Majda's model explained

Majda's model is a qualitative model (in mathematical physics) introduced by Andrew Majda in 1981 for the study of interactions in the combustion theory of shock waves and explosive chemical reactions.^[1]

The following definitions are with respect to a Cartesian coordinate system with 2 variables. For functions

u(x,t)

z(x,t)

of one spatial variable

representing the Lagrangian specification of the fluid flow field and the time variable

, functions

f(w)

\phi(w)

of one variable

, and positive constants

k,q,B

, the Majda model is a pair of coupled partial differential equations:^[2]

	\partialu(x,t)
	\partialt

+q ⋅

	\partialz(x,t)
	\partialt

	\partialf(u(x,t))
	\partialx

=B ⋅

	\partial^2u(x,t)
	\partialx²

	\partialz(x,t)
	\partialt

=-k ⋅ \phi(u(x,t)) ⋅ z(x,t)

^[2]

the unknown function

u=u(x,t)

is a lumped variable, a scalar variable formed from a complicated nonlinear average of various aspects of density, velocity, and temperature in the exploding gas;

the unknown function

z=z(x,t)\in[0,1]

is the mass fraction in a simple one-step chemical reaction scheme;

the given flux function

f=f(w)

is a nonlinear convex function;

the given ignition function

\phi=\phi(w)

is the starter for the chemical reaction scheme;

is the constant reaction rate;

is the constant heat release;

is the constant diffusivity.^[2]

Notes and References

Majda, Andrew. A qualitative model for dynamic combustion. SIAM J. Appl. Math.. 41. 1. 70–93. 10.1137/0141006. 1981.
Humphreys, Jeffrey. Lyng, Gregory. Zumbrun, Kevin. Stability of viscous detonations for Majda's model. Physica D: Nonlinear Phenomena. 259. 2013. 63–80 . 1301.1260. 10.1016/j.physd.2013.06.001. 2013PhyD..259...63H. 119301730.