Wang and Nur (1991) have published a series of measurements on ultrasonic velocities of liquid hydrocarbons and mixtures. They found that their data on hydrocarbon mixtures could be adequately explained using a simple mixing rule having the form of a volume average over the velocities of the pure hydrocarbon constituents
<V> = x_i V_i, where the xis are the volume fractions of the constituents and the Vis are their acoustic velocities. This result is somewhat surprising since the formula (1) has no known theoretical standing. Instead, I would have expected that a liquid mixture should obey Wood's formula (Wood, 1955)
V_W = (K_R/)^12, where the effective bulk modulus of the fluid mixture should be given by the harmonic mean (or Reuss average)
1K_R = x_iK_i of the constituents' adiabatic bulk moduli () and the average density is just
= x_i _i. Wood (1955) states that the formula (2) is `` applicable to a mixture of any two fluid media, which do not react chemically.''
Wood's formula (2) is based on the fact that (3) is an exact result for the effective bulk modulus of a fluid mixture during quasistatic (isothermal) deformations, while the corrections needed to change from isothermal to adiabatic moduli are generally negligible at moderate temperatures. Wood's formula is therefore known to be correct for mixtures of Newtonian fluids at low frequencies, and should also give an adequate description of the ultrasonic data on hydrocarbons if the acoustic wavelengths in the fluid mixtures are sufficiently large. The center frequency of the transmitter in the Wang and Nur (1991) experiments was 2.25 MHz and a typical value of the wave speed was 1.3 km/s, so the wavelength is approximately 0.6 mm. Whether this wavelength is large or small depends on the manner of the fluid mixing: If the fluid is well-mixed at the microscopic level, then this is a large wavelength. However, if the fluid mixture typically contains blobs of one fluid floating in the other liquid, then the relative size of the wavelength depends on the size of these blobs. The mixtures were described as being ``well-stirred'' by Wang and Nur (1991). How this translates into the microstructure of the fluid mixture is unknown to the author, but if the hydrocarbons are miscible [as is most likely (Orr, 1992)] then the blobs should be of molecular size and Wood's formula should apply.
Another well-known estimate that might be suspected to work well in describing these data is the time average formula of Wyllie et al. (1957)
<1V>^-1 = (x_iV_i)^-1. Although this formula is not exact in any nontrivial limit, it is based in part on Fermat's principle of least traveltime through a mixture. The average traveltime will generally be greater than the minimum traveltime. Therefore, we expect the velocity estimate (5) to be a lower bound on the true velocity, whenever this formula applies. Applicability of the formula depends again on the wavelength, but this time the wavelength should be short compared to the inclusion size - since Fermat's principle may be derived from the eikonal equation which is a high frequency approximation.
To the author's knowledge, general relationships among these three estimates of the mixture velocity have not been noted before. The main purpose of this paper is to point out that these estimates are ordered, i.e., Wood's formula always gives a smaller velocity than the time average, which is in turn always smaller than the volume average. The secondary purpose is to show that all three estimates give an adequate description (within 1%) of the ultrasonic data on hydrocarbon mixtures presented by Wang and Nur (1991).