Abstract
In Part II, our description of fluid behavior differs from our description of the dynamics of masses and springs, strings, bars, and two-dimensional vibrating surfaces. In Part I, it was reasonable to identify the coordinates of a specific point on such a vibrating system and write an equation for the time evolution of that point based on Newton’s Second Law of Motion and some description of the appropriate elastic restoring force. With fluids, it is rare that we identify a “specific point” in the fluid and try to track the motion of that parcel of fluid. With fluidic systems (gases and liquids), we generally adopt a different perspective, since it is inconvenient (and frequently impossible) to identify a specific “fluid particle” and track its flow under the influences of various forces and boundaries. Instead, we choose to identify a differential volume, dV = dx dy dz, specified in coordinates (x, y, and z) that are fixed in our laboratory frame of reference , while calculating the changes in the properties of the fluid within that differential volume (e.g., pressure, density, temperature, enthalpy , fluid velocity, mixture concentration, void fraction , dielectric polarization) as we keep track of the amount of fluid that enters or leaves that (fixed in space) differential volume. The fact that we no longer choose to identify individual fluid parcels requires the introduction of a mass conservation equation (also known as the continuity equation ), in addition to our dynamical equation (Newton’s Second Law) and our constitutive equation (Hooke’s Law or some other equation of state ).
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Notes
- 1.
Lagrange, who was of French/Italian descent (born Giuseppe Lodovico Lagrangia), was one of Euler’s doctoral students. Euler recommended that Lagrange succeed him as the Director of Mathematics at the Prussian Academy of Sciences in Berlin.
- 2.
Acoustics is particularly important to cosmologists. For the first 300,000 years after the “Big Bang,” all matter was ionized and therefore opaque to electromagnetic radiation. The only “channel” for wave propagation was acoustical. The 1/f noise in that cosmic background radiation is still evident as the distributed “lumpiness” in the distribution of matter observed in the universe today: http://www.astro.ucla.edu/~wright/CMB-DT.html.
- 3.
The principles and methods of Statistical Mechanics are treated clearly and systematically at the advanced undergraduate level in the textbook by Fred Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965); ISBN 07-051800-9.
- 4.
In an elastic collision, no energy is dissipated.
- 5.
Whether or not a particular degree of freedom is “available” will be a consequence of quantum mechanics. Such quantum restrictions will be addressed specifically in Sect. 7.2.2.
- 6.
The internationally accepted value of Universal Gas Constant was determined by an acoustics experiment done at the National Bureau of Standards (now called the National Institute for Standards and Technology), in Gaithersburg, MD. Michael Moldover, and his colleagues at NIST, evaluated ℜ by measuring the sound speed in argon as determined by the resonance frequencies of that gas contained within a spherical resonator. If we use the definition of standard conditions of temperature (T = 0 °C = 273.15 K) and pressure (P = 1 atm = 101.325 kPa), then the volume of one mole of an ideal gas, under those conditions, is given by Eq. (7.3) as 22.414 L = 22.414 × 10−3 m3.
- 7.
We will deal with the part of Pascal’s Law that addresses “points are at the same depth below the fluid’s surface” in Sect. 8.3.
- 8.
If the number of variables exceeds the number of conservation laws , the system is “underdetermined.” If the number of conservation laws exceeds the number of variables, then the system is “over determined.”
- 9.
How do we know that two variables are enough? The best answer is that when we assume that two are enough, we get results that are consistent with experiment. Although I’ve heard arguments that the number of variables can be connected with “spontaneously broken symmetries,” I do not understand (or necessarily believe) such arguments. Knowing the number of variables, a priori, is not necessary for a phenomenological theory. You can guess the number of variables, write the corresponding conservation laws, and then see if your theory explains your existing measurements and predicts some new behaviors that are testable.
- 10.
We’ll calculate how small the heat leakage will be later in Sect. 9.3 when we address thermal conduction.
- 11.
Although the “frictionless gas-tight piston” in a cylinder is convenient for pedagogical purposes, such frictionless gas pistons are approximated quite well by Airpot® Precision Air Dashpots . These have a very circular glass cylinder that is fitted to a graphite piston. Airpot Corp., Norwalk, CT 06852; www.airpot.com.
- 12.
This same calculation for the adiabatic temperature change in an ideal gas was done as an example of logarithmic differentiation in Sect. 1.1.3.
- 13.
Note that “degrees Kelvin” can be abbreviated [K] without a degree sign but “degrees Celsius” requires the degree sign, [°C]. This distinguishes it from the abbreviation for Coulomb, [C], the SI unit of electrical charge.
- 14.
In English units, the heat capacity is expressed in calories/gram where 1.0 cal = 4.184 J. The nominal heat capacity of liquid water is 1.0 cal/g, with the calorie defined as the amount of heat necessary to raise 1 g of water from 19.5 to 20.5 °C.
- 15.
This is quite fortunate for us, otherwise matter would not exist—electrons that orbit nuclei would radiate continuously and would spiral into their own nuclei.
- 16.
To be quantum mechanically correct, we should use the Planck Distribution :
\( {P}_{\mathrm{Planck}}(E)={\left[{\mathrm{e}}^{E/{k}_{\mathrm{B}} T}-1\right]}^{-1} \) . Expansion of the exponential using a Taylor series makes it easy to see that for sufficiently high temperatures, E ≪ k B T, the Planck Distribution reduces to the Boltzmann Factor .
- 17.
Another way to appreciate the fact that rotation about the axis joining the two atoms is negligible is to remember that the moment of inertia is proportional to the mass times the square of the length of the “lever arm.” For the two “dumbbell” rotational degrees of freedom that lever arm is about half the atomic separation, d. For N2, d ≅ 1.0976 × 10−10 m ≅ 1.1 Å. The diameter of the nitrogen nucleus is about 1 femtometer = 1 × 10−15 m = 10−5 Å (also called a Fermi), so the moment of inertia about the common axis is about 1010 times smaller than the moment of inertia for the “dumbbell” rotation.
- 18.
A. Gore, An Inconvenient Truth (Rodale Press, 2006); ISBN 1594865671.
- 19.
The temperature dependence of the specific heat of hydrogen was measured nearly half a century before the “photoelectric effect” and the “ultraviolet catastrophe” were understood through the introduction of quantum mechanics and Planck’s constant . With the benefit of hindsight, one can consider how the development of physics might have been altered if investigators had understood this macroscopic clue to the quantum character of the microscopic world.
- 20.
The “bulk viscosity ” or “second viscosity ” is a correction for the fact that there is a sixth “relaxing” variable in a system of phenomenological equations based on only five variables (see Sect. 14.5). The introduction of this “relaxation time ,” τ R, which is responsible for the delayed equilibration between internal degrees of freedom, is reflected in a time-dependent specific heat, introduced in a rigorous manner in L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, 1959), §78, entitled “Second Viscosity .”
- 21.
- 22.
The \( \rho \overrightarrow{g} \) term will be important for waves on the free surface of water (e.g., tsunamis, surf) or for acoustic oscillations of planetary atmospheres that are generated by seismic events, volcanic explosions, meteors, etc.
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Garrett, S.L. (2017). Ideal Gas Laws. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_7
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