The uniform stretching limit of the total viscosity without hydrodynamics is κP(0) = 12ηφ2b and with full hydrodynamics κB(0)=5.247ηφ2b. Therefore, we expect that there must also be a contribution to the total viscosity in the suspension.
Measurement of the bulk viscosity
The excess dissipation shows up as a change in the effective bulk viscosity of the material. Instead, one can exploit the compressibility of the particulate phase (not the particles themselves) to define a bulk viscosity.
Bulk viscosity of a suspension
The difference between the mechanical pressure in the system and the equilibrium pressure gives the effective mass viscosity times the average rate of expansion of the suspension. Thus we obtain the bulk viscosity correction as 43η/φ, a result first derived by Taylor [1954a,b].
The O(φ 2 ) correction to bulk viscosity
Pair-evolution equation
The small O(φ) concentration of bubbles produces a small expansion rate (hei= φep) throughout the dispersion, but the pressure field is disturbed over the entire surrounding fluid, which leaves a large O(1−φ) fraction of the dispersion in takes possession. total volume. The conditional probability density can be expressed in terms of the number density and the pair distribution function asP1/1 = n(t)g(r,t), and the resulting equation for the pair distribution function can be written as.
Hydrodynamic interactions
The scalar functionsTαβQ were defined for all separations and particles with different radii [Khairet al.2006]. A lubrication theory expression is derived for very small separations and the multipole expansion is used for all other separations following the method of Jeffrey and Onishi [1984] .
Expressions for the bulk viscosity
The hydrodynamic functions described here were used to determine the particle stresses resulting from Brownian, interparticle force, and strain-rate interactions between particles and the result was used to calculate the corresponding contributions to the effective bulk viscosity of the suspension. All two-particle contributions are positive and proportional to the shear viscosity η, since they arise from the incompressible disturbance flow caused by the presence of the stiff particles.
Results and discussion
Numerical solution of the perturbed microstructure
The perturbation of the microstructure occurs within this boundary layer and thus, f(r) decays very quickly within a very short distance of the order of the boundary layer thickness ∼ O(1/|Peb|). The contact value of f(r) (atr = 2) is shown in Figure 2.4 as a function of the Péclet number for different levels of hydrodynamic interactions depending on the value of ˆb.
The bulk viscosity contribution of the hard sphere without hydrodynamic interactions is shown in Figure 2.9 as a function of the Péclet number. The inter-particle force coefficient decays most slowly as 1/bˆ because the particle diffusivity D=kT/6πη depends on the actual or hydrodynamic size of the particle.
Concluding remarks
Here we determine the viscoelastic response for a third type of viscosity - the total viscosity. The frequency-dependent response can also be understood as the Fourier transform of the time response.
Bulk viscosity of a suspension
The normalized elastic bulk modulus of the suspensionK0 is related to the imaginary part of the bulk viscosity by K0(ω)= K0+ωκ00(ω) [Zwanzig and Mountain 1965], where K0 is the normalized zero-frequency bulk modulus (inverse compressibility) of the suspension given by. To isolate the frequency dependence of the bulk viscosity, it is useful to define reduced bulk viscosity functions.
Microstructure and bulk viscosity: No hydrodynamics
In the high frequency limit (α → ∞) the reduced mass and shear viscosities asymptote to zero with the same dependence on α; only the coefficient is different. Only the hard sphere stress contributes to the shear viscosity in the absence of hydrodynamics and thus the reduced shear viscosity functions depend only on f(2, α), as for the bulk viscosity.
Temporal response
Therefore, one can obtain the pressure autocorrelation function from an inverse Laplace transform of the frequency-dependent bulk viscosity. As t → ∞ the pressure autocorrelation decays as CPNH π)τ−3/2, obtained using an asymptotic expansion of the complementary error function [Abramowitz and Stegun 1964].
E ff ect of hydrodynamic interactions
Thus in the complete limit of hydrodynamics (δ → 0) the perturbation O(α−1/2) given by f0 becomes zero and the first order perturbation in the microstructure is O(α−1), given by. The high frequency limit of K0(α) was calculated numerically as described in the next section.
Numerical results
In the absence of hydrodynamics (ˆb−1= 105) the real part of the bulk viscosity decays as α−1/2asα→ ∞, but in full hydrodynamics the decay is almost O(α−2). In the absence of hydrodynamic interactions (ˆb 1), the bulk modulus has a scaling of α1/2 at high frequencies, as discussed previously.
Conclusions
The stable limit of bulk viscosity expansion without hydrodynamics is κP(0)=12ηφ2 band with full hydrodynamicsκB(0)=5.247ηφ2b. The principal viscosity of a suspension relates the deviation of the macroscopic stress trace from its equilibrium value to the average rate of expansion.
Suspension stress
The wake hei, given by hei ≡ h∇ ·ui, is the average rate of extension of the suspension. The particle tension is the symmetric part of the first moment of surface tension on the particle.
Stokesian Dynamics
- Review of the existing method
- Linear compressible flow
- The Mobility matrix with expansion flow
- Brownian motion
The remaining far-field contribution to the pressure moment will be calculated from the Stokes pressure corresponding to all the reflected disturbance currents by inverting the mobility matrix. The RS E part of the thus obtained far-field resistance matrix does not contain the hydrodynamic resistance functions for calculating the pressure moment due to the anomalous strain rate.
Accelerated Stokesian Dynamics
Expansion flow in ASD
The fluid velocity due to the particle sources is added to the velocity contribution from the forces and stresses acting on the particles and interpolated to find the particle velocities in the far field using Faxén's laws. The near-field contribution to the pressure moment is calculated from the hydrodynamic drag functions as in the SD method.
Conclusions
The computation of the far-field pressure moment has already been implemented in ASD, and the expressions for the wavespace and real-space contributions are given in the PhD thesis of Sierou [2002] . Since the only force in the problem is a scalar, the velocity must be of the form f(r)Sx.
Suspension stress and the bulk viscosity
Finally, as defined in this work, the effective bulk viscosity of a suspension is given by κe f f. The Brownian and interparticle force contributions to the stress have a finite mean value at equilibrium, independent of the strain rate, because they originate from the thermal motion of the particles.
Brownian Dynamics simulations
Simulation results
We simply fit the thin theory curve to the simulation data in the region before the data becomes too noisy in order to obtain long tails. The bulk Green-Kubo viscosity in the absence of hydrodynamic interactions is shown in Figure 5.9.
Scaling with volume fraction
Consequently, the stress disturbance is also isotropic and occurs only in the trace of the bulk stress. This simple argument would also suggest that the magnitude of the pressure autocorrelation function scales withφg0(2;φ).
Temporal scaling
The adjusted theoretical prediction for the total viscosity given by (5.20) is shown in Fig. 5.9 together with the results of the Brownian dynamics simulation. The scaled theory for shear viscosity is shown in Fig. 5.12 and agrees very well with the BD results.
E ff ect of hydrodynamic interactions
Scaling with volume fraction
It is instructive to look at how the φscaling of the pressure autocorrelation function changes in the presence of hydrodynamic interactions before analyzing the simulation results. The high-frequency bulk modulus is related to the zero-time limit of the pressure autocorrelation by K∞0 =V/kThδΠ(0)δΠ(0)i, from which we obtain the non-dimensional zero-time limit.
Temporal scaling
The SD simulations show a sudden break in the increase of the relaxation time scale for φ >. For large φ, one would expect the dull self-diffusivity with hydrodynamic interactions D∞,Hs (φ) to set the relaxation time scale, but we found that for 0.35≤φ≤0.5 the dull self-diffusivity without hydrodynamics Ds∞,NH( φ) provides a better temporal scaling of the simulation data.
Simulation results
This observation further supports the theory that the D0s(φ) scaling in the bulk viscosity mainly arises from the scaling of the strength of hydrodynamic interactions. Furthermore, because it is a measure of the moment of pressure in an equilibrium configuration, the scaling for higher volume fractions should be closest to the equilibrium osmotic pressure Π0, which would explain the increase in particle pressure with φ.
Conclusions
The Brownian contribution to both the bulk and shear viscosities scales asg0(2;φ)/DS∞,H(φ)D0s(φ) and therefore shows a stronger divergence with increasing φ, as shown in Figure 5.20 and Figure 5.25. The pressure autocorrelation data highlight the competition between long-range structural relaxation and the relaxation delay due to hydrodynamic interactions, which was not noticeable in the shear stress autocorrelation.
Introduction
This disturbance current causes the tension on the particles to change, which also changes the bulk tension in the suspension. We expect that the bulk viscosity calculated by this technique will be comparable to that for particle phase expansion alone in an incompressible fluid, because the contribution to the isotropic stress in both cases is due to incompressible disturbance currents.
Simulation procedure
The bulk viscosity of the suspension is then determined by calculating the deviation in the mean stress in the material in a manner analogous to that for shear viscosity [Batchelor and Green 1972b; Brady and Bossis 1988], and relate this to the average expansion rate. Thus, averaging over different time steps in each run serves to reduce fluctuations due to the Brownian motion of the particles, and averaging over all runs reduces the error due to variations in the microstructure.
Microstructure in compression
The imposed compression force is only compensated in the boundary layer by Brownian motion, so that particles have to diffuse over a distance of O. In the above discussion we have assumed that the initial configuration of particles is spatially homogeneous.
Brownian Dynamics
Results and scaling
The dependence of the stress on the volume fraction can be seen more clearly in the bulk viscosity κP shown in Figure 6.8 and calculated using (6.7), since the dependence is reduced. For smaller values of Pethe, scaled bulk viscosity data decrease with increasing φ due to reduced simulation cell size and longer times required to reach steady state.
Accelerated Stokesian Dynamics
Results and scaling
There is an excellent collapse of the data for φ≤ 0.35 and for higher φ the bulk viscosity is also larger. Simulation data for bulk hydrodynamic viscosity can be collapsed for allφ by scaling the increase in κH due to compressional flow (κH−κ0∞) with its equilibrium value (given byκH0 =κ0∞−43ηφ) times the function of the equilibrium pair distribution g0(2; φ) as shown in Figure 6.18.
E ff ect of shear on the hydrodynamic bulk viscosity
Thus, the normalized effect of shear on the bulk viscosity (shown in Figure 6.24) is given by. Such a correlation can be used to estimate the bulk viscosity of a displaced suspension from the known dynamic viscosity of the suspension.
Conclusions
The simulation results for allPesh collapse to a single curve with this normalization, indicating that the effect of shear on total viscosity is the same as the effect of shear on dynamic viscosity. In most practical situations, the suspension would undergo shear in addition to expansion, so it is important to consider the effect of shear on the effective total viscosity.
![Figure 5.9: The Green-Kubo bulk viscosity κ B from equilibrium Brownian Dynamics simulations (solid diamonds), the MD simulation results of Sigurgeirsson and Heyes [2003] for the bulk viscos-ity of hard sphere fluids ( + and *), and the scaled theoretical](https://thumb-ap.123doks.com/thumbv2/123dok/10415299.0/153.918.173.800.304.811/viscosity-equilibrium-brownian-dynamics-simulations-simulation-sigurgeirsson-theoretical.webp)
