Laplacian Growth Without Surface Tension in Filtration Combustion: Analytical Pole Solution

Abstract

Filtration combustion is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion in a complex plane. The main problem with such a solution is the existence of finite time singularities. To prevent such singularities, nonzero surface tension is usually used. However, nonzero surface tension does not exist in filtration combustion, and this destroys the analytical solutions. However, a more elegant approach exists for solving the problem. First, we can introduce a small amount of pole noise to the system. Second, for regularization of the problem, we throw out all new poles that can produce a finite time singularity. It can be strictly proved that the asymptotic solution for such a system is a single finger. Moreover, the qualitative consideration demonstrates that a finger with 1/2 of the channel width is statistically stable. Therefore, all properties of such a solution are exactly the same as those of the solution with nonzero surface tension under numerical noise. The solution of the Saffman-Taylor problem without surface tension is similar to the solution for the equation of cellular flames in the case of the combustion of gas mixtures.

Appendices

Appendix 1: Proof of Theorems

We need to prove that $\tau \mapsto \infty$ if $t\mapsto \infty$ and if no finite time singularity exists. The formula for $\tau$ is as follows:

$$\begin{aligned} \tau =t+\left[ -\frac{1}{2}\sum _{k=1}^{N}\sum _{l=1}^{N}\overline{\alpha _{k}}\alpha _{l}\log (1-\overline{a_{k}}a_{l})\right] +C_{0}, \end{aligned}$$

(5.62)

where $\mid a_{l}\mid <1$ for all l.

Let us prove that the second term in this formula is greater than zero:

$$\begin{aligned} -\frac{1}{2}\sum _{k=1}^{N}\sum _{l=1}^{N}\overline{\alpha _{k}}\alpha _{l}\log (1-\overline{a_{k}}a_{l})=\nonumber \\ -\frac{1}{2}\sum _{k=1}^{N}\sum _{l=1}^{N}\overline{\alpha _{k}}\alpha _{l}\sum _{n=1}^{\infty }\left( -\frac{(\overline{a_{k}}a_{l})^{n}}{n}\right) =\nonumber \\ \frac{1}{2}\sum _{n=1}^{\infty }\frac{1}{n}\left( \sum _{k=1}^{N}\overline{\alpha _{k}}(\overline{a_{k}})^{n}\right) \left( \sum _{l=1}^{N}\alpha _{l}(a_{l})^{n}\right) =\nonumber \\ \frac{1}{2}\sum _{n=1}^{\infty }\frac{1}{n}\overline{\left( \sum _{l=1}^{N}\alpha _{l}(a_{l})^{n}\right) }\left( \sum _{l=1}^{N}\alpha _{l}(a_{l})^{n}\right) >0 \end{aligned}$$

(5.63)

Therefore, the second term in (5.62) always greater than zero, and consequently, $\tau \mapsto \infty$ if $t\mapsto \infty$ for no finite time singularity .

Appendix 2: Filtration Combustion—Basic Equations

In the Appendix we mainly follow to the short derivation of Laplacian growth for Filtration Combustion [15]. More detailed derivation can be found in [19].

The filtration of a gas via the combustion products at a constant speed in the direction of combustion area propagation. It is postulated that the temperature in the front and gas flow speed corresponds to the Filtration Combustion regime, where the oxidizer is totally utilized in the reaction with the fuel component of the porous medium [146, 147]. In such case, the burning speed is driven by the oxidizer flow rate, and, therefore, the thermal task can be decoupled from the hydrodynamic one. In the hydrodynamic estimation, where the combustion area thickness is viewed as infinitesimally smaller than the combustion channel width, the task of the steady propagation of a Filtration Combustion front is formulated with the help of equations of conservation of mass of oxidizer, gas and fuel:

$$\begin{aligned} \textit{div}\ \overline{G\ }=\ {\mu }_g\ {\rho }_{f0}W \end{aligned}$$

(5.64)

$$\begin{aligned} \textit{div}\ \overline{G}a\ =\ -\mu \ {\rho }_{f0}W \end{aligned}$$

(5.65)

$$\begin{aligned} U\ \partial \nu /\ \partial x\ =\ -W \end{aligned}$$

(5.66)

$$\begin{aligned} \overline{G}\ =\rho \overline{V},\overline{V} = -\textit{fgrad}(p), W=W_{0}\delta \left( x-x_f \right) ,\ p=\rho \textit{RT} \end{aligned}$$

(5.67)

in which $\overline{G}$ and $\overline{V}$ are the flux and speed of the gas; $\rho$, p, and T are the molar density, pressure, and temperature of gas;$\ {\rho }_{f0}$ is the density of unburnt fuel; $f\$is the filtration coefficient, $\nu$ is the extent of fuel burnout; a is the oxidizer fraction; W is the rate of the reaction in the combustion front $x_f$; ${\mu }_g$ is the coefficients of generation of net gaseous products (${\mu }_g={\mu }_{gp}-\mu$; ${\mu }_{gp}$—full gas production) and $\mu$ is the coefficients of generation of oxidizer per unit mass of fuel, R—constant.

Equations (5.64)–(5.67) are written in the coordinates going in the direction at the reaction front speed U, which is much small in comparison to the filtration speed. In front of (unburnt material region, 1) and behind (burned material region, 2) the combustion front, the right-hand sides of (5.64)–(5.66) are zeros, and the task can be decreased to one equation:

$$\begin{aligned} \Delta P\ =\ 0,\ \ P\ =\ p^2. \end{aligned}$$

(5.68)

Equations (5.65) and (5.66) are satisfied at constant a and $\nu$. In region 1, the reaction does not occur because of the absence of the oxidizer ($a_1 =0,{\nu }_1= 1$). In region 2, the reaction ceases because of a complete burnout of the fuel ($a_2\ =\ a_0,\ {\nu }_2=\ 0$).

The pressure is the same on either side of the front:

$$\begin{aligned} x\ =\ x_f,\ {\ P}_1=P_2 \end{aligned}$$

(5.69)

The normal components of the flux $G_n$ and speed $U_n$ are determined by integration of (5.64)–(5.66) over the cross sectional area of the front:

$$\begin{aligned} G_{1n}-G_{2n}={\mu }_g\ U_n\ {\rho }_{f0}\ \end{aligned}$$

(5.70)

$$\begin{aligned} a_0G_{2n}=\mu \ U_n{\rho }_{f0}\ \end{aligned}$$

(5.71)

This declaration takes into account the basic hydro-dynamic aspects of the task: the difference in the permeability of the burned ($f_2$) and unburnt ($f_1$) materials and the change in the gas mass during the interaction of the oxidizer with the fuel ingredient of the porous medium. As showed in [19], the trivial solution to the task such as a flat front occupying the full cross section of the channel may be unstable. It will become unstable when

$$\begin{aligned} F_1<F_2(1 +{\mu }_ga_0/\mu ),\ \ \ F =f/2\textit{RT}, \end{aligned}$$

(5.72)

in which $F_1$ and $F_2$ are the effective permeability of the initial medium and products, correspondingly; $a_0$ is the oxidizer concentration at the channel inlet.

A grow in the permeability of the products and gas release at the time of the burnout of the combustible mixture is common for real situations; therefore, a planar Filtration Combustion wave is basically unstable in wide enough channels. In [19], it was also proven that, together with the trivial solution in the form of a planar wave, the Filtration Combustion task has a solution equivalent to the solution received by Saffman and Taylor in examining the displacement of a high-viscosity liquid by a low-viscosity liquid in a narrow space between two plates. In the traditional formulation of this task [148], the pressure in the two liquids must be identified by the Laplace equation, while, at the interface between phases, the conditions of continuity of the pressure and normal component of the speed are established. For these boundary conditions, Saffman and Taylor received the following analytical solution for the interface between phases:

$$\begin{aligned} x=\frac{1-\lambda }{\pi }\ln \left\{ \frac{1}{2}\left[ 1+{\cos \left( \pi y/\lambda \right) \ }\right] \right\} \end{aligned}$$

(5.73)

Right here, x and y are the coordinates in the longitudinal and transverse directions normalized by the channel width in the coordinate frame set at the foremost point of the propagating boundary. The interface position described by expression (5.73) possesses the form of a finger symmetric regarding the channel axis. The reduced width of the finger $\lambda$ far from the tip is a free parameter; i.e., expression (5.66) presents a set of solutions to the problem related to numerous possible values of $\lambda \ (0<\lambda <1)$. The speed of propagation of the finger U increases as its width $\lambda$ decreases. At a big ratio among the viscosities of the displaced and displacing liquids, we may write

$$\begin{aligned} U\ =\ V/\lambda . \end{aligned}$$

(5.74)

It was showed [149] that, besides the set of solutions (5.73) for symmetrical fingers, there exists a set of asymmetrical solutions that corresponds to fingers attracted to one wall of the channel. The task of choosing of an real solution to the Saffman-Taylor (SF) problem from a continuum set of possible solutions has attracted the interest of both theoreticians and experimentalists during the last 50 years. The experimentally known solution under conditions in accordance with the assumptions underlying the formation of the problem [148] is a symmetrical finger occupying half the channel width $\lambda \ =\ 1/2$. Tries to establish this outcome by invoking extremal properties of this solution, like the maximum entropy production as well as minimum work done, were unsuccessful.

The authors of [18] suggested a rule of selection of solutions based on comparing of the scales of waves in the spectrum of possible values. The propagation of a finger can be viewed as as a travelling wave of the flowing liquid in the transverse direction. The transverse flow field is localized near the tip. Estimates of the scale L of transverse flow waves for a set of symmetrical fingers [18] shows that the quantity $L\ \sim \lambda (1\ -\lambda )$ has its maximum at $\lambda \ =\ 1/2$.

Application of this criterion to the full spectrum of solutions to the ST problem provides an asymmetrical finger having $\lambda \ =\ 1/2$, which moves along a wall of the channel, a finger that has never been discovered in experiments on liquid displacement.

In this book and the paper [14], we propose more relevant and dynamically motivated principle of selection of solutions based on minimum of finite time singularities (see Fig. 5.3).

Currently, the majority of scientists focused on the ST problem suppose that a key criterion in selecting the solution is the surface tension at the interphase boundary, which provides an increase to a pressure jump proportionate to the local curvature of the finger. Quantitatively, the effect of surface tension is small but when accounted for, it radically modifies the spectrum of problem’s solutions. It was showed [150] that, at an arbitrary nonzero value of the surface tension , asymmetrical fingers and fingers with a width of $\lambda < 1/2$ turn out to be forbidden. Experimental data demonstrating the absence of such fingers show that the surface tension has an important role in the mechanism of choosing the solution.

For Filtration Combustion , the hydrodynamic approximation produces the same spectrum of solutions as that obtained for the ST problem; however, the task of choosing the real solution remains unsolved. In this case, the interface is the combustion front, which has no such a feature as surface tension . As a result, both symmetrical and asymmetrical fingers from the full set of possible solutions can be came to the realization.

The detailed numerical investigation of front instability for the basic equations of Filtration Combustion [19] was carried out in [151–153].