- Stokes' law (sound attenuation)
Stokes derived law for
attenuation ofsound inNewtonian liquid [ Stokes, G.G. "On the theories of the internal friction in fluids in motion, and of the equilibrium and motion of elastic solids", "Transaction of the Cambridge Philosophical Society", vol.8, 22, pp. 287-342 (1845] . According to this law attenuation of sound α is proportional to thedynamic viscosity η, square of the soundfrequency ω, and reciprocally proportional to the liquiddensity ρ and cubic power ofsound speed V:alpha = frac{2 etaomega^2}{3 ho V^3}
Attenuation is expressed in
neper permeter in this equation. The author of this law is the same famousStokes who derived well knownStokes' law for the friction force in fluids. It is 160 years old. This remarkable law does not contain unknown or un-measurable parameters.:It is convenient to convert attenuation into other units, by normalizing attenuation with respect to frequency, because the attenuation typically increases rapidly with frequency. These new units make more adequate presentation of attenuation within a wide frequency range.It is seen that attenuation of the Newtonian liquid, presented in these units, becomes a linear function of frequency.
There has been substantial theoretical development in this field since Stokes’ pioneering work. It has brought one important correction to the Stokes law. It turns out that in addition to the
dynamic viscosity the parameter ofvolume viscosity ηv also affects the total attenuation according to the following relationship::alpha = frac{2 (eta+eta^v)omega^2}{3 ho V^3}
The parameter
volume viscosity is surprisingly little known despite its fundamental role forfluid dynamics at highfrequencies . This parameter appears inNavier-Stokes equation if it is written forcompressible fluid , as described in the most books on general hydrodynamics [ Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", "Prentice-Hall", (1965)] , [ Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", "Pergamon Press",(1959)] , and the acoustics [ Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", "Princeton University Press"(1986)] , [ Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", "Elsevier", (2002) ] .This "volume viscosity" coefficient becomes important only for such effects where fluid compressibility is essential and, importantly, ultrasound propagation is one such effect. Indeed, many rheological texts just assume the fluid to be incompressible and the volume viscosity therefore plays no role.
The only values for the volume viscosity of simple Newtonian liquids known to us come from the old Litovitz and Davis review [ Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, "Academic Press", NY, (1964)] . They report a "volume viscosity" of water at 15 Co equals 3.09
centipoise :More recent studies have established that Stokes's law is actually a low frequency
asymptotic of the more general relationship that describes sound attenuation at very high frequencies::2(frac{alpha V}{omega})^2 = frac{1}{sqrt{1+omega^2 au^2 -frac{1}{1+omega^2 au^2}
where
relaxation time τ equals::au = frac{1}{ ho V^3}(4 eta/3 + eta^v)
Corresponding relaxation frequency is about 1000
GHz . It is extremely high. For all practical purposes of describing sound attenuation in Newtonian liquids Stokes' law is clearly sufficient.References
External links
* [http://www.dispersion.com/ Dispersion Technology]
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