Majda's model (original) (raw)

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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 ) {\displaystyle u(x,t)} $u(x,t)$ , z ( x , t ) {\displaystyle z(x,t)} $z(x,t)$ of one spatial variable x {\displaystyle x} $x$ representing the Lagrangian specification of the fluid flow field and the time variable t {\displaystyle t} $t$ , functions f ( w ) {\displaystyle f(w)} $f(w)$ , ϕ ( w ) {\displaystyle \phi (w)} $\phi (w)$ of one variable w {\displaystyle w} $w$ , and positive constants k , q , B {\displaystyle k,q,B} $k,q,B$ , the Majda model is a pair of coupled partial differential equations:[2]

∂ u ( x , t ) ∂ t + q ⋅ ∂ z ( x , t ) ∂ t + ∂ f ( u ( x , t ) ) ∂ x = B ⋅ ∂ 2 u ( x , t ) ∂ x 2 {\displaystyle {\frac {\partial u(x,t)}{\partial t}}+q\cdot {\frac {\partial z(x,t)}{\partial t}}+{\frac {\partial f(u(x,t))}{\partial x}}=B\cdot {\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}} ${\frac {\partial u(x,t)}{\partial t}}+q\cdot {\frac {\partial z(x,t)}{\partial t}}+{\frac {\partial f(u(x,t))}{\partial x}}=B\cdot {\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}$

∂ z ( x , t ) ∂ t = − k ⋅ ϕ ( u ( x , t ) ) ⋅ z ( x , t ) {\displaystyle {\frac {\partial z(x,t)}{\partial t}}=-k\cdot \phi (u(x,t))\cdot z(x,t)} ${\frac {\partial z(x,t)}{\partial t}}=-k\cdot \phi (u(x,t))\cdot z(x,t)$ [2]

the unknown function u = u ( x , t ) {\displaystyle u=u(x,t)} $u=u(x,t)$ 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 ) ∈ [ 0 , 1 ] {\displaystyle z=z(x,t)\in [0,1]} $z=z(x,t)\in [0,1]$ is the mass fraction in a simple one-step chemical reaction scheme;

the given flux function f = f ( w ) {\displaystyle f=f(w)} $f=f(w)$ is a nonlinear convex function;

the given ignition function ϕ = ϕ ( w ) {\displaystyle \phi =\phi (w)} $\phi =\phi (w)$ is the starter for the chemical reaction scheme;

k {\displaystyle k} $k$ is the constant reaction rate;

q {\displaystyle q} $q$ is the constant heat release;

B {\displaystyle B} $B$ is the constant diffusivity.[2]

Since its introduction in the early 1980s, Majda's simplified "qualitative" model for detonation ... has played an important role in the mathematical literature as test-bed for both the development of mathematical theory and computational techniques. Roughly, the model is a 2 × 2 {\displaystyle 2\times 2} $2\times 2$ system consisting of a Burgers equation coupled to a chemical kinetics equation. For example, Majda (with Colella & Roytburd) used the model as a key diagnostic tool in the development of fractional-step computational schemes for the Navier-Stokes equations of compressible reacting fluids ...[3]

^ Majda, Andrew (1981). "A qualitative model for dynamic combustion". SIAM J. Appl. Math. 41 (1): 70–93. doi:10.1137/0141006.
^ a b c Humphreys, Jeffrey; Lyng, Gregory; Zumbrun, Kevin (2013). "Stability of viscous detonations for Majda's model". Physica D: Nonlinear Phenomena. 259: 63–80. arXiv:1301.1260. Bibcode:2013PhyD..259...63H. doi:10.1016/j.physd.2013.06.001. S2CID 119301730.
^ Lyng, Gregory D. (2015). "Spectral and nonlinear stability of viscous strong and weak detonation waves in Majda's qualitative model" (PDF).