Ingo Müller's research while affiliated with Technische Universität Berlin and other places

The temperature gradient in the upper troposphere is –0.65 \( \frac{\text{K}}{{ 1 0 0 {\text{m}}}} \). We suggest that the thermal diffusion factor of air in the gravitational field determines this gradient.

The gravitational equations of Einstein are solved for a sphere filled with a dust gas and floating in infinite empty space. It turns out that the radial acceleration of the gas has negative and positive contributions which may be interpreted as attractive and repulsive gravitational forces respectively. Two cases are considered: the collapse of a gas initially at rest and with uniform density, and the expansion of the gas with initial conditions appropriate to the observed Hubble diagram of Type Ia supernovae. The latter case may be seen as a proposal for a new type of cosmology.

The Navier-Stokes-Fourier theory of viscous, heat-conducting fluids provides parabolic equations and thus predicts infinite pulse speeds. Naturally this feature has disqualified the theory for relativistic thermodynamics which must insist on finite speeds and, moreover, on speeds smaller than c. The attempts at a remedy have proved heuristically important for a new systematic type of thermodynamics: Extended thermodynamics. That new theory has symmetric hyperbolic field equations and thus it provides finite pulse speeds. Extended thermodynamics is a whole hierarchy of theories with an increasing number of fields when gradients and rates of thermodynamic processes become steeper and faster. The first stage in this hierarchy is the 14-field theory which may already be a useful tool for the relativist in many applications. The 14 fields — and further fields — are conveniently chosen from the moments of the kinetic theory of gases. The hierarchy is complete only when the number of fields tends to infinity. In that case the pulse speed of non-relativistic extended thermodynamics tends to infinity while the pulse speed of relativistic extended thermodynamics tends to c, the speed of light. In extended thermodynamics symmetric hyperbolicity — and finite speeds — are implied by the concavity of the entropy density. This is still true in relativistic thermodynamics for a privileged entropy density which is the entropy density of the rest frame for non-degenerate gases.

A complete set of equations for the description of the properties of monatomic ideal gases is formed by the balance equations of all moments of the distribution function which satisfies the Boltzmann equation. In a manner of speaking these balance equations describe a continuum and the elements of its microstructure are the moments. There are infinitely many moments and for rapidly changing processes with steep gradients they are all needed. However, for slow and smooth processes the necessary set of balance equations may be cut off at some point and may then still be useful. The most drastic cut-off provides the Euler equations in which no dissipation occurs, so that its applicability is limited to isentropic flows. A less rigorous cut-off leads to the equations of Navier-Stokes and Fourier which permit the mathematical treatment of viscous flow and heat conduction. Those curtailed sets of balance equations have been studied at great length: Their study is the subject of ordinary thermodynamics. A still less drastic cut-off leads to Grad’s 13- and 14-moment equations. These provide some improvement upon both Euler and Navier-Stokes-Fourier. Thus they forbid rigid rotation of a gas in the presence of heat conduction. Even so, the Grad theory is not suitable for high frequency sound propagation and high frequency light scattering and for the study of shock structures. The study of Grad’s equations and of the balance equations for higher moments is the subject of extended thermodynamics. Mathematically speaking the set of balance equations of moments in the kinetic theory of gases provides a prototype of the hyperbolic balance laws of continuum physics. Indeed, the equations are hyperbolic and the entropy principle makes them symmetric hyperbolic. High frequency light scattering in monatomic gases proves the applicability and usefulness of extended thermodynamics, because it furnishes results that are in full agreementwith experiments at any frequency, while ordinary thermodynamics merely describes the low-frequency limit. In hyperbolic equations there is a competition between non-linearity and dissipation: Non-linearity attempts to steepen a field to a shock while dissipation smoothes it out. And if dissipation is big enough, no shock singularities will appear. In extended thermodynamics singularities can be prevented by adding more equations, hence more dissipative terms. The study of shock structures makes this evident. Hyperbolic equations have as many sound modes as there are equations and all their speeds are different from the ordinary sound speed. Extended thermodynamics proves that the highest sound speed increases monotonically as more equations are added. It is when a shock structure moves more rapidly than the highest sound speed that discontinuities appear in the theory. One may thus say that the flow is truely supersonic only when its speed is quicker than the highest characteristic speed. However, that will generally happen at Mach numbers well beyond Ma equal to 1. It is a clear sign that more equations are needed.

When I first met Krzysztof Wilmanski in 1977 he was one of the bright young scientists in the Institute of Fundamental Technological Research of the Polish Academy of Sciences. This was the time of the cold war and it was not altogether easy for us westerners to meet colleagues from beyond the iron curtain. But among all people from the east it was still easiest to meet Polish scientists. Because, indeed, the wise elder scientists at the helm of the Polish Academy—among them Professors Nowacki, Olszak, Fiszdon, and Sawczuk—held some influence in political circles. And they knew that good science requires free and easy communication between scientists. Also they believed in mechanics as an essential part of the natural sciences.

The universe is viewed as a dust gas filling a sphere and floating in infinite empty space. Einstein's gravitational equations are applied to this case together with appropriate boundary values. The equations are solved for initial conditions chosen so as to describe the observed Hubble diagram. We find that the solution is not unique so that more astronomical observations are needed. However, those solutions which were found do not exhibit an accelerated expansion of the universe, nor -- obviously then -- do they need the notion of a dark energy driving such an expansion. We present this study as an alternative to the prevailing Robertson-Walker cosmology.

This brief note revisits an idea originally expressed by Mach according to which the inertial forces are a consequence of the far away masses of the universe. As it is well known this idea lead to Einstein’s field equations of gravity. An attempt is made to derive the classical inertial acceleration of a mass from the field equations.

The socio-thermodynamics of a population of two competing species exhibits strong analogies with the thermodynamics of solutions and alloys of two constituents. In particular we may construct strategy diagrams akin to the phase diagrams of chemical thermodynamics, complete with regions of homogeneous mixing and miscibility gaps.

The laws of FOURIER and NAVIER-STOKES We recall that – at the very beginning of this book – we have defined the determination of the five fields mass density \(\rho(\underline{x},t)\) , velocity \(w_i(\underline{x},t)\) , temperature \(T(\underline{x}, t)\) (12.1) as the objective of thermodynamics of fluids.

Observations of pseudo-elastic hysteresis loops in the shape memory alloy CuAlNi are presented. Particular emphasis is laid on the interior of the overall loop and the phenomena of internal yield and recovery and internal loops are discussed. A thermodynamic argument is presented which may afford an interpretation of the observed phenomena in terms of interfacial energies.