Becker–Morduchow–Libby solution

The solution was discovered in a restrictive form by Richard Becker in 1922, which was generalized by Morris Morduchow and Paul A. Libby in 1949.

Before these works, Lord Rayleigh obtained solutions in 1910 for fluids with viscosity but without heat conductivity and for fluids with heat conductivity but without viscosity.

[5] Following this, in the same year G. I. Taylor solved the whole problem for weak shock waves by taking both viscosity and heat conductivity into account.

In this frame, the steady Navier–Stokes equations for a viscous and heat conducting gas can be written as where

and an expression for the energy in terms of any two thermodynamics variables, say

Let us denote properties pertaining upstream of the shock with the subscript "

The first integral of the governing equations, after imposing the condition that all gradients vanish upstream, are found to be By evaluating these on the downstream side where all gradients vanish, one recovers the familiar Rankine–Hugoniot conditions,

Two assumptions has to be made to facilitate explicit integration of the third equation.

so that the terms inside the parenthesis becomes a total derivative, i.e.,

is the sound speed, the above equation provides the relation between the ratio

and the corresponding velocity (or density or specific volume) ratio i.e.,[8] where

Combining this with momentum and continuity integrals, we obtain the equation for

The notable feature is that the entropy does not monotonically increase across the shock wave, but it increases to a larger value and then decreases to a constant behind the shock wave.

Such scenario is possible because of the heat conduction, as it will become apparent by looking at the entropy equation which is obtained from the original energy equation by substituting the thermodynamic relation

, analytical solution is possible only in the weak shock-wave limit, as first shown by G. I. Taylor in 1910.

is found to be related to the steady travelling-wave solution of the Burgers' equation and is given by[10] where in which

is a constant which when multiplied by some characteristic frequency squared provides the acoustic absorption coefficient.

is a third-order small quantity as can be inferred from the above expression which shows that

Validity of continuum hypothesis: since the thermal velocity of the molecules is of the order

is the mean free path of the gas molecules,, we have

; an estimation based on heat conduction gives the same result.

, shows that i.e., the shock-wave thickness is of the order the mean free path of the molecules.

However, in the continuum hypothesis, the mean free path is taken to be zero.

It follows that the continuum equations alone cannot be strictly used to describe the internal structure of strong shock waves; in weak shock waves,

[8] Neglect of viscosity means viscous forces in the momentum equation and the viscous dissipation in the energy equation disappear.

Hence the first integral of the governing equations are simply given by All the required ratios can be expreses in terms of

It turns out there is no continuous solution for strong shock waves, precisely when[8] for

Here continuous solutions can be found for all shock wave strengths.

Further, here the entropy increases monotonically across the shock wave due to the absence of heat conduction.

Here the first integrals are given by One can eliminate the viscous terms in the last two equations and obtain a relation between