# Kinematics of a Fluid Ellipse in a Linear Flow

## Abstract

**:**

## 1. Introduction

## 2. Definitions and Notation

#### 2.1. A Linear Velocity Field

**∇**be the horizontal gradient operator represented as a two-vector, $\mathbf{\nabla}\equiv {\left[\frac{\partial}{\partial x}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial y}\right]}^{T}$, as opposed to the basis-free representation ∇. In this notation, the velocity gradient matrix is given by $\nabla \mathbf{u}={\left(\mathbf{\nabla}{\mathbf{u}}^{T}\right)}^{T}$, with the extra transpose required in order that the operator

**∇**acts from the left and at the same time recovering the correct arrangement of terms as seen in Equation (2). The velocity divergence and the vertical component of vorticity are then

#### 2.2. A Matrix Basis

**∇**as in ${\mathbf{\nabla}}^{T}{\mathbf{G}}^{T}\mathbf{u}={\left(\mathbf{G}\mathbf{\nabla}\right)}^{T}\mathbf{u}$. The first two such operators in Equation (17) are recognized from Equation (5) as the divergence and the vertical component of the curl, respectively. Writing out all four quantities explicitly leads to

#### 2.3. The Kinetic Energy, Stream Function. and Angular Velocity

#### 2.4. Measures of Ellipse Size and Shape

#### 2.5. Stream Function and Energy Ellipses

## 3. Ellipse Kinematics

#### 3.1. A Kinematic Model for Fluid Particles in an Ellipse

#### 3.2. The Ellipse Flow Matrix

#### 3.3. The Ellipse Evolution Equations and Ellipse/Flow Equivalence

#### 3.4. The Kinematic Boundary Condition Approach

## 4. Integrals of a Fluid Ellipse

#### 4.1. An Elliptical Ring of Fluid

#### 4.2. Moment of Inertia, Angular Momentum, and Circulation

#### 4.3. Physical Properties of an Elliptical Disk of Fluid

#### 4.4. Kinetic Energy

#### 4.5. Computing the Integrals of the Ellipse

#### 4.6. Computing the Kinetic Energy Integral

#### 4.7. A Partitioning of the Ellipse Kinetic Energy

## 5. The Extended Stokes’ Theorem

#### 5.1. Moment Matrices

#### 5.2. An Extended Stokes’ Theorem

## 6. Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Kinematic Boundary Condition

## References

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**Figure 1.**Matrix products associated with the $\mathbf{I}\mathbf{J}\mathbf{K}\mathbf{L}$ matrices presented as quiver plots. From left to right, plots of $\mathbf{I}\mathbf{x}$, $\mathbf{J}\mathbf{x}$, $\mathbf{K}\mathbf{x}$, and $\mathbf{L}\mathbf{x}$ are shown. These are the same as a velocity field of pure divergence, pure vorticity, pure normal strain and pure shear strain, respectively.

**Figure 2.**Quadratic forms associated with the $\mathbf{I}\mathbf{J}\mathbf{K}\mathbf{L}$ matrices. From left to right, contour plots of ${\mathbf{x}}^{T}\mathbf{I}\mathbf{x}$, ${\mathbf{x}}^{T}\mathbf{J}\mathbf{x}$, ${\mathbf{x}}^{T}\mathbf{K}\mathbf{x}$, and ${\mathbf{x}}^{T}\mathbf{L}\mathbf{x}$ are shown, with positive contours shown as black solid lines and negative contours as dashed-dotted gray lines. Note that ${\mathbf{x}}^{T}\mathbf{J}\mathbf{x}=0$ identically.

**Figure 3.**Schematics for (

**a**) an ellipse and (

**b**) an elliptical ring or annulus. In both panels, the semi-major and semi-minor axes a and b are denoted by heavy solid and heavy dashed lines, respectively, while the location of one particular particle along the periphery is marked with a thin solid line. The ellipse orientation angle $\theta $, measured counterclockwise from the x-axis, and the phase angle $\varphi $ to the particle location from the major axis are also shown. The twenty-four “spokes” around the ellipse mark uniform increments of $\Delta \varphi =\pi /48$ radians in the phase angle $\varphi $. In (b), two concentric ellipses of identical shape and orientation, but slightly different sizes, are drawn. The space between these two ellipses forms an elliptical ring or annulus, with the phase angle $\varphi $ now taken to mark parcels, shown later to have the same area, as opposed to simply a location around the ellipse periphery. The parcel between the indicated phase angle $\varphi $ and $\varphi +\Delta \varphi $ is filled in with dark shading.

**Table 1.**Multiplication rules for the $\mathbf{I}\mathbf{J}\mathbf{K}\mathbf{L}$ matrices, giving the result of multiplying the row matrix by the column matrix. For example, the terms in the second row are the values of ${\mathbf{J}}^{T}\mathbf{I}$, ${\mathbf{J}}^{T}\mathbf{J}$, ${\mathbf{J}}^{T}\mathbf{K}$, and ${\mathbf{J}}^{T}\mathbf{L}$.

I | J | K | L | |
---|---|---|---|---|

$\mathbf{I}$ | $\mathbf{I}$ | $\mathbf{J}$ | $\mathbf{K}$ | $\mathbf{L}$ |

${\mathbf{J}}^{T}$ | $-\mathbf{J}$ | $\mathbf{I}$ | $-\mathbf{L}$ | $\mathbf{K}$ |

$\mathbf{K}$ | $\mathbf{K}$ | $-\mathbf{L}$ | $\mathbf{I}$ | $-\mathbf{J}$ |

$\mathbf{L}$ | $\mathbf{L}$ | $\mathbf{K}$ | $\mathbf{J}$ | $\mathbf{I}$ |

**Table 2.**A comparison of four different quantities describing ellipse shape. Each quantity is expressed in terms of the first three, as well as in terms of the major and minor semi-axis lengths a and b.

Name | Symbol | Range | $(\mathit{a},\mathit{b})$ | $(\mathit{\eta})$ | $(\mathit{\lambda})$ | $(\mathit{\mu})$ |
---|---|---|---|---|---|---|

Aspect ratio | $\eta $ | $(1,\infty ]$ | $\frac{a}{b}$ | $\eta $ | $\sqrt{\frac{1+\lambda}{1-\lambda}}$ | $\sqrt{\frac{\mu +\sqrt{{\mu}^{2}-1}}{\mu -\sqrt{{\mu}^{2}-1}}}$ |

Linearity | $\lambda $ | $(0,1)$ | $\frac{{a}^{2}-{b}^{2}}{{a}^{2}+{b}^{2}}$ | $\frac{{\eta}^{2}-1}{{\eta}^{2}+1}$ | $\lambda $ | $\frac{\sqrt{{\mu}^{2}-1}}{\mu}$ |

Extension | $\mu $ | $(1,\infty ]$ | $\frac{{a}^{2}+{b}^{2}}{2ab}$ | $\frac{{\eta}^{2}+1}{2\eta}$ | $\frac{1}{\sqrt{1-{\lambda}^{2}}}$ | $\mu $ |

— | $\frac{\lambda}{\mu}$ | $(0,\infty ]$ | $\frac{{a}^{2}-{b}^{2}}{2ab}$ | $\frac{{\eta}^{2}-1}{2\eta}$ | $\frac{\lambda}{\sqrt{1-{\lambda}^{2}}}$ | $\sqrt{{\mu}^{2}-1}$ |

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Lilly, J.M. Kinematics of a Fluid Ellipse in a Linear Flow. *Fluids* **2018**, *3*, 16.
https://doi.org/10.3390/fluids3010016

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Lilly JM. Kinematics of a Fluid Ellipse in a Linear Flow. *Fluids*. 2018; 3(1):16.
https://doi.org/10.3390/fluids3010016

**Chicago/Turabian Style**

Lilly, Jonathan M. 2018. "Kinematics of a Fluid Ellipse in a Linear Flow" *Fluids* 3, no. 1: 16.
https://doi.org/10.3390/fluids3010016