Parameterisations of slow invariant manifolds: application to a spray ignition and combustion model

Abstract

A wide range of dynamic models, including those of heating, evaporation and ignition processes in fuel sprays, is characterised by large differences in the rates of change of variables. Invariant manifold theory is an effective technique for investigation of these systems. In constructing the asymptotic expansions of slow invariant manifolds, it is commonly assumed that a limiting algebraic equation allows one to find a slow surface explicitly. This is not always possible due to the fact that the degenerate equation for this surface (small parameter equal to zero) is either a high degree polynomial or transcendental. In many problems, however, the slow surface can be described in a parametric form. In this case, the slow invariant manifold can be found in parametric form using asymptotic expansions. If this is not possible, it is necessary to use an implicit presentation of the slow surface and obtain asymptotic representations for the slow invariant manifold in an implicit form. The results of development of the mathematical theory of these approaches and the applications of this theory to some examples related to modelling combustion processes, including those in sprays, are presented.

Appendix

1.1 Effective methods of parameterisation

1.1.1 Case $m=n$

Let us assume that the degenerate equation (3) can be solved with respect to x in the form $x = \chi _0(y)$ and matrices $A(y)={\partial \chi _0(y)}/{\partial y}$ and $B(y)=g_x (\chi _0(y),y)$ are invertible with bounded norms of inverse matrices. In this case, the fast variable y can be chosen as a parameter, and the slow invariant manifold of the system (2) can be found in the following parametric form:

$$\begin{aligned} x=\chi (y,\varepsilon )=\chi _0(y)+\varepsilon \chi _1(y)+\cdots +\varepsilon ^k \chi _k(y)+\cdots . \end{aligned}$$

(47)

From (2) and (47), we obtain the invariance equation:

$$\begin{aligned} \frac{\partial \chi }{\partial y}g(\chi , y)=\varepsilon f(\chi , y). \end{aligned}$$

(48)

Using the asymptotic representations

$$\begin{aligned} f(\chi _0+\varepsilon \chi _1 +\varepsilon ^2\ldots ,y)= & {} f(\chi _0,y)+\varepsilon \ldots ,\\ g(\chi _0+\varepsilon \chi _1 +\varepsilon ^2\ldots ,y)= & {} g(\chi _0,y)+\varepsilon g_x (\chi _0,y) \chi _1 +\varepsilon ^2\cdots =\varepsilon g_x (\chi _0,y) \chi _1 +\varepsilon ^2\ldots , \end{aligned}$$

and the assumption that $g(\chi _0,y)=0$, Eq. (48) allows us to obtain the following equation:

$$\begin{aligned} A(y)B(y) \chi _1=f(\chi _0,y). \end{aligned}$$

Hence

$$\begin{aligned} \chi _1=B^{-1} (y) A^{-1}(y)f(\chi _0,y). \end{aligned}$$

Thus, we obtain the first-order approximation to the slow invariant manifold in the following form:

$$\begin{aligned} x=\chi (y,\varepsilon )=\chi _0(y)+\varepsilon \chi _1(y)= B^{-1} (y) A^{-1}(y)f(\chi _0,y). \end{aligned}$$

The higher-order approximations can be obtained in a similar way.

Returning to the classical combustion problem, described by Eqs. (17) and (18), consider the degenerate equation (19) which implies that

$$\begin{aligned} \eta =\chi _0 (\theta ) =\alpha \theta \mathrm{e}^{-\theta }. \end{aligned}$$

Fast variable $\theta$ is used as parameterising variable v, see Sect. 2.2.

In the general case when

$$\begin{aligned} \eta =\chi (\theta )= \chi _0 (\theta )+\varepsilon \chi _1 (\theta )+\varepsilon ^2 \chi _2 (\theta )+ \cdots , \end{aligned}$$

we can rewrite Eq. (18) as

$$\begin{aligned} \varepsilon \frac{\mathrm{d}\theta }{\mathrm{d}\tau }= \varepsilon (\chi _1 (\theta )+\varepsilon \chi _2 (\theta )+ \cdots )\mathrm{e}^\theta \equiv \varepsilon F. \end{aligned}$$

(49)

Remembering that $\eta =\chi (\theta )$ does not explicitly depend on time, we have ${\partial \chi }/{\partial t}=0$. This allows us to simplify Eq. (24) to

$$\begin{aligned} \frac{\partial \chi }{\partial \theta }F = -\chi (\theta ,\varepsilon )\mathrm{e}^\theta . \end{aligned}$$

(50)

Remembering the definition of F in (49), we can rewrite Eq. (50) as

$$\begin{aligned} \left( \frac{\partial \chi _0}{\partial \theta }+\varepsilon \frac{\partial \chi _1}{\partial \theta }+\varepsilon ^2\frac{\partial \chi _2}{\partial \theta }+ \cdots \right) \left( \chi _1 (\theta )+\varepsilon \chi _2 (\theta )+ \cdots \right) \mathrm{e}^\theta = -\left( \chi _0 (\theta )+\varepsilon \chi _1 (\theta )+\varepsilon ^2 \chi _2 (\theta )+ \cdots \right) \mathrm{e}^\theta . \end{aligned}$$

Equating the coefficients before powers of $\varepsilon$, we find

$$\begin{aligned} \frac{\partial \chi _0}{\partial \theta } \chi _1=-\chi _0, \quad \frac{\partial \chi _0}{\partial \theta } \chi _2+\frac{\partial \chi _1}{\partial \theta } \chi _1=-\chi _1, \quad \frac{\partial \chi _0}{\partial \theta }=\alpha (1-\theta ) \mathrm{e}^{-\theta }. \end{aligned}$$

Hence, explicit formulae for $\chi _1$ and $\chi _2$ can be presented as

$$\begin{aligned} \chi _1=\frac{\theta }{\theta -1}, \quad \chi _2=\mathrm{e}^\theta \frac{\theta ^2 (\theta -2)}{\alpha (\theta -1)^4}. \end{aligned}$$

This allows us to obtain the following expression for $\eta$:

$$\begin{aligned} \eta = \chi (\theta ,\varepsilon )=\alpha \theta \mathrm{e}^{-\theta } +\varepsilon \frac{\theta }{\theta -1} +\varepsilon ^2 \mathrm{e}^\theta \frac{\theta ^2 (\theta -2)}{\alpha (\theta -1)^4}+ O(\varepsilon ^3). \end{aligned}$$

This representation is correct outside a certain neighbourhood of $\theta =1$. It gives us an approximation of the attractive (repulsive) one-dimensional slow invariant manifold if $0\le \theta <1$ ($\theta >1$).

1.2 Case $m<n$

Let us present vector y in the form $y=(y_1,y_2)^T$, where $\dim y_1=n-m$ and $\dim y_2=m$, and vector g in the form $g=(g_1,g_2)^T$, where $\dim g_1=n-m$ and $\dim g_2=m$. In this case, the system (2) can be rewritten as

$$\begin{aligned}&\dot{x}=f(x,y_1,y_2),\nonumber \\&\varepsilon \dot{y}_1=g_1(x,y_1,y_2),\nonumber \\&\varepsilon \dot{y}_2=g_2(x,y_1,y_2). \end{aligned}$$

(51)

Let us assume that the solution to the degenerate equation $g(x,y)=0$ can be presented as

$$\begin{aligned} x=\chi _0(y_2), \quad y_1=\psi _0(y_2). \end{aligned}$$

In this case, the slow invariant manifold can be found in the following parametric form:

$$\begin{aligned} x=\chi (y_2,\varepsilon ), \quad y_1=\psi (y_2,\varepsilon ). \end{aligned}$$

(52)

From (52) and (51), we have the following invariance equations:

$$\begin{aligned} \frac{\partial \chi }{\partial y_2}g_2(\chi , \psi , y_2)= & {} \varepsilon f(\chi ,\psi , y_2),\\ \frac{\partial \psi }{\partial y_2}g_2(\chi , \psi , y_2)= & {} g_1(\chi ,\psi , y_2). \end{aligned}$$

These equations can be rewritten as

$$\begin{aligned} A_1(y_2) (K_3(y_2)\chi _1(y_2)+K_4(y_2)\psi _1(y_2))= & {} f(\chi _0(y_2),\psi _0(y_2),y_2), \end{aligned}$$

(53)

$$\begin{aligned} A_2(y_2)(K_3(y_2)\chi _1(y_2)+K_4(y_2)\psi _1(y_2))= & {} K_1(y_2)\chi _1(y_2)+ K_2(y_2)\psi _1(y_2), \end{aligned}$$

(54)

where $A_1(y_2)={\partial \chi _0(y_2)}/{\partial y_2}$, $A_2(y_2)= {\partial \psi _0(y_2)}/{\partial y_2}$.

When deriving (53) and (54) the following asymptotic representations for $\chi (y_2,\varepsilon )$, $\psi (y_2,\varepsilon )$, $f(\chi ,\psi , y_2)$, $g_1(\chi ,\psi , y_2)$, and $g_2(\chi ,\psi , y_2)$ were used:

$$\begin{aligned}&\chi (y_2,\varepsilon )=\chi _0(y_2)+\varepsilon \chi _1(y_2)+\cdots ,\\&\psi (y_2,\varepsilon )=\psi _0(y_2)+\varepsilon \psi _1(y_2)+\cdots ,\\&f(\chi _0(y_2)+\varepsilon \chi _1(y_2)+\cdots ,\psi _0(y_2)+\varepsilon \psi _1(y_2)+\cdots ,y_2)=f(\chi _0(y_2),\psi _0(y_2),y_2)+\varepsilon \ldots ,\\&g_1(\chi _0(y_2)+\varepsilon \chi _1(y_2)+\cdots ,\psi _0(y_2)+\varepsilon \psi _1(y_2)+\cdots ,y_2)\\&\quad =g_1(\chi _0(y_2),\psi _0(y_2),y_2))+\varepsilon K_1(y_2)\chi _1(y_2)+\varepsilon K_2(y_2)\psi _1(y_2)+\varepsilon ^2\ldots ,\\&g_2(\chi _0(y_2)+\varepsilon \chi _1(y_2)+\cdots ,\psi _0(y_2)+\varepsilon \psi _1(y_2)+\cdots ,y_2)\\&\quad =g_2(\chi _0(y_2),\psi _0(y_2),y_2))+\varepsilon K_3(y_2)\chi _1(y_2)+\varepsilon K_4(y_2)\psi _1(y_2)+\varepsilon ^2\ldots , \end{aligned}$$

where

$$\begin{aligned} K_1(y_2)= & {} \frac{\partial g_{1}}{\partial x}(\chi _0,\psi _0,y_2), \quad K_2(y_2)=\frac{\partial g_{2}}{\partial x}(\varphi _0,\psi _0,y_2),\\ K_3(y_2)= & {} \frac{\partial g_{1}}{\partial y_1}(\varphi _0,\psi _0,y_2), \quad K_4(y_2)=\frac{\partial g_{2}}{\partial y_1}(\varphi _0,\psi _0,y_2). \end{aligned}$$

The system (53) and (54) is a linear algebraic system for $\chi _1(y_2)$ and $\psi _1(y_2)$. If the determinant of this system is not equal to zero, we can find the first approximation to the slow invariant manifold in the following form:

$$\begin{aligned} x=\chi _0(y_2)+\varepsilon \chi _1(y_2), \quad y_1=\psi _0(y_2)+\varepsilon \psi _1(y_2). \end{aligned}$$

The higher-order approximations can be obtained in a similar way.

1.3 Case $m>n$

Let us present vector x in the form $x=(x_1,x_2)^T$, where $\dim x_1=n$ and $\dim x_2=m-n$, and vector f in the form $f=(f_1,f_2)^T$, where $\dim f_1=n$ and $\dim f_2=m-n$. In this case, the system (2) can be rewritten as

$$\begin{aligned} \dot{x}_1= & {} f_1(x_1,x_2,y),\nonumber \\ \dot{x}_2= & {} f_2(x_1,x_2,y),\nonumber \\ \varepsilon \dot{y}= & {} g(x_1,x_2,y). \end{aligned}$$

(55)

Let us assume that the solution to the degenerate equation $g(x,y)=0$ can be presented as $x_1=\chi _0(x_2,y).$ In this case, the slow invariant manifold can be found in the following parametric form:

$$\begin{aligned} x_1=\chi (x_2,y,\varepsilon ). \end{aligned}$$

(56)

From (56) and (55), we obtain the invariance equation

$$\begin{aligned} \varepsilon \frac{\partial \chi }{\partial x_2}f_2(\chi ,x_2, y,\varepsilon )+ \frac{\partial \chi }{\partial y}g(\chi ,x_2, y)=\varepsilon f_1(\chi ,x_2, y). \end{aligned}$$

(57)

From (57), we obtain as follows:

$$\begin{aligned} \frac{\partial \chi _0}{\partial x_2} f_2(\chi _0(x_2,y),x_2,y)+L(x_2,y)M(x_2,y)\chi _1(x_2,y)=f_1(\chi _0(x_2,y),x_2,y). \end{aligned}$$

(58)

When deriving (58), the following asymptotic representations for $\chi$, $f_1$, $f_2$, and g were used:

$$\begin{aligned}&\chi (x_2,y,\varepsilon )=\chi _0(x_2,y)+\varepsilon \chi _1(x_2,y)+\varepsilon ^2\ldots ,\\&f_1(\chi _0(x_2,y)+\varepsilon \chi _1(x_2,y)+\varepsilon ^2\ldots ,x_2,y)=f_1(\chi _0(x_2,y),x_2,y)+\varepsilon \ldots ,\\&f_2(\chi _0(x_2,y)+\varepsilon \chi _1(x_2,y)+\varepsilon ^2\ldots ,x_2,y)=f_2(\chi _0(x_2,y),x_2,y)+\varepsilon \ldots ,\\&g(\chi _0(x_2,y)+\varepsilon \chi _1(x_2,y)+\varepsilon ^2\ldots ,x_2,y),x_2,y)=\varepsilon M(x_2,y) \chi _1(x_2,y)+\varepsilon ^2\ldots , \end{aligned}$$

where $L(x_2,y)= {\partial \chi _0}/{\partial y}$, $M(x_2,y)=g_{x_1}(\chi _{0},x_2,y)$. Assuming that matrices L and M are invertible, function $\chi _1=\chi _1(x_2,y)$ can be found in the following form:

$$\begin{aligned} \chi _1(x_2,y)=M^{-1}(x_2,y)L^{-1}(x_2,y)\left( f_1(\chi _0(x_2,y),x_2,y)-\frac{\partial \chi _0}{\partial x_2} f_2(\chi _0(x_2,y),x_2,y)\right) . \end{aligned}$$

The higher-order approximations can be obtained in a similar way.

Let us illustrate this approach in the case of a system of three differential equations, which can be considered as a simplified version of the system describing spray combustion analysed in Sect. 3.

$$\begin{aligned} \dot{x}_1 = x_2, \quad \dot{x}_2 = y, \quad \varepsilon \dot{y} = -y-\mathrm{e}^y-x_1-x_2. \end{aligned}$$

This system has an attractive slow invariant manifold since (see the condition (16))

$$\begin{aligned} \frac{\partial }{\partial y}(-y-\mathrm{e}^y-x_1-x_2)=-1-\mathrm{e}^y<0. \end{aligned}$$

The degenerate equation

$$\begin{aligned} 0= -y-\mathrm{e}^y-x_1-x_2, \end{aligned}$$

cannot be solved with respect to the fast variable y, but it can be solved with respect to one of the slow variables $x_1$ or $x_2$. Thus, the fast variable y and the slow variable $x_2$ can be chosen as parameters and the slow invariant manifold can be represented in the form

$$\begin{aligned} x_1=\chi (x_2,y,\varepsilon )=\chi _0(x_2,y) +\varepsilon \chi _1(x_2,y)+\varepsilon ^2 \chi _2(x_2,y) +O(\varepsilon ^3), \end{aligned}$$

where $\chi _0(x_2,y)=-y-\mathrm{e}^y-x_2$. The flow on this manifold is described by the equations

$$\begin{aligned} \dot{x}_2 = y, \quad \varepsilon \dot{y} = -\chi _1(x_2,y)-\varepsilon \chi _2(x_2,y) +O(\varepsilon ^2). \end{aligned}$$

The invariance equation

$$\begin{aligned} \frac{\mathrm{d} \chi }{\mathrm{d} t}=\frac{\partial \chi }{\partial x_2} \frac{\mathrm{d} x_2}{\mathrm{d} t} + \frac{\partial \chi }{\partial y} \frac{\mathrm{d} y}{\mathrm{d} t}, \end{aligned}$$

in this case can be presented as

$$\begin{aligned} \frac{\partial \chi }{\partial x_2} y +\frac{\partial \chi }{\partial y} \frac{1}{\varepsilon }\left( -y-\mathrm{e}^y-\chi -x_2\right) =x_2. \end{aligned}$$

This equation can be rewritten as

$$\begin{aligned} \left( \frac{\partial \chi _0}{\partial x_2}+\varepsilon \frac{\partial \chi _1}{\partial x_2}+\cdots \right) y +\left( \frac{\partial \chi _0}{\partial y} +\varepsilon \frac{\partial \chi _1}{\partial y} +\cdots \right) \left( -\chi _1-\varepsilon \chi _2-\cdots \right) =x_2. \end{aligned}$$

Equating the coefficients before the powers of $\varepsilon$ we obtain

$$\begin{aligned} \chi _1=\frac{x_2+y}{1+\mathrm{e}^y}, \quad \chi _2=\left( -\frac{\partial \chi _1}{\partial x_2}y+\frac{\partial \chi _1}{\partial y}\chi _1\right) /(1+\mathrm{e}^y). \end{aligned}$$

When deriving these equations, we took into account that ${\partial \chi _0}/{\partial y}=-1-\mathrm{e}^y.$

Note that in all cases considered so far, some system variables are used for the parameterisation of slow invariant manifolds. In some cases, however, this approach turned out to be impossible or ineffective. This is illustrated for the following system of equations:

$$\begin{aligned} \dot{x}=y, \quad \varepsilon \dot{y}= 4x^2+y^2-9, \end{aligned}$$

(59)

The parametric form of the slow invariant manifold for this system can be presented as

$$\begin{aligned} x=\frac{r}{2}\cos \theta -\frac{\varepsilon }{2}, \quad y={r}\sin \theta , \end{aligned}$$

where $r=\sqrt{9-\varepsilon ^2}$, $\theta$ is the polar angle. The flow on this slow invariant manifold is described by the equation $\dot{\theta }=-2$.

The implicit form of this slow invariant manifold can be described by the following equation:

$$\begin{aligned} 4\left( x+\frac{\varepsilon }{2} \right) ^2+y^2=9-\varepsilon ^2. \end{aligned}$$

The part of this ellipse, shown in Fig. 3, with $y<0$ is attractive, and the part with $y>0$ is repulsive.

Fig. 3

The slow curve (dashed) and the exact slow invariant manifold (solid) of (59); $\varepsilon =0.1$