# Hydrodynamics of Highly Viscous Flow past a Compound Particle: Analytical Solution

## Abstract

**:**

## 1. Introduction

## 2. Formulation of Problem

## 3. Analytical Solution for Uniform Flow past Spherical Objects

#### 3.1. Uniform Flow past a Rigid Sphere

#### 3.2. Uniform Flow past a Fluid Drop

#### 3.3. Uniform Flow past a Rigid-Kernel Sphere with a Fluid Coating

- The boundary condition at infinity implies$$C=\frac{1}{2}U\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}D=0.$$
- The zero normal velocity at the interface of fluids ${v}_{r}({R}_{-},\theta )={\tilde{v}}_{r}({R}_{+},\theta )=0$ sets$$2cos\theta \left(\frac{A}{{R}^{3}}+\frac{B}{R}+\frac{1}{2}U\right)=2cos\theta \left(\frac{\tilde{A}}{{R}^{3}}+\frac{\tilde{B}}{R}+\tilde{C}+\tilde{D}{R}^{2}\right)=0.$$
- With the continuity of tangential velocity at the interface of fluids, the following equation is satisfied$$\begin{array}{c}\hfill -\frac{A}{{R}^{3}}+\frac{B}{R}+U=-\frac{\tilde{A}}{{R}^{3}}+\frac{\tilde{B}}{R}+2\tilde{C}+4\tilde{D}{R}^{2}.\end{array}$$
- The no-slip boundary condition at $r=a$ requires$$\begin{array}{ccc}\hfill \frac{\tilde{A}}{{a}^{3}}+\frac{\tilde{B}}{a}+\tilde{C}+\tilde{D}{a}^{2}=0& \mathrm{and}& -\frac{\tilde{A}}{{a}^{3}}+\frac{\tilde{B}}{a}+2\tilde{C}+4\tilde{D}{a}^{2}=0.\hfill \end{array}$$
- Continuity of tangential stress at the interface of the fluids ${\sigma}_{r\theta}\left({R}_{-},\theta \right)={\tilde{\sigma}}_{r\theta}\left({R}_{+},\theta \right)$ implies$$\begin{array}{ccc}\hfill -6\mu sin\theta \left(\frac{A}{{R}^{4}}+DR\right)& =& -6\overline{\mu}sin\theta \left(\frac{\tilde{A}}{{R}^{4}}\phantom{\rule{4.pt}{0ex}}+\tilde{D}R\right).\hfill \end{array}$$

#### 3.4. Hydrodynamic Drag Force and Terminal Velocity

**viscosity ratio**as $\lambda =\frac{\overline{\mu}}{\mu}$ and the

**volume fraction**in terms of ratio $\gamma =\frac{a}{R}$. After nondimensionalizing the drag with a dimensional factor $4\pi \mu R\phantom{\rule{0.166667em}{0ex}}U$, the non-dimensional drag force is

#### 3.5. Multiple-Layer Fluid Coating

**one layer**of fluid coating discussed before ($n=1$, see Figure 1), the linear system is the combination of Equations (31)–(35):

**two-layer**fluid coating, Figure 8a shows the diagram of the problem. Radii for the fluid coating satisfy $a\le {R}_{1}\le R$. Keep $\{A,B,C,D\}$ for the stream function outside the particle and $\{\tilde{A},\tilde{B},\tilde{C},\tilde{D}\}$ for the inner layer next to the rigid kernel, as the previous case. Note $\{{A}_{1},{B}_{1},{C}_{1},{D}_{1}\}$ as the coefficients for the stream function of the outer layer (${R}_{1}\le r\le R$). The viscosity of fluid outside the particle is μ; $\tilde{\mu}$ is the viscosity of the fluid in the layer $a\le r\le {R}_{1}$, and the new parameter ${\mu}_{1}$ is for the fluid in the layer ${R}_{1}\le r\le {R}_{2}$. To attain the analytical solution, we collect the boundary conditions in terms of the unknowns. The linear system for the unknowns is

**three-layer**fluid coating as shown in Figure 8b, radii of the fluid coating satisfy $a\le {R}_{1}\le {R}_{2}\le R$. Compared to the linear system (47), four more unknowns are introduced. The coefficient matrix for the corresponding linear system is

## 4. Summary and Future Directions

## Conflicts of Interest

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**Figure 1.**Uniform flow past a rigid sphere with a fluid coating. The background uniform flow is $\mathbf{U}=U{\mathbf{e}}_{z}$. The dark colored region indicates the rigid spherical kernel with radius a. The light colored region shows the fluid coating with thickness d. In total, the radius of the obstacle is $R=a+d$.

**Figure 2.**Streamlines in the symmetry plane for a uniform flow past a rigid sphere. The radius of the rigid sphere is $a=1$.

**Figure 3.**Streamlines in the symmetry plane for uniform flow past a fluid drop. The radius of the fluid drop is $d=1$ and the viscosity ratio $\overline{\mu}/\mu =10$.

**Figure 4.**Streamlines in the symmetry plane for the flow field of a uniform flow past a fluid-coated rigid-kernel sphere. The black region is the rigid kernel and the gray region with circulation is the fluid coating. The viscosity ratio is $\overline{\mu}/\mu =5$. The radius of the rigid kernel and the thickness of the fluid coating are set as $a=d=1$.

**Figure 5.**Loglog plot of the dimensionless drag force $F(\lambda ,\gamma )$ as a function of the viscosity ratio λ with fixed values of the volume fraction, ${\gamma}^{3}$.

**Figure 6.**The dimensionless drag force $F(\lambda ,\gamma )$ as a function of γ with fixed values of the viscosity ratio λ. ${\gamma}^{3}$ is the volume fraction of the object.

**Figure 7.**The dimensionless terminal velocity $\mathrm{Un}(\lambda ,\gamma )$ as a function of the viscosity ratio λ with fixed volume fraction ${\gamma}^{3}$. The densities of the fluid in the coating and the rigid kernel are set be the same as the mean density ${\rho}_{\mathrm{mean}}$. The radii of the two-layer object, the rigid sphere, the fluid drop, and the air bubble are the same. The air bubble result is obtained by taking the viscosity ratio $\lambda =\overline{\mu}/\mu =0$ for the fluid drop.

**Figure 8.**The uniform background flow $\mathbf{U}=U{\mathbf{e}}_{z}$ past a rigid sphere with a multiple-layer fluid coating. The gray region with radius a indicates the rigid kernel. The light colored regions show the fluid coating with radii ${R}_{1}$, ${R}_{2}$, and R for the three-layer case. (

**a**) The rigid kernel covered with a two-layer fluid coating. $a\le {R}_{1}\le R$; (

**b**) The rigid kernel covered with a three-layer fluid coating. $a\le {R}_{1}\le {R}_{2}\le R$.

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**MDPI and ACS Style**

Zhao, L.
Hydrodynamics of Highly Viscous Flow past a Compound Particle: Analytical Solution. *Fluids* **2016**, *1*, 36.
https://doi.org/10.3390/fluids1040036

**AMA Style**

Zhao L.
Hydrodynamics of Highly Viscous Flow past a Compound Particle: Analytical Solution. *Fluids*. 2016; 1(4):36.
https://doi.org/10.3390/fluids1040036

**Chicago/Turabian Style**

Zhao, Longhua.
2016. "Hydrodynamics of Highly Viscous Flow past a Compound Particle: Analytical Solution" *Fluids* 1, no. 4: 36.
https://doi.org/10.3390/fluids1040036