Shocks

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Burgers Equation ◽  
Constant Pressure ◽  
Initial Density ◽  
One Dimension ◽  
Constant Density ◽  
Problem Of Time ◽  
Small Viscosity ◽  
Maxwell Construction ◽  
Zero Viscosity ◽  
Basic Intuition

When the speed of a fluid exceeds that of sound, discontinuities in density occur, called shocks.The opposite limit from incompressibility (constant density) is constant pressure. In this limit, we get Burgers equation. It can be solved exactly in one dimension using the Cole–Hopf transformation. The limit of small viscosity is found not to be the same as zero viscosity: there is a residual drag no matter how small it is. The Maxwell construction of thermodynamics was adapted by Lax and Oleneik to derive rules for shocks in this limit. The Riemann problem of time evolution with a discontinuous initial density is solved in one dimension. These simple solutions provide the basic intuition for more complicated shocks.

The Interpretation of EUV Spectra of Sunspots

1983 ◽  
Vol 71 ◽  
pp. 325-326
Author(s):  
J.G. Doyle ◽  
J.C. Raymond ◽  
R.W. Noyes ◽  
A.E. Kingston
Keyword(s):  
Observational Data ◽  
Electron Density ◽  
Constant Pressure ◽  
Wavelength Region ◽  
Constant Density ◽  
Ionization Equilibrium ◽  
Sensitive Line ◽  
Temperature Diagnostic

We report here on EUV observations of a sunspot observed by the Harvard instrument on Skylab. The observational data used here have been presented in a previous paper by Noyes et al. (1982), in which line identifications and intensities for the wavelength region 350 – 1350 A were given. Several electron density sensitive line ratios suggest a constant density, rather than constant pressure, emitting region, while temperature diagnostic line ratios of several ions yield temperatures below the temperatures expected in ionization equilibrium.

Frontogenesis in a fluid with horizontal density gradients

1989 ◽  
Vol 202 ◽  
pp. 1-16 ◽  
Author(s):  
J. E. Simpson ◽  
P. F. Linden
Keyword(s):  
Density Distribution ◽  
Density Gradient ◽  
Initial Density ◽  
Vertical Shear ◽  
Horizontal Gradient ◽  
Constant Density ◽  
Piecewise Constant ◽  
Vertical Density ◽  
Step Density ◽  
Horizontal Density Gradient

The adjustment under gravity of a fluid containing a horizontal density gradient is described.’ The fluid is initially at rest and the resulting motion is calculated as the flow accelerates, driven by the baroclinic density field. Two forms of the initial density distribution are considered. In the first the initial horizontal gradient is constant. A purely horizontal motion develops as the isopycnals rotate towards the horizontal. The vertical density gradient increases continually with time but the horizontal density gradient remains unchanged. The horizontal velocity has a uniform vertical shear, and the gradient Richardson number is constant in space and decreases monotonically with time to ½. The second density distribution consists of a piecewise constant gradient with a jump in the gradient along a vertical isopycnal. The density is continuous. In this case frontogenesis is predicted to occur on the isopycnal between the two constant-density-gradient regions, and the timescale for the formation of a front is determined. Laboratory experiments are reported which confirm the results of these calculations. In addition, lock exchange experiments have been carried out in which the horizontal mean gradient is represented by a series of step density differences separated by vertical gates.

FIRST SOUND IN LIQUID HELIUM AT HIGH PRESSURES

1953 ◽  
Vol 31 (7) ◽  
pp. 1156-1164 ◽  
Author(s):  
K. R. Atkins ◽  
R. A. Stasior
Keyword(s):  
Specific Heat ◽  
Liquid Helium ◽  
Carrier Frequency ◽  
High Pressures ◽  
Constant Pressure ◽  
Constant Density ◽  
Pulse Technique ◽  
Temperature Range ◽  
Specific Heats ◽  
First Sound

The velocity of ordinary sound in liquid helium has been measured in the temperature range from 1.2 °K. to 4.2 °K. at pressures up to 69 atm. A pulse technique was used with a carrier frequency of 12 Mc.p.s. Curves are given for the variation of velocity with temperature at constant pressure and also at constant density. There is no detectable discontinuity along the λ-curve. The results are used to discuss the ratio of the specific heats, the coefficient of expansion below 0.6 °K., and the specific heat above 3 °K.

Properties of a Lattice Gas Model for 3He–4He Mixtures II: Analysis of High Temperature Series for the f.c.c. Lattice

1975 ◽  
Vol 53 (10) ◽  
pp. 987-1002 ◽  
Author(s):  
M. Plischke ◽  
D. D. Betts
Keyword(s):  
High Temperature ◽  
Specific Heat ◽  
Lattice Gas ◽  
Chemical Potential ◽  
Constant Pressure ◽  
Temperature Series ◽  
Adjustable Parameter ◽  
Constant Density ◽  
Constant Concentration ◽  
Chemical Potential Difference

For the Cheng–Schick model of 3He–4He mixtures high temperature series expansions at (a) constant density and constant concentration and (b) constant pressure and constant chemical potential difference are presented for the f.c.c. lattice for the fluctuation in the superfluid order parameter, the concentration susceptibility, and the specific heat at constant chemical potential. Analysis of the fluctuation series yields well defined lambda temperatures. In addition analysis of the concentration susceptibility series provides a less precise estimate of the tricritical concentration. The specific heat series have not proved very amenable to analysis. Upon fixing a single adjustable parameter the lambda curve of the model agrees precisely with experiment for all 3He concentrations. Estimates of tricritical exponents could not be obtained.

scholarly journals Heat Flow Birefringence in Liquids and Liquid Crystals

1985 ◽  
Vol 40 (1) ◽  
pp. 3-7
Author(s):  
D. R. Baalss ◽  
S. Hess
Keyword(s):  
Heat Flow ◽  
Constant Pressure ◽  
Planck Equation ◽  
Pressure Variation ◽  
Flow Birefringence ◽  
Isotropic Phase ◽  
Constant Density ◽  
Collision Term ◽  
Nonspherical Particles ◽  
Physical Mechanisms

The alignment of nonspherical particles is inferred from the solution of a Fokker-Planck equation where a thermal torque has been taken into account which is proportional to the second spatial derivative of the temperature field. A pretransitional enhancement of the effect is predicted for the isotropic phase of a liquid crystal. Two distinct physical mechanisms are considered in order to estimate the magnitude of the thermal torque. One of them is due to the pressure variation at constant density. For constant pressure, the torque is inferred from the collision term of an Enskog-Boltzmann equation generalized to (strongly) nonspherical particles. In both cases, the resulting heat flow birefringence is of measurable size.

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