Combustion theory

Definition

Consider a gas mixture that fills a region of $\R^3$ and moves with a velocity field $\mathbf{u}(x, t)$. Let $T(x, t)$ be its temperature and $Y(x, t) \in [0, 1]$ the mass fraction of a fuel that burns in one irreversible step. Combustion theory is the study of the reacting flow equations, of which the basic model is $$\frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T = \kappa \Delta T + \frac{q}{c_p} \, \omega, \qquad \frac{\partial Y}{\partial t} + \mathbf{u} \cdot \nabla Y = D \Delta Y - \omega,$$ with the Arrhenius reaction rate $$\omega = B \, Y \exp\!\left( -\frac{E}{R T} \right).$$ Here $\kappa > 0$ is the thermal diffusivity, $D > 0$ the diffusivity of the fuel and $c_p$ the specific heat. The constant $q > 0$ is the heat released per unit mass of fuel, and $B > 0$ is a rate constant. Finally, $E > 0$ is the activation energy and $R$ is the gas constant. In the full theory, $\mathbf{u}$ and the density solve the compressible flow equations, coupled to $T$ through the heat release.