \mathbf{f} \,\, \mathbf{r} \left[ \mathbf{a}\_{2} + \mathbf{b}\_{2} \,\, \Phi\_{2}(\mathbf{r}) \right] \sinh(\beta\_{2} \mathbf{r}) \,\, \mathbf{d}\mathbf{r} \text{ and } \mathbf{v}\_{4} = -\frac{1}{\beta\_{2}} \mathbf{f} \,\, \mathbf{r} \left[ \mathbf{a}\_{2} + \mathbf{b}\_{2} \,\, \Phi\_{2}(\mathbf{r}) \right] \cosh(\beta\_{2} \mathbf{r}) \mathrm{d}\mathbf{r} \end{aligned}$$

These expressions contain the following notations:

$$\begin{aligned} a\_1 &= \beta\_1^2 \,\mathrm{T}\_\mathrm{B}^1 \text{ and } a\_2 = \beta\_2^2 \,\mathrm{T}\_\mathrm{B}^2 \\\\ \mathbf{b}\_1 &= \frac{\beta\_1^2 \,\mathrm{P}}{\omega\_\mathrm{b}^1 \,\mathrm{c}\_\mathrm{b} \,\,\rho\_\mathrm{b}} \text{ and } \mathbf{b}\_2 = \frac{\beta\_2^2 \,\mathrm{P}}{\omega\_\mathrm{b}^2 \,\mathrm{c}\_\mathrm{b} \,\rho\_\mathrm{b}} \end{aligned}$$

The solutions of Eq. (A2.1) are:

$$T\_1(\mathbf{r}) = \mathbf{c}\_1 \frac{\cosh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{c}\_2 \frac{\sinh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{v}\_1 \frac{\cosh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{v}\_2 \frac{\sinh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} \tag{A2.3}$$

and

$$T\_2(\mathbf{r}) = \mathbf{c}\_3 \frac{\cosh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{c}\_4 \frac{\sinh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{v}\_3 \frac{\cosh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{v}\_4 \frac{\sinh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} \tag{A2.4}$$

c1 , *c*2 , *c*3 , *c*4 are the integration constants and *β<sup>i</sup>* (i = 1, 2) have the expressions:

$$\beta\_1^2 = \frac{\alpha\_{\rm b}^1 c\_{\rm b}}{k\_1} ; \beta\_2^2 = \frac{\alpha\_{\rm b}^2 c\_{\rm b}}{k\_2}$$

### **Author details**

Iordana Astefanoaei\* and Alexandru Stancu

\*Address all correspondence to: iordana@uaic.ro

Faculty of Physics, Alexandru Ioan Cuza University of Iasi, Romania
