Band Fluctuation for the Average Temperature Describing a Front Wave Propagating in a Random Porous Medium

Abstract

In this work, we have formulated a stochastic model to describe the dynamics of a wave of heat, which propagates through an inhomogeneous one-dimensional porous medium. Using a previously studied analytic but deterministic model for describing a combustion tube, we have stochastically extended it. To represent the undetermined heterogeneity of the transport properties of the medium, we have allowed simultaneously the diffusive and convective terms to be randomly functions of the position. We have calculated analytically the two first momenta of the spatially distributed temperature profile, which have permitted us to discern the effects of randomness on both the expected value of temperature profiles and its corresponding standard deviation. Our results have shown that it is possible to estimate characteristic periodic lengths associated to the spatial variations of both the convective and diffusive properties. Finally, in the limit case when the noise intensity is null, we have consistently recovered the deterministic model from which we have departed.

Appendix

Here we display the explicit expresions of the integrals involved in Eqs. (54–56) and Eqs. (76–81) in the main text,

$$\begin{aligned} I_1= & {} \int _{0}^{\infty }\frac{2 {\varGamma }_{h}(\chi )\left( \cosh \omega \chi -1\right) }{\omega ^{2}}\mathrm{d}\chi \end{aligned}$$

(104)

$$\begin{aligned} I_2= & {} \int _{0}^{\infty }\frac{{\varGamma }_{h}(\chi )\left( \beta _{0} -\beta _{0}\cosh \omega \chi -\omega \sinh \omega \chi \right) }{ \omega ^{2}}\mathrm{d}\chi \end{aligned}$$

(105)

$$\begin{aligned} I_3= & {} \int _{0}^{\infty }\frac{{\varGamma }_{\beta }(\chi )\left( 2h_{0}+\beta _{0}^{2}+2h_{0}\cosh \omega \chi \right) }{\omega ^{2}}\mathrm{d}\chi \end{aligned}$$

(106)

$$\begin{aligned} I_4= & {} \int _{0}^{\infty }\frac{{\varGamma }_{\beta }(\chi )h_{0}\left( \beta _{0}\cosh \omega \chi +\omega \sinh \omega \chi -\beta _{0}\right) }{\omega ^{2}} \mathrm{d}\chi \end{aligned}$$

(107)

$$\begin{aligned} I_5= & {} \int _{0}^{\infty }\frac{{\varGamma }_{h}(\chi ){\varOmega } _{21}\left( 1-\mathrm{e}^{-\frac{\beta _{0}}{2}\chi }{\varOmega }_{22}\right) }{\omega }\mathrm{d}\chi \end{aligned}$$

(108)

$$\begin{aligned} I_6= & {} \int _{0}^{\infty }\frac{{\varGamma }_{\beta }(\chi )h_0\left( \beta _{0}-\beta _{0}\cosh \omega \chi -\omega \sinh \omega \chi \right) }{\omega ^{2} }\mathrm{d}\chi \end{aligned}$$

(109)

$$\begin{aligned} I_7= & {} \int _{0}^{\infty }\frac{{\varGamma }_{h}(\chi ){\varOmega }_{21}}{\omega }\mathrm{d}\chi , \end{aligned}$$

(110)

such that, $I_{8}=4I_{3}$, $I_{9}=4I_{6}$ and

$$\begin{aligned} I_{10}= & {} \int _{0}^{\infty }\frac{2{\varGamma }_{h}(\chi )\left( 2h_{0}+2h_{0} \cosh (\omega \chi )+g\right) }{\omega ^{2}}\mathrm{d}\chi ;\nonumber \\ g\equiv & {} \beta _{0}^{2}\cosh (\omega \chi )+\beta _{0} \omega \sinh (\omega \chi )\end{aligned}$$

(111)

$$\begin{aligned} I_{11}= & {} \int _{0}^{\infty }\frac{2{\varGamma }_{h}(\chi )(2-2\cosh (\omega \chi ))}{\omega ^{2}}\mathrm{d}\chi \end{aligned}$$

(112)

$$\begin{aligned} I_{12}= & {} \int _{0}^{\infty }\frac{{\varGamma }_{\beta }(\chi )(\beta _{0}-\beta _{0} \cosh (\omega \chi )+\omega \sinh (\omega \chi ))}{\omega ^{2}}\mathrm{d}\chi \end{aligned}$$

(113)

$$\begin{aligned} I_{13}= & {} \int _{0}^{\infty }\frac{{\varGamma }_{\beta }(\chi )(4h_{0}+\beta _{0}^{2})}{\omega ^{2}}\mathrm{d}\chi \end{aligned}$$

(114)

$$\begin{aligned} I_{14}= & {} \int _{0}^{\infty }\frac{2{\varGamma }_{h}(\chi ) (\beta _{0}\cosh (\omega \chi )-\beta _{0}+\omega \sinh (\omega \chi ))}{\omega ^{2}}\mathrm{d}\chi , \end{aligned}$$

(115)

here,

$$\begin{aligned} {\varOmega }_{21}(\chi )\equiv \left( \mathrm{e}^{\omega \chi }-1\right) \mathrm{e}^{\frac{1}{2}\left( \beta _0-\omega \right) \chi } \end{aligned}$$

(116)

and

$$\begin{aligned} {\varOmega }_{22}(\chi )\equiv \left( \cosh \left[ \frac{\omega }{2}\chi \right] +\frac{\beta _0 \sinh \left[ \frac{\omega }{2}\chi \right] }{\omega }\right) , \end{aligned}$$

(117)

where $\omega =\sqrt{4h_0+\beta _0^2}$.