# Fundamental Rheology of Disperse Systems Based on Single-Particle Mechanics

## Abstract

**:**

## 1. Introduction

- Rigid-solid spherical particles: uncharged and electrically charged
- Rigid-porous spherical particles
- Non-rigid (soft) solid particles
- Liquid droplets: without and with surfactant
- Bubbles: without and with surfactant
- Capsules
- Core-shell particles: solid core-porous shell, solid core-liquid shell, liquid core-liquid shell
- Rigid-solid non-spherical particles
- Ferromagnetic rigid-solid spherical and non-spherical particles

## 2. Bulk Stress and Bulk Rate of Strain in Particulate Fluids

^{1/3}, is large compared with the average spacing between the particles ℓ, but small compared with the apparatus length scale L, that is, ℓ << V

^{1/3}<< L. Under the condition ℓ << V

^{1/3}<< L, the volume of the element V is large enough to contain a statistically significant number of particles but small relative to the characteristic macro-scale of the system. Thus, the bulk or volume-average stress tensor $\langle \overline{\overline{\sigma}}\rangle $ in particulate fluids can be defined as:

## 3. Rheological Constitutive Equation for Particulate Fluids

## 4. Rheology of Suspensions of Rigid-Solid Spherical Particles

#### 4.1. Electrically-Neutral Particles

#### 4.2. Electrically-Charged Particles

^{−5}M and $[\eta ]=5.5$ at KCl concentration of 1.2 × 10

^{−4}M. The corresponding $p$ values (primary electroviscous coefficient) are: 1.76 and 1.20, respectively.

^{2}and the particle diameter was 106 nm. The surface charge on the particles of less viscous latex suspension (bottom curve) was −0.20 μC/cm

^{2}and the particle diameter was 210 nm. With the increase in electrolyte concentration, the electrical double layer thickness decreases ($R\kappa $ increases) and therefore, the coefficient $p$ decreases. For the same electrolyte concentration, the suspension of weakly-charged particles gives lower $p$ values, as expected. Furthermore, the $R\kappa $ value of the suspension of weakly-charged particles is also larger (hence thinner double layer) due to large particle size.

## 5. Rheology of Suspensions of Rigid-Porous Spherical Particles

## 6. Rheology of Suspensions of Non-Rigid (Soft) Solid Particles

## 7. Rheology of Emulsions

^{−4}. The emulsions are produced by mixing the two liquids in an agitated vessel at different agitator speeds. The intrinsic viscosity (Equation (68)) of these emulsions is 1.0013. The experimental data show good agreement with the Taylor viscosity equation when $\varphi \le 0.05$. At higher concentrations of dispersed-phase, the Taylor equation under predicts the emulsion viscosity.

#### 7.1. Influence of Electric Charge Present on the Surface of Emulsion Droplets

#### 7.2. Influence of Surfactant on Emulsion Rheology

## 8. Rheology of Bubbly Suspensions

#### 8.1. Influence of Electric Charge Present on the Surface of Bubbles

#### 8.2. Influence of Surfactant Present on the Surface of Bubbles

#### 8.3. Influence of Capillary Number

## 9. Rheology of Suspensions of Capsules

## 10. Rheology of Suspensions of Core-Shell Particles

#### 10.1. Solid Core—Porous Shell

#### 10.2. Solid Core—Liquid Shell

#### 10.3. Liquid Core—Liquid Shell

## 11. Rheology of Suspensions of Rigid Non-Spherical Particles

## 12. Rheology of Suspensions of Ferromagnetic Particles

## 13. Concluding Remarks

## Conflicts of Interest

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**Figure 1.**Comparison of Einstein viscosity equation with experimental data for viscosity of suspensions of hard spheres.

**Figure 2.**Charged particle surrounded by an ionic cloud [9]. Variations of ionic concentration and electric potential are also shown. (

**a**) Charged particle surrounded by an ionic cloud; (

**b**) Variation of ionic concentration with distance from the particle surface; (

**c**) Variation of electric potential with distance from the particle surface.

**Figure 3.**Variation of primary electroviscous coefficient $p$ with dimensionless surface potential ${\tilde{\psi}}_{0}$ for different values of $R\kappa $.

**Figure 4.**Plots of $1/{\eta}_{r}$ versus $\varphi $ data for polystyrene latex suspensions consisting of negatively charged particles of diameter 250 nm suspended in KCl solutions.

**Figure 5.**Variation of primary electroviscous coefficient $p$ with electrolyte concentration for electrically charged polystyrene latex suspensions.

**Figure 6.**Variation of [η] with Pe for suspensions of charged particles at different values of Hartmann number (Rk fixed at 8).

**Figure 7.**Variation of [η] with Pe for suspensions of charged particles at different values of Rk (Ha fixed at 0.5).

**Figure 8.**Reduced first normal-stress difference ${N}_{1}/\left(\epsilon \text{}{\psi}_{0}^{2}/{R}^{2}\right)$ as a function of ion Peclet number.

**Figure 9.**Reduced second normal-stress difference $-{N}_{2}/\left(\epsilon \text{}{\psi}_{0}^{2}/{R}^{2}\right)$ as a function of ion Peclet number.

**Figure 10.**Comparison of bulk and shear primary electroviscous coefficients at a small Hartmann number of 0.1.

**Figure 11.**Intrinsic viscosity $[\eta ]$ as a function of dimensionless permeability ($\alpha /{R}^{2}$ ).

**Figure 12.**Influence of particle permeability on the intrinsic viscosity of suspension of electrically-charged porous particles.

**Figure 13.**Intrinsic viscosity $[\eta ]$ as a function of $Se$ for suspensions of soft solid particles.

**Figure 14.**Reduced first normal-stress difference ${N}_{1r}/\varphi $ as a function of $Se$ for suspensions of soft-solid particles.

**Figure 15.**Reduced second normal-stress difference $-{N}_{2r}/\varphi $ as a function of $Se$ for suspensions of soft-solid particles.

**Figure 16.**Relative viscosity ${\eta}_{r}$ versus droplet concentration $\varphi $ for dilute emulsions for different values of the viscosity ratio $\lambda $.

**Figure 17.**Comparison between experimental viscosity data and theoretical predictions for emulsions.

**Figure 18.**Intrinsic viscosity of emulsion as a function of capillary number for different values of viscosity ratio $\lambda $.

**Figure 19.**Reduced first normal stress difference ${N}_{1r}/\varphi $ versus $Ca$ , keeping $\lambda =1$.