Sound temperature gradient

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

Ultrasonic velocity has been almost exclusively measured in ultrasonic studies of fat crystallization, but the attenuation coefficient also can reveal interesting information. As the sound wave passes, the fluid is alternately compressed and rarefied which results in the formation of rapidly varying temperature gradients. Heat energy is lost because the conduction mechanisms are inefficient (thermal losses) and together with molecular friction (viscous losses) cause an attenuation of the sound given by classical scattering theory (5) ... 

Urick (1948) measured ultrasonic attenuation in aqueous kaolin as well as sound dispersion. The results were in good agreement with the losses predicted from viscous drag at the particle surface. Sound propagation in a suspension can also produce temperature gradients at the particle/suspending-fluid interface, and thus, results on attenuation via thermal diffusion. 

An important special case is that of incompressible flow. As discussed in Section 1.2, the term incompressible is something of a misnomer, since what is generally meant in fluid mechanics is constant density. However, a flow in which there are temperature gradients is not quite one of constant density since the density varies with temperature. But the criterion for a constant-density flow is that the flow velocity be small compared with the sound speed in the fluid that is, the Mach number must be small. For a small Mach number the pressure changes are small. Therefore when evaluating the derivatives of thermodynamic quantities for an incompressible flow with an imposed spatial variation in temperature, we must hold the pressure, not the density, constant (Landau Lifshitz 1987), whence... 

A finite wave may be thought of as a succession of infinitesimal pressure pulses. When the velocity and temperature gradients are not too steep, viscous and heat conduction effects are negligible. Thus, the wave is isentropic, and each elementary part of the wave travels at the local speed of sound with respect to the fluid in which it is propagating. The change in the propagation velocity of a part of the wave with respect to pressure, in a fixed reference frame is given as,... 

To start from the dissipation function, may lead to erroneous results, see Ref 5 pages 55 and 56. The assumption of local equilibrium is sound, as tested by applications of temperature gradients much larger than realizable in industrial contexts (c/. Table 14.1). 

In addition to the T measurements, those of the transverse lelaxation time, T2, and of the self diffusion coefficient, D, have been peifoimed on die low molecular wei FE 2440. In good agreement with the sound velocity data in Hg. 9, all of these quantities indicate a of 169 C (see Figure 20). The behavior of the temperature gradient of the self diffusion coefficient, D, su esis an intNBHdecuIar contribution to the T transition. 

Table II summarizes the pertinent time and space scales in this problem. Assuming the speed of sound is 105cm/sec, a time-step of about 10-3 sec would be required to resolve the motion of sound waves bouncing across the chamber. Chemical timescales, as mentioned above, are about 10-6 sec. This number may be reduced drastically if the reaction rates or density changes are very fast. It takes a sound wave about 10 3 seconds to cross the 1 meter system and it takes the flame front about one second to cross. We further assume that the flame zone is about 10-2 cm wide and that it takes grid spacings of 10 3 cm to resolve the steep gradients in density and temperature in this flame zone.

VanKrevelen-Hoftijzer viscosity-temperature relationship, 539 Van t Hoff equation, 751 Velocities of sound waves, 390 Velocity gradient, 526 Vertical Burning Test, 854 Vicat softening temperature, 849 Vickers hardness test, 837 Viscoelasticity, 405... 

It only affects the temperature held and only has to be taken into account when friction gives rise to a noticeable warming of the fluid. This only occurs at high velocities, of the order of the speed of sound, and with large velocity gradients, such as those that appear in flow through narrow slots. 

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