I looked at energy conservation in e.g. a pipe, assuming internal energy proportional to temperature. Then with the ideal gas law, the static solution is a mode shape, with wavelength depending on density, heat flux factor and the chemical constant in the gas law. (Invoking convective terms but not time dependency, a velocity gives damping or the opposite.)
If including time dependency, these are transients multiplied on the mode shape(which now also dep on a new constant). If the physics is such that heating at one side, this will change the boundary condition, f.i. with a time dependency, but that cannot be modelled with the exact solution. Is there something missing? If I did a weak formulation and FE, or CFD there would be solutions, because linear, and these will be the transient solutions probably(?). But exact traveling waves appear more physical, but are such solutions to the temperature field in CFD?
Is it possible to derive the wave equation from continuum mechanics, and then use the ideal gas law, to express it in temperature? I did something like that years ago, but cannot recall how.
The transients could give a timedependency that agrees with the boundary condition, if suffieciently small increments, maybe?
And considering the entire time derive could give traveling wave solutions by a kind of first order wave equation, but I don't know if that is how CFD-solvers do, or if it is physical. Answers appreciated, if not too large a field.
Yes, I looked at both flow and the static case.
When flow, I think there are traveling wave solutions with the flow velocity as wave velocity. At static condition (or coexisting, there are the mode shape). We started with a small report, to be attached later on. For heat conduction, the transverse direction are most important, but possibly (hopefully since then determined) correlated to the other, that is why I started there.
(I seldom used the complex notation, except for inplane spatial problems. Then there are the Cauchy Riemann equations and what not, but that is for a solid displacement field)
Those are solutions, (not the same as for the heat equation), instead the temperature distributes in mode shape, and also with traveling wave. Since temperature is a scalar, the transverse solution could be related and maybe derived.
The usual heat equation is obtained when neglecting internal work from pressure and volumetric change.
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