Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1963, Volume 149, No. 2
Physical Chemistry
Ya. I. Kanel’
On a Stationary Solution for a System of Equations of Combustion Theory
(Presented by Academician Ya. B. Zel’dovich on 6 IX 1962)
The one-dimensional process of combustion of a combustible gas mixture is described by the system (see (¹))
\[ \begin{aligned} \frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2} &= f(u)v,\\ \frac{\partial v}{\partial t}-\lambda\frac{\partial^2 v}{\partial x^2} &= -f(u)v. \end{aligned} \tag{1} \]
Here \(u\) is the temperature of the mixture; \(v\) is the concentration of the active substance; \(f(u)v\) is the reaction rate; \(f(u)=0\) for \(u<\alpha\); \(f(u)>0\) for \(u>\alpha\); \(\lambda = D\rho c/k\); \(D\) is the diffusion coefficient; \(\rho\) is the density of the substance; \(c\) is its heat capacity, \(k\) is the coefficient of thermal conductivity.
A solution of system (1) of the form
\[ u=\tilde u(x+mt+C),\qquad v=\tilde v(x+mt+C), \tag{2} \]
where
\[ m=\mathrm{const}>0,\qquad C=\mathrm{const},\qquad \tilde u(-\infty)=u_-<\alpha,\qquad \tilde u(+\infty)=u_+>\alpha, \]
\[ \tilde v(-\infty)=v_->0,\qquad \tilde v(+\infty)=0, \]
is called stationary. For \(\lambda=1\), system (1) reduces to a single equation. The existence of a stationary solution for this equation was proved in (¹). There, the stationary solution for system (1) is also considered. In the present paper, the existence of the latter is proved for arbitrary \(\lambda>0\), and uniqueness for \(0<\lambda<1\).
Let \(\xi=x+mt\). Then the stationary solution will satisfy the system
\[ m\frac{du}{d\xi}-\frac{d^2u}{d\xi^2}=f(u)v,\qquad m\frac{dv}{d\xi}-\lambda\frac{d^2v}{d\xi^2}=-f(u)v. \tag{1'} \]
This system has the first integral
\[ m(u+v)-\frac{du}{d\xi}-\lambda\frac{dv}{d\xi}=C=\mathrm{const}. \tag{3} \]
It is natural to assume that the stationary solution also satisfies the conditions
\[ \left.\frac{du}{d\xi}\right|_{\xi=\pm\infty} = \left.\frac{dv}{d\xi}\right|_{\xi=\pm\infty} =0. \tag{2'} \]
From (3), (2), and (2′) it follows that the numbers \(u_+\), \(u_-\), \(v_-\) must satisfy the additional restriction \(u_+=u_-+v_-\).
Thus, it is required to prove that, for some \(m=m_0\), there exists a solution of system (1′) satisfying conditions (2) and (2′). Here it is assumed that \(u_-<\alpha<u_+\), \(v_->0\), \(u_+=u_-+v_-\). We replace the second equation of system (1′) by the integral (3), where \(C=mu_+\) (by virtue of the boundary condi-
…wise). Put \(du/d\xi=p\). We shall regard \(u\) as the independent variable. Then instead of system \((1')\) we obtain the system
\[ \frac{dp}{du}=m-\frac{f(u)v}{p}, \qquad \frac{dv}{du}=\frac{m}{\lambda p}(u+v-u_+)-\frac{1}{\lambda}. \tag{4} \]
Instead of conditions (2) and \((2')\) we obtain the conditions
\[ v\big|_{u=u_-}=v_-,\qquad v\big|_{u=u_+}=0,\qquad p\big|_{u=u_-}=p\big|_{u=u_+}=0. \tag{5} \]
It can be shown that through the singular point \(u=u_+\), \(v=0\), \(p=0\) there pass three integral curves of system (4). Moreover, for two of them \(p<0\) for \(u<u_+\) in a sufficiently small neighborhood of \(u_+\), and for one \(p>0\) for the same \(u\). Only the last of these curves is meaningful to consider. Let \(p=p(u,m)\), \(v=v(u,m)\) be its equations. From the second equation of system (4), where \(p=p(u,m)\), we find
\[ v(u,m)=u_+-u+\left(\frac{1}{\lambda}-1\right) \int_u^{u_+}\exp\left[-\frac{m}{\lambda}\int_u^{u_2}\frac{du_1}{p(u_1,m)}\right]du_2. \tag{6} \]
It follows from (6) that for \(u<u_+\), in some neighborhood of \(u_+\), the inequalities
\[ u_+-u<v<\frac{1}{\lambda}(u_+-u) \qquad \text{for } \lambda<1; \tag{7} \]
\[ \frac{1}{\lambda}(u_+-u)<v<u_+-u \qquad \text{for } \lambda>1. \tag{7'} \]
hold.
Let \(p=p_1(u,m)\) be the separatrix of the saddle at the point \(u=u_+\), \(p=0\) for the equation \(dp/du=m-f(u)(u_+-u)/p\), passing above the \(u\)-axis for \(u<u_+\); let \(p=p_2(u,m)\) be the separatrix of the same kind for the equation \(dp/du=m-f(u)(u_+-u)/\lambda p\). For \(u<u_+\) the inequalities
\[ p_1(u,m)<p(u,m)<p_2(u,m) \qquad \text{for } \lambda<1; \tag{8} \]
\[ p_2(u,m)<p(u,m)<p_1(u,m) \qquad \text{for } \lambda>1. \tag{8'} \]
hold.
Indeed, we have, for example,
\[ \frac{d\Delta p}{du}-\frac{f(u)(u_+-u)}{p p_1}\Delta p = -\frac{f(u)}{p}\,[v-(u_+-u)], \qquad \Delta p=p-p_1. \]
Solving this equation under the condition \(\Delta p|_{u=u_+}=0\) and using (7) and \((7')\), we obtain \(p_1(u,m)<p(u,m)\) for \(\lambda<1\); \(p(u,m)<p_1(u,m)\) for \(\lambda>1\). From inequalities (8) and \((8')\) it follows that the solution \(p=p(u,m)\), \(v=v(u,m)\) can be continued to the whole interval \(u_-<u<u_+\). Direct calculation shows that the functions \(p_1(u,m)\) and \(p_2(u,m)\) are strictly positive for \(m=0\), \(u_-\le u<u_+\). As \(m\) increases, the quantities \(p_1(u_-,m)\) and \(p_2(u_-,m)\) decrease, and there exist such positive numbers \(m_1\) and \(m_2\)\(^{(1)}\) that \(p_1(u_-,m_1)=0\), \(p_2(u_-,m_2)=0\), \(p_1(u,m_1)>0\) for \(u_-<u<u_+\); \(p_2(u,m_2)>0\) for \(u_-<u<u_+\). Then, by virtue of (8) and \((8')\), there is an \(m=m_0>0\) such that \(p(u_-,m_0)=0\), \(p(u,m_0)>0\) for \(u_-<u<u_+\).
The equality \(v(u_-,m_0)=v_-\) is then obtained automatically from (6). Thus, the curve \(p=p(u,m_0)\), \(v=v(u,m_0)\) will be the desired one. Solving the equation \(du/d\xi=p(u,m_0)\), we find \(u=\tilde u(\xi+C)\). \(v=v(\tilde u(\xi+C),m_0)=\tilde v(\xi+C)\), where \(C\) is an arbitrary constant—the shift of the stationary wave along the \(\xi\)-axis.
We shall prove that for \(\lambda<1\) the number \(m_0\) is determined uniquely, i.e., there is uniqueness of the stationary solution up to the shift \(C\).
Suppose the contrary: let a solution of problem (4)—(5) exist for \(m=m_1\) and for \(m=m_2\), where \(0<m_1<m_2\). Let \(p_1, v_1\) be the solution corresponding to \(m_1\); \(p_2, v_2\) the solution corresponding to \(m_2\).
Subtracting from the equality \(\dfrac{dp_2}{du}=m_2-\dfrac{f(u)v_2}{p_2}\) the equality \(\dfrac{dp_1}{du}=m_1-\dfrac{f(u)v_1}{p_1}\), and from the equality \(\dfrac{dv_2}{du}=\dfrac{m_2}{\lambda p_2}(u+v_2-u_+)-\dfrac{1}{\lambda}\) the equality \(\dfrac{dv_1}{du}=\dfrac{m_1}{\lambda p_1}(u+v_1-u_+)-\dfrac{1}{\lambda}\), we obtain
\[ \frac{d\Delta p}{du}-\frac{f(u)v_1}{p_1p_2}\Delta p = \Delta m-\frac{f(u)}{p_2}\Delta v; \tag{9} \]
\[ \frac{d\Delta v}{du}-\frac{m_2}{\lambda p_2}\Delta v = \frac{u-u_+ + v_1}{\lambda p_1} \left(\Delta m-\frac{m_2}{p_2}\Delta p\right), \tag{10} \]
where
\[ \Delta p=p_2-p_1,\qquad \Delta v=v_2-v_1,\qquad \Delta m=m_2-m_1. \]
Considering equality (9) as a linear equation with respect to \(\Delta p\), and equality (10) as a linear equation with respect to \(\Delta v\), and taking into account that \(\Delta p=\Delta v=0\) at \(u=u_+\), we obtain, for \(u_-<u<u_+\),
\[ \Delta p = -\int_u^{u_+} \left[ \Delta m-\frac{f(w)}{p_2(w)}\Delta v \right] \exp\left[ -\int_u^w \frac{f(z)v_1(z)}{p_1(z)p_2(z)}\,dz \right]dw; \tag{11} \]
\[ \Delta v = -\int_u^{u_+} \left[w+v_1(w)-u_+\right] \frac{1}{\lambda p_1} \left(\Delta m-\frac{m_2\Delta p}{p_2}\right) \exp\left[ -\int_u^w \frac{m_2\,dz}{\lambda p_2(z)} \right]dw. \tag{12} \]
Investigating the singular point \(u=u_+\), \(v=0\), \(p=0\) of system (4), one can show that, as \(u\to u_+-0\),
\[ p(u,m)=k(u_+-u)+o(u_+-u), \qquad \text{where } k=\sqrt{\frac{m^2}{4\lambda^2}+\frac{f(u_+)}{\lambda}}-\frac{m}{2\lambda}, \]
\[ v(u,m)=l(u_+-u)+o(u_+-u), \qquad \text{where } l=\frac{m+k}{m+\lambda k}. \]
Hence, from the inequality \(dk/dm<0\) and the inequality \(dl/dm<0\), valid for \(\lambda<1\), it follows that
\[ \Delta p<0,\qquad \Delta v<0 \quad \text{for } \quad u_+-\delta<u<u_+, \tag{13} \]
where \(\delta>0\) is sufficiently small. In view of (6), \(u+v_1(u)-u_+>0\) for \(\lambda<1\), \(u_-<u<u_+\).
Hence, from equalities (11) and (12) it follows that inequalities (13) can be continued to the whole half-segment \(u_-\le u<u_+\). But this contradicts the assumption that, at \(u=u_-\), the equalities \(p_1=p_2=0\), \(v_1=v_2=v_-\) hold.
The author expresses gratitude to G. I. Barenblatt for his attention to the work.
Received
11 IV 1962
CITED LITERATURE:
- Ya. B. Zel’dovich, ZhFKh, 22, No. 1, 27 (1948).