Effective dynamic properties of random complex media with spherical particles

Abstract

The effective dynamic bulk modulus and density are presented for random media consisting of particles in a viscous host fluid, using a core-shell, self-consistent effective medium model, under the large compressional wavelength assumption. These properties are relevant to acoustic or dynamic processes in nano-and micro-particle fluids including particle density determination, resonant acoustic mixing, and acoustic characterisation. Analytical expressions are obtained for the effective bulk modulus and mass density, incorporating the viscous nature of the fluid host into the core-shell model through wave mode conversion phenomena. The effective density is derived in terms of particle concentration, particle and host densities, particle size, and the acoustic and shear wavenumbers of the liquid host. The analytical expressions obtained agree with prior known results in the limit of both static and inviscid cases; the ratio of the effective bulk modulus to that of the fluid is found to be quasi-static. Numerical calculations demonstrate the dependence of the effective mass density on frequency, particle size (from nano-to micro-regime), and concentration. Herein it is demonstrated both theoretically and numerically that the viscosity, often neglected in the literature, indeed plays a significant role in the effective properties of nanofluids.

Ultrasonic propagation in concentrated colloidal dispersions: improvements in a hydrodynamic model

“…Hydrodynamic models, being unable to calculate the effective bulk modulus of a suspension, assume a volumeaveraged compressibility or a harmonic (or Reuss) average of bulk modulus, [2,23] . By contrast, scattering theory shows that the effective bulk modulus for a viscous suspension slightly differs from its inviscid counterpart [21,35] and thereby affects ultrasonic attenuation and velocity to a noticeable extent. To calculate the effective wavenumber, we, therefore, choose to use the viscosity-corrected, effective bulk modulus expression of Ref.…”

“…To calculate the effective wavenumber, we, therefore, choose to use the viscosity-corrected, effective bulk modulus expression of Ref. [21] which is given by…”

“…(8), when expressed in terms of the Lamé parameters through R Q , R U , and P : QQ , takes the well-known effective bulk modulus form obtained by Kuster and Toksöz for a solidin-solid system. [21,36] We prefer the form of Eq. (8), because it captures more information about attenuation, such as the bulk viscosity, through the monopole scattering coefficient and the wavenumbers, and can effectively model attenuation even for small particles, as we shall see.…”

“…Alam et al, based on the partial wave analysis, have recently developed a core-shell, selfconsistent effective medium model for a suspension of solid spheres in a viscous fluid. [21] They have incorporated the viscosity of the base fluid through shear waves into the model and have found the viscosity to produce an appreciable effect on the effective bulk modulus and density. Since this model…”

“…Experimental findings by Forrestal et al [20] for a suspension of silica spheres in water corroborated attenuation values predicted by this model. In an attempt to understand the effect of wave mode conversion on the effective properties of a suspension, Alam et al, [21] based on the coherent potential approximation (CPA), developed a core-shell effective medium model incorporating shear wave contributions into the model and calculated the effective bulk modulus and density. This work, however, has yet to investigate ultrasonic speed and attenuation.…”