We present a deterministic continuum mechanics foundation of thermodynamics for slightly viscous fluids or gases based on a 1st Law in the form of the Euler equations expressing conservation of mass, momentum and energy, and a 2nd Law formulated in terms of kinetic energy, internal (heat) energy, work and shock/turbulent dissipation, without reference to entropy.
Heat, a quantity which functions to animate, derives from an internal fire located in the left ventricle. (Hippocrates, 460 B.C.)
Thermodynamics is fundamental in a wide range of phenomena from macroscopic to microscopic scales. Thermodynamics essentially concerns the interplay in a fluid or gas between kinetic energy and heat energy, also referred to as internal energy. Kinetic energy, or mechanical energy, may generate heat/internal energy by compression or turbulent dissipation. Heat energy may generate kinetic energy by expansion, but not through a reverse process of turbulent dissipation .
Newcomen´s engine for converting heat enerygy to mechanical energy from 1712.
- thermodynamics concerns the interplay between kinetic energy and heat energy in turbulent flow.
- computational solution of the Euler and Navier-Stokes equations
- by stabilized finite element methods with a posteriori error estimation.
Basic facts are:
- heat energy is small scale microscopic kinetic energy
- turbulence is irreversible conversion of large scale macroscopic kinetic energy into small scale kinetic energy in the form of heat energy.
Joule’s 1845 experiment
Basic aspects of thermodynamics can be illustrated in Joule’s experiment from 1845 with a gas compressed to high pressure (20 atmospheres) in a chamber R connected to another empty chamber L through a tube with a initially closed valve, both containers being submerged in a larger container filled with water. At initial time the valve was opened and the gas was allowed to expand into the double volume R+L while the temperature change in the water was carefully measured by Joule. The experiment can be followed in movies from computational simulation of density and momentum with snap-shots below of density and temperature in a model with cubical chambers.
- gas flowing through the valve from R into L
- the gas is put into motion by the high pressure in R and thus picks up kinetic energy
- since the total energy (sum of kinetic energy and heat energy) is constant the temperature drops in R
- as the gas expands into L turbulence develops and shocks bounce back and forth in R
- eventually the gas comes to rest in the double volume R+L at initial temperature
- the turbulent motion is converted into heat as the gas comes to rest.
- finite precision computation
- instability of slightly viscous flow generating turbulence
- macroscopic kinetic energy can generate microscopic kinetic energy by turbulent dissipation
- microscopic kinetic energy can only generate macroscopic kinetic energy in expansion
- an inverse process of turbulent dissipation is impossible because infinite precision would be required to coordinate microscopic motion into macroscopic motion.
- breaking into pieces is possible with low precision
- the reverse process of putting pieces together requires infinite precision.
Joule measured the temperature in the surrounding water to record the temparture drop, but to his surprise dissappointment, could not notice any temperature drop. Thus thermodynamics got an unsuccessful start which prepared for the coming development of an incomprehensible theory.
With our experience from above we can explain Joule´s observation from the rapidity of the expansion and related temperature drop, which could not be followed by the slow experimental set-up with a thermometer in the surrounding water. Since the temperature after expansion to rest is the same as the initial temperature, there is no final termperature drop and this was what Joule observed, but could not explain.
The laws of thermodynamics
1st Law of thermodynamics
2nd Law of thermodynamics
The 2nd Law has the form of an inequality dS ≥ 0 for a scalar quantity named entropy denoted by S, with dS denoting change thereof, supposedly expressing a basic feature of real thermodynamic processes. The classical 2nd Law states that the entropy cannot decrease; it may stay constant or it may increase, but it can never decrease (for an isolated system).
- Were it not for the existence of irreversible processes, the entire edifice of the 2nd Law would crumble.
Those who have talked of chance are the inheritors of antique superstition and ignorance…whose minds have never been illuminated by a ray of scientific thought. (T. H. Huxley)
- If the 2nd Law is a new independent law of Nature, how can it be justified?
- What is the physical significance of that quantity named entropy, which Nature can only get more of and never can get rid of, like a steadily accumulating heap of waste? What mechanism prevents Nature from recycling entropy?
The basic objective of statistical mechanics as the basis of classical thermodynamics, is to (i) give entropy a physical meaning, and (ii) to motivate its tendency to (usually) increase. Before statistical mechanics, the 2nd Law was viewed as an experimental fact, which could not be rationalized theoretically. The classical view on the 2nd Law is thus either as a statistical law of large numbers or as a an experimental fact, both without a rational deterministic mechanistic theoretical foundation.
Explaining Joule´s experiment by statistical mechanics
Does anybody understand thermodynamics?
- Every mathematician knows it is impossible to understand an elementary course in thermodynamics. (V. Arnold)
- …no one knows what entropy is, so if you in a debate use this concept, you will always have an advantage. (von Neumann to Shannon)
- As anyone who has taken a course in thermodynamics is well aware, the mathematicsused in proving Clausius’ theorem (the 2nd Law) is of a very special kind, having only the most tenous relation to that known to mathematicians. (S. Brush )
- Where does irreversibility come from? It does not come form Newton’s laws. Obviouslythere must be some law, some obscure but fundamental equation. perhaps in electricty,maybe in neutrino physics, in which it does matter which way time goes. (Feynman )
- For three hundred years science has been dominated by a Newtonian paradigm presentingthe World either as a sterile mechanical clock or in a state of degeneration and increasing disorder…It has always seemed paradoxical that a theory based on Newtonian mechanics can lead to chaos just because the number of particles is large, and it is subjectivly decided that their precise motion cannot be observed by humans… In the Newtonian world of necessity, there is no arrow of time. Boltzmann found an arrow hidden in Nature’smolecular game of roulette. (Paul Davies )
- The goal of deriving the law of entropy increase from statistical mechanics has so far eluded the deepest thinkers. (Lieb ])
- There are great physicists who have not understood it . (Einstein about Boltzmann’s statistical mechanics)
- Yet, when we went with the odds and imagined that everything popped into existence by a statistcal fluke, we found ourselves in a quagmire: that route called into question the laws of physics themselves. And so we are inclined to buck the boggies and go with low-entropy big bang as the explanation of the arrow of time. The puzzle then is to explain how the universe began in such an unlikely, highly ordered configuration. That is the question to which the arrow of time points. It all comes down to cosmology. (Greene ).
Euler and Navier-Stokes equations
Computational thermodynamics: EG2
We can thus a posteriori assess the quality of EG2 solutions as solutions of the Euler equations and identify what outputs are wellposed and converge with decreasing mesh size. We prove that EG2 solutions satisfy a basic form of the 2nd Law formulated without the concept of entropy, and we discover a connection between the 2nd Law and wellposednes with satisfaction of the 2nd Law being necessary for wellposedness of some outputs.
EG2 as LES with automatic turbulence model
From probable to necessary
New 2nd Law without entropy
EG2 satisfies a 2nd Law formulated without the concept of entropy, in terms of the basic physical quantities of kinetic energy K, heat energy E, rate of work W and shock/turbulent dissipation D > 0. We refer to this law as the new 2nd Law, which reads
Here K is the total kinetic energy as the integral in space of the pointwise kinetic energy, with E, W and D
Irreversibility and direction of time
The new 2nd Law shows that processes with turbulent dissipation with D>0 cannot be reversed in time, that is are irreversible, and thus define a direction of time. Slightly viscous flow is always turbulent and thus defines a direction of time.
Compression and expansion
The work W is positive in expansion and negative in compression, since
W=p∇ · u
The classical 2nd Law is often described as a general tendence of heat to spread or temperature gradients to decrease in a steady march to a heat death with uniform temperature, or more generally a tendency in physical processes of decreasing differences with increasing time. In this form the classical 2nd Law seems to contradict all forms of emergence of ordered structures in the form of crystals, waves and life, characterized by increasing difference. The new 2nd Law does not contradict emergence of ordered structures, only states that increasing difference and creating order cannot be done quickly 
Comparison with Classical Thermodynamics
Classical thermodynamics is based on the relation
TdS = dT + pdV,
where dS represents change of entropy s per unit mass, dV change of volume, p is pressure and dT
denotes the change of temperature T per unit mass, combined with a classical 2nd Law in the
form dS ≥ 0. The new 2nd Law in the form dE/dt +W = D ≥ 0 takes the symbolic form
dT + pdV ≥ 0,
effectively expressing that TdS ≥ 0, which is the same as dS ≥ 0 since T > 0. The new 2nd Law thus effectively expresses the same inequality as the classical 2nd Law, without reference to entropy.
Integrating the classical 2nd Law for a perfect gas with p = (γ − 1)ρT, where 1<γ<2 is a gas constant,
S = log(Tρ1−γ) .
- What is the physical significance of S?
- Why is dS ≥ 0?
The Euler equations
We consider the Euler equations  for an inviscid perfect gas enclosed in a volume Ω in three-dimensional spacewith boundary Γ with outward unit normal n over a time interval I = (0, 1],expressing conservation of mass density ρ, momentum m and total energy e: Find U = (ρ, m, e) depending on (x, t) ∈ Q ≡ Ω × I such that
Rρ(U) ≡ dρ/dt + ∇ · (ρu) = 0 in Q,
Rm(U) ≡ dm/dt + ∇ · (mu + p) = f in Q,
Re(U) ≡ de/dt + ∇ · (eu+pu) = g in Q,
u · n = 0 on Γ × I
u(·, 0) =u0 in Ω,
Output uniqueness and stability
Defining a mean-value output M(U) in the form of a space-time integral of U defined by a smooth weight function, it follows by duality-based a posteriori error estimation that
|((U, ψ)) − ((W, ψ))| ≤ S(||hR(U)|| + ||hR(U)||)
We show below compressible flow around a sphere with a bow shock and turbulent wake both generating