# A Unified, One Fluid Model for the Drag of Fluid and Solid Dispersals by Permeate Flux towards a Membrane Surface

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

**S**represents the surface force per unit area due to the motion of the fluid relative to the object, then the net surface force in the direction of the bulk motion may be expressed as:

**e**is the unit vector in the direction of the bulk flow field. The surface force per unit area,

**S**, is correlated with the stress tensor at the surface, such that $S=Tn,$ where $T$ is the stress tensor and

**n**is the outwardly unit normal vector to the area. The stress tensor for incompressible Newtonian fluids can be written as $T=-pI+\tau $, where $\tau =\mu \left(\nabla v+\nabla {v}^{T}\right)$ is the viscous stress tensor. This formulation suggests that there exist two contributions into the hydrodynamic drag—one contribution from the pressure field (this is called form drag) and another contribution from viscous force (this is called skin drag). In order to accurately determine the drag force, both the velocity and the pressure fields around the object need to be resolved.

^{3}], $\mu $ is the viscosity [M/LT],$T$ is the stress tensor [M/LT

^{2}], p is the pressure [M/LT

^{2}], and $g$ is the gravity.

## 3. Framework for Validation

## 4. The Numerical Scheme

## 5. Benchmarking and Comparisons

_{x}. In the middle of this domain, an extended cylinder of radius a is initially inserted and is released to settle under the effect of gravity. The object will initially accelerate until reaching a terminal velocity, which will be maintained for the rest of the motion until the disc encounters the bottom boundary. The contours of some hydrodynamic parameters, such as the vorticity as well as the y-direction velocity profiles at different sections, will be generated.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. (Results of Benchmarking)

**Parameters:**Radius = a, Diameter = d, Width = L

_{x}, Height= L

_{y}. The colored cells represent the variables investigated.

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}=\mathbf{d}/{\mathbf{L}}_{\mathbf{x}}$ | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) | Mesh (nx,ny) |
---|---|---|---|---|---|---|

0.2 | 0.4 | 1:4 | 10 | 10^{−3} | 10^{3} | 40,80 |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % Relative Error |
---|---|---|---|

${k}_{x}$ = d/L_{x} = 1:2 | −0.0702567 | −0.0889332 | 26.5833 |

1:4 | −0.0999891 | −0.0984975 | 1.49169 |

1:6 | −0.071729 | −0.0704685 | 1.75721 |

1:8 | −0.0520517 | −0.0511041 | 1.82054 |

1:10 | −0.0392939 | −0.38567 | 1.85011 |

1:20 | −0.0145841 | −0.014307 | 1.89984 |

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) | Mesh (nx,ny) |

0.2 | 0.4 | 1:4 | 10 | 10^{−3} | 10^{3} | 40,80 |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | |||

With density Indicator | −0.0999891 | −0.0999057 | 0.0833489 | |||

Weighted Arithmetic Avg (all) | −0.0999891 | −0.0811345 | 18.8566 | |||

Weighted Arithmetic Avg XX only | −0.0999891 | −0.0929154 | 7.0744 | |||

Weighted Arithmetic Avg XY only | −0.0999891 | −0.0811345 | 18.8566 | |||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) | Mesh (nx,ny) |

0.2 | 0.4 | 1:6 | 10 | 10^{−3} | 10^{3} | 40,80 |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | |||

With density Indicator | −0.071729 | −0.0714543 | 0.382922 | |||

Weighted Arithmetic Avg | −0.071729 | −0.0617425 | 13.9225 | |||

Weighted Arithmetic Avg XX only | 0.071729 | −0.0675415 | 5.8379 | |||

Weighted Arithmetic Avg XY only | −0.0999891 | −0.0617425 | 13.922 |

Case I | ||||||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) | Mesh (nx,ny) |

0.2 | 0.4 | 1:4 | 10 | 10^{−3} | 10^{5} | 40,80 |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | |||

With density Indicator | −0.0999891 | −0.0984977 | 1.49151 | |||

Case II | ||||||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) | Mesh (nx,ny) |

0.2 | 0.4 | 1:4 | 10 | 10^{−3} | 10 | 40,80 |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | |||

With density Indicator | −0.0999891 | −0.0985065 | 1.49151 |

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:2 | 20 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0702567 | −0.088169 | 25.4956 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:2 | 40 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0702567 | −0.0883669 | 25.7772 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:2 | 80 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0702567 | −0.0883825 | 25.7994 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:2 | 160 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0702567 | −0.0884388 | 25.8796 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:6 | 20 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.071729 | −0.0712175 | 0.712979 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:6 | 40 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.071729 | −0.0715578 | 0.23855 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:6 | 80 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.071729 | −0.0716596 | 0.0967284 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:10 | 20 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0392939 | −0.0390103 | 0.721781 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:10 | 40 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0392939 | −0.0391964 | 0.248302 | ||

L_{x} (m) | L_{y} (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | d/∆x | μ_{f} (Pa.s) | μ_{s} (Pa.s) |

0.2 | 0.4 | 1:10 | 80 | 10^{−3} | 10^{5} |

Case | U_{c}, Analytical (m/s) | U_{p}, Simulated (m/s) | % relative error | ||

With density Indicator | −0.0392939 | −0.0392554 | 0.0981869 |

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**Figure 1.**Surface forces at the two sides of the interface between two phases. Jump in the normal stress is correlated with the curvature of the interface. Tangential stress does not experience jump under the no-slip condition.

**Figure 2.**Schematic diagram of the gradient function of the phase function, $\phi $. The gradient is zero in every way except at the interface, where it jumps as a positive delta function on half the interface and a negative delta function over the second half. Therefore, its integral over the whole interface is zero.

**Figure 3.**Schematic of the domain where Faxén [39] expansion apply.

**Figure 5.**Discrete determination of the average velocity over the cells that are completely inside the boundary.

**Figure 15.**Comparisons between the terminal settling velocity and the simulated one for different diameter to width ratios. In these simulations, the following parameters are held fixed: μ

_{f}= 0.001 Pa·s, μ

_{p}= 1000 Pa·s, Mesh n

_{x}= 40, n

_{y}= 80.

**Figure 16.**Comparisons between the terminal settling velocity and the simulated one for different algorithms for calculating the density in the cells encompassing the interface. In these simulations, the following parameters are held fixed: ${k}_{x}$ = d/L

_{x}= 4, μ

_{f}= 0.001 Pa·s, μ

_{p}= 1000 Pa·s, Mesh n

_{x}= 40, n

_{y}= 80.

**Figure 17.**Comparisons between the terminal settling velocity and the simulated one for different viscosity ratios. In these simulations, the following parameters are held fixed: ${k}_{x}$ = d/L

_{x}= 4, μ

_{f}= 0.001 Pa·s, Mesh density n

_{x}= 40, n

_{y}= 80.

**Figure 18.**Comparisons between the terminal settling velocity and the simulated one for different spatial resolutions. In these simulations, the following parameters are held fixed: ${k}_{x}$ = d/L

_{x}= 4, μ

_{f}= 0.001 Pa·s, μ

_{p}= 1000 Pa·s.

μ | $\rho $ | |

Harmonic average | x | |

Weighted arithmetic average, only $({\mathsf{\tau}}_{\mathrm{xx}},{\mathsf{\tau}}_{\mathrm{yy}})$ | x | |

Weighted arithmetic average, only $({\mathsf{\tau}}_{\mathrm{xy}},{\mathsf{\tau}}_{\mathrm{yx}})$ | x | |

Weighted arithmetic average, all (${\mathsf{\tau}}_{\mathrm{xx}},{\mathsf{\tau}}_{\mathrm{yy}},{\mathsf{\tau}}_{\mathrm{xy}},{\mathsf{\tau}}_{\mathrm{yx}}$) | x | |

Weighted volume average | x | |

Density indicator | x |

L_{x} (m) | L_{y} (m) | d (m) | ${\mathit{k}}_{\mathit{x}}$ = d/L_{x} | ρ_{s} | ρ_{f} | μ_{f} (Pa.s) | μ_{s} (Pa.s) | Mesh (nx,ny) |

0.2 | 0.4 | 0.05 | 1:4 | 3000 | 1000 | 17.52 | 10^{3} | 200,400 |

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**MDPI and ACS Style**

Salama, A.; Sun, S.; Zhang, T.
A Unified, One Fluid Model for the Drag of Fluid and Solid Dispersals by Permeate Flux towards a Membrane Surface. *Membranes* **2021**, *11*, 154.
https://doi.org/10.3390/membranes11020154

**AMA Style**

Salama A, Sun S, Zhang T.
A Unified, One Fluid Model for the Drag of Fluid and Solid Dispersals by Permeate Flux towards a Membrane Surface. *Membranes*. 2021; 11(2):154.
https://doi.org/10.3390/membranes11020154

**Chicago/Turabian Style**

Salama, Amgad, Shuyu Sun, and Tao Zhang.
2021. "A Unified, One Fluid Model for the Drag of Fluid and Solid Dispersals by Permeate Flux towards a Membrane Surface" *Membranes* 11, no. 2: 154.
https://doi.org/10.3390/membranes11020154