appendix.tex (4233B)
1 \appendix 2 \section{Appendix} 3 \subsection{Leibniz Rule of Integration} 4 \label{appendix:leibniz} 5 The Leibniz integral rule for differentiation under the integral sign 6 initiates with an integral 7 \begin{align} 8 \mathcal{I}(t, x) = \int_{a(t)}^{b(t)} f(t, x) dx = \mathcal{I}(t, a(t, 9 a(t), b(t))). 10 \end{align} 11 And upon differentiation w.r.t. $t$, utilizes the chain rule on $a(t)$ and 12 $b(t)$ respectively, by 13 \begin{align} 14 \frac{d\mathcal{I}}{dt} = 15 \frac{\partial \mathcal{I}}{\partial t}+ 16 \frac{\partial \mathcal{I}}{\partial b}\frac{\partial b}{\partial t}+ 17 \frac{\partial \mathcal{I}}{\partial a}\frac{\partial a}{\partial t}. 18 \end{align} 19 Which in integral representation reads 20 \begin{align} 21 \frac{d\mathcal{I}}{dt} = \int_{a(t)}^{b(t)}\frac{\partial f(t, 22 x)}{\partial t} dx + f(t, b(t)) \frac{\partial b(t)}{\partial t} 23 - f(t, a(t)) \frac{\partial a(t)}{\partial t} 24 \end{align} 25 26 \subsection{Identity for Vorticity} 27 \label{appendix:diff identity} 28 We start off with the standard material derivative 29 \begin{align} 30 \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t} 31 +(\mathbf{u}\nabla)\mathbf{u}. 32 \end{align} 33 We will use Einstein's Summation Convention, where we sum over indices 34 appearing at the bottom and the top. To rewrite the second part of the 35 material derivative $(\mathbf{u}\nabla)\mathbf{u}$ into 36 \begin{align} 37 (\mathbf{u}\times (\nabla \times \mathbf{u}))^i 38 &= \varepsilon^{ijk}u_j(\nabla \times \mathbf{u})_k \\ 39 &= \varepsilon^{ijk}u_j\varepsilon_{klm}\partial^l u^m\\ 40 &=(\delta^i_l\delta^j_m-\delta^i_m\delta^j_l)u_j\partial^l u^m\\ 41 &=u_m\partial^i u^m - u_l \partial^l u^i.\label{eq:identity split} 42 \end{align} 43 Now the first part in equation \ref{eq:identity split} can be rewritten into 44 \begin{align} 45 u_m\partial^i u^m =\partial^i (\frac{1}{2}u_mu^m) . 46 \end{align} 47 Thus we get 48 \begin{align} 49 (\mathbf{u}\times (\nabla \times \mathbf{u}))^i 50 = \frac{1}{2}\partial^i(u_m u^m) + u_l \partial^l u^i, 51 \end{align} 52 which is 53 \begin{align} 54 (\mathbf{u}\nabla)\mathbf{u} = \nabla(\frac{1}{2}\mathbf{u}\mathbf{u}) - 55 \left(\mathbf{u}\times (\nabla \times \mathbf{u})\right) 56 \end{align} 57 \subsection{Middle Curvature of an Implicit Function} 58 \label{appendix:curvature} 59 In our case the implicit function for fixed time reads 60 \begin{align} 61 z-h\left(x_1,x_2\right) = 0. 62 \end{align} 63 The parametric representation is 64 \begin{align} 65 \vec{\sigma} = \begin{pmatrix} x_1 \\ x_2 \\ h \end{pmatrix} . 66 \end{align} 67 The middle curvature of the surface parametrized by $\vec{\sigma}$ is 68 \begin{align} 69 \frac{1}{R} = \text{Tr}(G^{-1}B), 70 \end{align} 71 where $G$ and $B$ are given by 72 \begin{align} 73 G_{ij} = \frac{\partial \vec{\sigma}}{\partial x_i} \frac{\partial 74 \vec{\sigma}}{\partial x_j}, \\ 75 B_{ij} = -\mathbf{N} \frac{\partial^2 \vec{\sigma}}{\partial 76 x_i\partial x_j}, 77 \end{align} 78 where $i, j = 1, 2$ and $\mathbf{N}$ is the normal, normalized surface vector given by 79 \begin{align} 80 \mathbf{N} &= \frac{\frac{\partial \vec{\sigma}}{\partial x_1}\times 81 \frac{\partial \vec{\sigma}}{\partial x_2}}{\|\frac{\partial \vec{\sigma}}{\partial x_1}\times 82 \frac{\partial \vec{\sigma}}{\partial x_2}\|} \\ 83 &= \frac{1}{\sqrt{h_x^2 + h_y^2 +1}} \begin{pmatrix} 84 -h_x\\-h_y\\1 \end{pmatrix}. 85 \end{align} 86 Thereby the matrices $B$ and $G$ are calculated to be 87 \begin{align} 88 G = \begin{pmatrix} 1+h_x^2 & h_xh_y\\h_xh_y & 1+h_y^2 \end{pmatrix} 89 \qquad 90 B =\frac{1}{\sqrt{h_x^2 +h_y^2 +1} } \begin{pmatrix}h_{x x} & 91 h_{yx}\\h_{x y} & h_{yy} \end{pmatrix}. 92 \end{align} 93 The inverse of $G$ is 94 \begin{align} 95 G^{-1} 96 &= \frac{1}{\det(G)} \text{adj}(G)\\ 97 &= \frac{1}{h_x^2+h_y^2 +1} \begin{pmatrix}1+h_y^2 & -h_xh_y \\-h_xh_y & 98 1+h_x^2\end{pmatrix} . 99 \end{align} 100 Hence the middle curvature is given by the follwing 101 \begin{align} 102 \frac{1}{R} & 103 = \text{Tr}(G^{-1}B)\\ 104 &= \frac{1}{(h_x^2 + h_y^2+1)^{\frac{3}{2}}} 105 \text{Tr}\begin{pmatrix} (1+h_y)^2 h_{x x} - h_x h_y h_{xy} & *\\ 106 * & (1+h_x^2)h_{yy}-h_xh_yh_{xy}\end{pmatrix}\\ 107 &=\frac{(1+h_y^2)h_{x x}+(1+h_y^2)h_{yy} - 108 2h_xh_yh_{xy}}{\left( h_x^2+h_y^2+1 \right)^{\frac{3}{2}} }. 109 \end{align}