Kinematics of Fluid Flow

Abstract

The study of kinematics has flourished as a subject where one may consider displacements and motions without imposing any restrictions on them; that is, there is no need to ask whether they are dynamically feasible in the physical world.

Appendix

In this Appendix, we list some basic results which are used in this monograph as required.

• 1. Divergence and curl of vectors. We list below the divergence and curl of vectors in the three coordinate systems.

  Divergence: div $\mathbf{v} = \nabla \cdot \mathbf{v}.$

  Cartesian: Physical components: $u, v, w.$

  $$\begin{aligned} {\frac{\partial u}{\partial x}} + {\frac{\partial v}{\partial y}} + {\frac{\partial w}{\partial z}}. \end{aligned}$$

  (A2.1)

  Cylindrical: Physical components: $u, v, w.$

  $$\begin{aligned} {\frac{\partial u}{\partial r}} + {\frac{u}{r}} + {\frac{1}{r}}{\frac{\partial v}{\partial \theta }} + {\frac{\partial w}{\partial z}}. \end{aligned}$$

  (A2.2)

  Spherical: Physical components: $u, v, w.$

  $$\begin{aligned} {\frac{\partial u}{\partial r}} + {\frac{2u}{r}} + {\frac{1}{r}}{\frac{\partial v}{\partial \theta }} + {\frac{v}{r}}\cot \theta + {\frac{1}{r\sin \theta }}{\frac{\partial w}{\partial \phi }}. \end{aligned}$$

  (A2.3)

  curl: $\pmb {\omega }$ = curl $\mathbf{v} = \nabla \times \mathbf{v}.$

  Cartesian:

  $$\begin{aligned} \pmb {\omega } =\ \begin{bmatrix} w_{,y} - v_{,z}\\ u_{,z} - w_{,x}\\ v_{,x} - u_{,y}\end{bmatrix}, \end{aligned}$$

  (A2.4)

  where $w_{,y} = \partial w/\partial y,$ etc.

  Cylindrical:

  $$\begin{aligned} \pmb {\omega }\ =\ \begin{bmatrix} {\frac{1}{r}}w_{,\theta } - v_{,z}\\ u_{,z} - w_{,r}\\ (v/r) + v_{,r} - (1/r)u_{,\theta } \end{bmatrix}. \end{aligned}$$

  (A2.5)

  Spherical:

  $$\begin{aligned} \pmb {\omega }\ =\ \begin{bmatrix}(1/r)w_{,\theta } + (w/r)\cot \theta - (1/r \sin \theta )v_{,\phi }\\ (1/r \sin \theta )u_{,\phi } - (w/r) - w_{,r}\\ (v/r) + v_{,r} - (1/r)u_{,\theta } \end{bmatrix}. \end{aligned}$$

  (A2.6)

• 2. Components of the Rivlin-Ericksen Tensor. The physical components of the Rivlin-Ericksen tensor $\mathbf{A}$ are listed in Cartesian, cylindrical and spherical coordinates below, in terms of the physical components of the velocity field $\mathbf{v}$.

  Cartesian:

  $$\begin{aligned} \mathbf{A}\ =\ \begin{bmatrix} 2 u_{,x}&(u_{,y} + v_{,x})&(u_{,z} + w_{,x}) \\ \cdot&2 v_{,y}&(v_{,z} + w_{,y})\\ \cdot&\cdot&2 w_{,z}\end{bmatrix}. \end{aligned}$$

  (A2.7)

  Cylindrical:

  $$\begin{aligned} \mathbf{A}\ =\ \begin{bmatrix} 2 u_{,r}&[(1/r)u_{,\theta } + v_{,r} - (v/r)]&(u_{,z} + w_{,r})\\ \cdot&(2/r)(u + v_{,\theta })&(v_{,z} + (1/r) w_{,\theta })\\ \cdot&\cdot&2 w_{,z}\end{bmatrix}. \end{aligned}$$

  (A2.8)

  Spherical:

  $$\begin{aligned} \mathbf{A} = \begin{bmatrix} 2 u_{,r}&[(1/r)u_{,\theta } + v_{,r} - (v/r)]&[(1/r \sin \theta ) u_{,\phi } + w_{,r} - (w/r)]\\ \cdot&[(2/r)(u + v_{,\theta })]&[(1/\sin \theta )v_{,\phi } + w_{,\theta } - w \cot \theta ]/r \\ \cdot&\cdot&2/r \sin \theta \left( w_{,\phi } + u\sin \theta + v\cos \theta \right) \end{bmatrix}. \end{aligned}$$

  (A2.9)

  In (A2.7)–(A2.9), the dots denote the symmetry of the tensor.

• 3. Components of the Spin Tensor. Finally, we note that the velocity field $\mathbf{v}$ gives rise to the vorticity $\pmb {\omega }$ through curl $\mathbf{v} = \pmb {\omega }.$ Since we know the physical components of $\pmb {\omega }$ in various coordinates, the physical components of the spin tensor $\mathbf{W} = \left( \mathbf{L} - \mathbf{L}^T\right) \!{/}2$, where $\mathbf{L}$ is the velocity gradient, can be found from

  $$\begin{aligned} \mathbf{W} = {\frac{1}{2}} \begin{bmatrix} 0&-\omega _3&\omega _2 \\\omega _3&0&-\omega _1 \\-\omega _2&\omega _1&0 \end{bmatrix} . \end{aligned}$$

  (A2.10)