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Calculating the Differential Equations

The basic equations for the gas are simple. The equation of motion is

equation31

With the self-similar functions (gif), this becomes

equation39

In keeping with the expected scaling law, Taylor lets

  equation54

which gives an equation for A:

equation59

This is the first of the final equations of motion.

The second equation comes from the continuity equation, expressing mass conservation,

equation71

or

equation77

The final equation derives from the equation of state. At this point, a further approximation is needed; the gas must have a polytropic equation of state, tex2html_wrap_inline254 . This approximation is often very accurate. The number tex2html_wrap_inline220 is the ratio of specific heats of the gas, tex2html_wrap_inline258 , and it represents the number n of degrees of freedom available, according to tex2html_wrap_inline262 . For a monoatomic gas, there are only the three translational degrees of freedom, so tex2html_wrap_inline264 . For a diatomic gas, there are two additional rotation axes with nonzero moments of inertia, so n=5, and tex2html_wrap_inline268 . For air, which is 99% diatomic (with N tex2html_wrap_inline270 and O tex2html_wrap_inline270 ), the measured value of tex2html_wrap_inline220 is 1.40. However, there are other values of interest. The interstellar medium contains mostly atomic hydrogen gas, H tex2html_wrap_inline276 , so the monoatomic value of tex2html_wrap_inline220 is more useful there. (In fact, Taylor did numerical calculations for both these values, as well as for tex2html_wrap_inline280 and tex2html_wrap_inline282 )

For whatever value of tex2html_wrap_inline220 is appropriate, the equation of state may be written in the alternative form that Taylor uses

equation97

With the equations above, this can be written in the (rather more complicated) form

equation102

This is the last of the three dynamical equations.

Taylor's last step in analyzing these equations is to nondimensionalize tex2html_wrap_inline236 and tex2html_wrap_inline240 , using the resting sound speed tex2html_wrap_inline290 . The dimensionless functions f and tex2html_wrap_inline294 are then

eqnarray117

This substitution entirely eliminates A from the equations of motion. (If A did not fall out, it would imply the existence of some other dimensional quantity in the problem.) The nondimensional equations of motion finally become

eqnarray125

This system of differential equations can be solved by an ordinary procedure, steppping through values of tex2html_wrap_inline242 in either direction.


next up previous
Next: Solving the Differential Equations Up: The formation of a Previous: The Basic Scaling Argument

brenner@math.mit.edu