main.tex (15796B)
1 \documentclass[fleqn]{beamer} 2 \beamertemplatenavigationsymbolsempty 3 4 \usepackage[T1]{fontenc} 5 \usepackage[utf8]{inputenc} 6 7 \usepackage{amsmath,amssymb} 8 \usepackage{nccmath} 9 \usepackage{graphicx} 10 \usepackage{mathptmx} 11 \usepackage{mathtools} 12 \usepackage{subcaption} 13 \usepackage{amsthm} 14 \usepackage{tikz} 15 \usepackage{enumitem} 16 %\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref} 17 \usetikzlibrary{patterns,decorations.pathmorphing,positioning, arrows, chains} 18 19 \newcommand{\tabitem}{% 20 \usebeamertemplate{itemize item}\hspace*{\labelsep}} 21 22 \usepackage[backend=biber, sorting=none]{biblatex} 23 \addbibresource{uni.bib} 24 25 \setbeamertemplate{endpage}{% 26 \begin{frame} 27 \centering 28 \Large \emph{Thank you for listening!} 29 \end{frame} 30 } 31 32 \AtEndDocument{\usebeamertemplate{endpage}} 33 34 % vertical separator macro 35 \newcommand{\vsep}{ 36 \column{0.0\textwidth} 37 \begin{tikzpicture} 38 \draw[very thick,black!10] (0,0) -- (0,7.3); 39 \end{tikzpicture} 40 } 41 \setlength{\mathindent}{0pt} 42 43 % Beamer theme 44 \usetheme{UniVienna} 45 \usefonttheme[onlysmall]{structurebold} 46 \mode<presentation> 47 \setbeamercovered{transparent=10} 48 49 \title 50 {Mathematical Modeling of Water-Wave Problems} 51 \subtitle{Applied PDE Seminar} 52 %\author[Popović Milutin] 53 %{Popović Milutin \inst{1}\\[1ex] {\small supervisor\inst{1,2}} 54 55 \author[Popović Milutin]{Popović Milutin\\[10mm]{\small Supervisor: Sabine Hittmeir}} 56 \date{29. June 2022} 57 58 \begin{document} 59 \begin{frame} 60 \titlepage 61 \nocite{johnson_1997} 62 \nocite{vallis_2017} 63 \nocite{constantin_tsunami} 64 \nocite{rupert_2009} 65 \nocite{mathe-physik} 66 \end{frame} 67 68 69 \begin{frame} 70 \frametitle{Fluid Description} 71 \begin{columns}[T] 72 \column{0.34\textwidth} 73 \hspace{50cm} 74 75 The fluid is described by 76 \begin{itemize} 77 \item[$\circ$] Fluid density\\ $\rho(\vec{x}, t)$ 78 \item[$\circ$] Velocity Field\\ $\vec{u}(\vec{x}, t) = (u, v, 79 w)$ 80 \end{itemize} 81 \column{0.57\textwidth} 82 83 \begin{figure}[H] 84 \centering 85 \caption{Control volume of the fluid} 86 \begin{tikzpicture}[>=latex,scale=1, xscale=1, opacity=.8] 87 % second sphere 88 \begin{scope}[rotate=10, xscale=3, yscale=2, shift={(2.3,-0.2)}] 89 \coordinate (O) at (0,0); 90 \shade[ball color=gray!10!] (0,0) coordinate(Hp) circle (1) ; 91 92 \draw[thick] (O) circle (1); 93 \draw[rotate=5] (O) ellipse (1cm and 0.66cm); 94 \draw[rotate=90] (O) ellipse (1cm and 0.33cm); 95 \node[circle, fill=black, inner sep=1pt] at (0.15, 0.25) {} ; 96 \draw[-latex, thick] (0.15, 0.25) -- (1, 1) ; 97 \node[right] at (1, 1) {$\vec{u}(\vec{x}, t)$}; 98 99 \node[] at (O) {$V$}; 100 \node[] at (0.55, -0.25) {$\rho(\vec{x}, t)$}; 101 102 \draw[-] (0.76, -0.66) -- (1.2, -0.7); 103 \node[right] at (1.2, -0.7) {$S$}; 104 105 \draw[-latex, thick] (-0.25, -0.65) -- (-1, -1); 106 \node[left] at (-1, -1) {$\vec{n}$}; 107 \end{scope} 108 \end{tikzpicture} 109 \end{figure} 110 \end{columns} 111 \end{frame} 112 113 \begin{frame} 114 \frametitle{Mass Conservation} 115 \begin{itemize} 116 \item[$\circ$] Mass: 117 \begin{align} 118 \hspace{0.3\linewidth} 119 m(t) = \int_V \rho(\vec{x}, t)\ dV \nonumber 120 \end{align} 121 \item[$\circ$] Rate of change: 122 \begin{align} 123 \hspace{0.15\linewidth} 124 \int_V \frac{\partial \rho(\vec{x}, t)}{\partial t}\ dV = \frac{dm}{dt} = 125 -\int_{S} \rho(\vec{x}, t) \vec{u}\cdot\vec{n}\ dS\nonumber 126 \end{align} 127 \item[$\circ$] Use Gauss's law to get the \textbf{Equation of Mass 128 conservation} 129 \begin{align} 130 \hspace{0.3\linewidth} \frac{\partial \rho}{\partial t} 131 +\nabla \cdot (\rho \vec{u}) = 132 0 \nonumber 133 \end{align} 134 \end{itemize} 135 \end{frame} 136 137 \begin{frame} 138 \frametitle{Euler's Equation of Motion} 139 \centering 140 $\rightarrow$ Apply Newton's second law to the Fluid 141 \begin{columns} 142 \column{0.4\textwidth} 143 \begin{block}{\centering\textbf{Body Force}} 144 \centering 145 $\vec{F} = (0, 0, -g)$ 146 \end{block} 147 \column{0.4\textwidth} 148 \begin{block}{\centering\textbf{Local/Short-range Force}} 149 \centering 150 Stress tensor\\ 151 For inviscid fluid: $P(\vec{x},t)$ 152 \end{block} 153 \end{columns} 154 \begin{align} 155 \hspace{0.3\linewidth}\Rightarrow\int_V \rho \frac{D\vec{u}}{Dt}\ dV = 156 \int_V \left( \rho \vec{F} - \nabla P \right)\ dV\nonumber 157 \end{align} 158 \centering 159 $\rightarrow$ Leads us to \textbf{Euler's Equation of Motion} 160 \begin{align} 161 \hspace{0.4\linewidth} \frac{D\vec{u}}{Dt} =- 162 \frac{1}{\rho}\nabla P + \vec{F} \nonumber 163 \end{align} 164 \end{frame} 165 166 \begin{frame} 167 \frametitle{Vorticity} 168 \begin{columns} 169 \column{0.4\textwidth} 170 \begin{block}{\centering\textbf{Vorticity}} 171 \centering 172 $\vec{\omega} = \nabla \times \vec{u}$ 173 \end{block} 174 \column{0.4\textwidth} 175 \begin{block}{\centering\textbf{Irrotational Flow}} 176 \centering 177 $\vec{\omega} = 0$ 178 \end{block} 179 \end{columns} 180 \vspace{0.5cm} 181 182 \centering $\rightarrow$ Vorticity pops up in the acceleration of 183 the fluid particles 184 \begin{align} 185 \hspace{0.25\linewidth} 186 \frac{D\vec{u}}{Dt} = \frac{\partial \vec{u}}{\partial t}+ 187 \nabla \left( \frac{1}{2} \vec{u} \cdot \vec{u}\right) - \left( 188 \vec{u}\times \vec{\omega} \right)\nonumber 189 \end{align} 190 191 \centering $\rightarrow$ We can incorporate vorticity into Euler's 192 Equation of Motion 193 \begin{align} 194 \hspace{0.25\linewidth}\frac{\partial \vec{u}}{\partial t} + \nabla\left( 195 \frac{1}{2}\vec{u}\cdot\vec{u} + \frac{P}{\rho} + \Omega \right) 196 = \vec{u} \times \vec{\omega} \nonumber 197 \end{align} 198 \end{frame} 199 200 \begin{frame} 201 \frametitle{Perfect Fluid} 202 \begin{itemize} 203 \item \textbf{inviscid} $\mu = 0$ 204 \item \textbf{incompressible} $\rho = \text{const}.$, then 205 $\nabla \vec{u} = 0$ 206 \end{itemize} 207 208 \end{frame} 209 210 \begin{frame} 211 \frametitle{Boundary Conditions for Water Waves} 212 \begin{center} 213 \begin{tabular}{@{}l@{}} 214 \tabitem \textbf{Kinematic Condition}: Fluid particles at the 215 surface\\ 216 \tabitem \textbf{Dynamic Condition}: Atmospheric Pressure on 217 the surface\\ 218 \tabitem \textbf{Bottom Condition}: Rigid and fixed bottom\\ 219 \tabitem (\textbf{Integrated Mass Condition}): Combination 220 \end{tabular} 221 \end{center} 222 \end{frame} 223 224 \begin{frame} 225 \frametitle{Nondimensionalisation} 226 \begin{center} 227 \begin{tabular}{@{}l@{}} 228 \tabitem $h_0$ for the typical water depth\\ 229 \tabitem $\lambda$ for the typical wavelength\\ 230 \tabitem $\sqrt{g h_0}$ velocity scale of waves in 231 $(x, y)$\\ 232 \tabitem $\frac{\lambda}{\sqrt{g h_0}}$ time scale 233 of wave propagation\\ 234 \tabitem $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in $z$ 235 \end{tabular} 236 \end{center} 237 \centering 238 $\rightarrow$ \textbf{Shallowness parameter} $\delta = 239 \frac{h_0}{\lambda}$ 240 \\ 241 \centering 242 $\rightarrow$ \textbf{Amplitude parameter} 243 $\varepsilon=\frac{a}{h_0}$ 244 \end{frame} 245 246 \begin{frame} 247 \frametitle{Nondimensionalisation} 248 $\rightarrow$ Nondimensionalisation 249 \begin{ceqn} 250 \begin{align} 251 &x \rightarrow\ \lambda x, \quad u \rightarrow \sqrt{gh_0} u, 252 \nonumber \\ 253 &y \rightarrow\ \lambda y, \quad v \rightarrow \sqrt{gh_0} v, \qquad 254 t\rightarrow \frac{\lambda}{\sqrt{gh_0}}t,\nonumber\\ 255 &z \rightarrow\ h_0 z, \quad w \rightarrow 256 \frac{h_0\sqrt{gh_0}}{\lambda} w.\nonumber 257 \end{align} 258 \end{ceqn} 259 \centering 260 $\rightarrow$ Top and Bottom conditions 261 \begin{ceqn} 262 \begin{align} 263 h = h_0 + a \eta(\vec{x}_\perp,t), \qquad b \rightarrow h_0 264 b(\vec{x}_\perp, t)\nonumber 265 \end{align} 266 \end{ceqn} 267 \centering 268 $\rightarrow$ Rewrite Pressure 269 \begin{ceqn} 270 \begin{align} 271 P = P_a + \rho g(h_0 -z) + \rho g h_0 p(\vec{x}) \nonumber 272 \end{align} 273 \end{ceqn} 274 \end{frame} 275 276 \begin{frame} 277 \frametitle{Scaling} 278 \centering 279 $\rightarrow$ $w$, $p$ and the free surface $z$ are $\propto$ 280 $\varepsilon$, leading to the scaling 281 \begin{ceqn} 282 \begin{align} 283 p \rightarrow \varepsilon p, \quad w \rightarrow \varepsilon w, \quad 284 \vec{u}_\perp \rightarrow \varepsilon 285 \vec{u}_\perp\nonumber 286 \end{align} 287 \end{ceqn} 288 \end{frame} 289 290 \begin{frame} 291 \frametitle{Results} 292 \centering 293 $\rightarrow$ Nondimensionalized Euler's Equation of motion 294 \begin{ceqn} 295 \begin{align} 296 \frac{Du}{Dt} = - p_x \quad 297 &\frac{Dv}{Dt} = - p_y \quad 298 \delta^2\frac{Dw}{Dt} = - p_z \nonumber\\ 299 \nonumber\\ 300 &\nabla \cdot \vec{u} = 0\nonumber 301 \end{align} 302 \end{ceqn} 303 \centering 304 $\rightarrow$ With boundary conditions 305 \begin{ceqn} 306 \begin{align} 307 \begin{drcases} 308 p = \eta - \frac{\delta^2\varepsilon h_0}{\lambda^2} \frac{W_e}{R}\\ 309 w = \frac{1}{\varepsilon}\eta_t + (\mathbf{u}_\perp \nabla_\perp)\eta 310 \end{drcases} \quad 311 \text{on}\;\; z = 1+\varepsilon\eta\\ 312 w =\frac{1}{\varepsilon}b_t + (\mathbf{u}_\perp \nabla_\perp)b \quad 313 \text{on}\;\; z=b 314 \end{align} 315 \end{ceqn} 316 \end{frame} 317 318 \begin{frame} 319 \frametitle{History of the Soliton} 320 \begin{itemize} 321 \item[$\circ$] John Scott Russell discovered the solitary wave in 1834, 322 firstly calling it the \textbf{wave of translation} 323 \item[$\circ$] a \textbf{soliton} is a solitary wave that resists 324 dispersion, maintaining its shape while it propagates at constant 325 velocity\\ 326 \end{itemize} 327 \end{frame} 328 329 \begin{frame} 330 \frametitle{Korteweg-de Vries equation (KdV)} 331 Korteweg-de Vries equation: nonlinear PDE 332 \begin{ceqn} 333 \begin{align} 334 \eta_t + 6K \eta \eta_{x} + \eta_{x x x} = 0\nonumber 335 \end{align} 336 \end{ceqn} 337 With Solution 338 \begin{ceqn} 339 \begin{align} 340 \eta(x, t) = 2c^2 \text{sech}^2\Big( c 341 \left(x - 4c^2t\right) \Big)\nonumber 342 \end{align} 343 \end{ceqn} 344 \end{frame} 345 346 \begin{frame} 347 \frametitle{KdV Regime} 348 \begin{itemize} 349 \item[$1$)] The KdV equation arises in the $\varepsilon = O(\delta^2)$ 350 \item[$2$)] by rescaling $\delta$ in favor of $\varepsilon$ in 351 Euler's Equations of motion 352 \item[$3$)] going into the frame of the moving wave $(\xi = x- t, \tau = \varepsilon t)$ 353 \item[$4$)] conducting an Asymptotic expansion of $u , w, p$ and $\eta$. 354 \item[$5$)] KdV equation is present in the $\varepsilon^1$ term 355 \end{itemize} 356 \end{frame} 357 358 \begin{frame} 359 \frametitle{2004 Tsunami: Description} 360 361 \begin{figure}[htpb] 362 \centering 363 \includegraphics[width=0.35\textwidth]{./pics/water-surface.png} 364 \caption{Earthquake generating a tsunami with $\lambda = 100\ 365 \text{km},\;\; a = 1\ \text{m}$ (found in \cite{graph_meter})} 366 \end{figure} 367 \end{frame} 368 369 \begin{frame} 370 \frametitle{2004 Tsunami: $\varepsilon=O\left(\delta^2\right)$} 371 372 \begin{itemize} 373 \item[$\circ$] $\varepsilon = \frac{a}{h_0}$ and $\delta = 374 \frac{h_0}{\lambda}$ need to enter the regime 375 $\varepsilon=O(\delta^2)$ for the KdV equation to become 376 relevant 377 \item[$\circ$] But also the geophysical scales need to be $\xi = O(1)$ and $\tau = 378 O(1)$ for the KdV dynamics to become relevant, that is 379 \begin{ceqn} 380 \begin{align} 381 x = O\left(\varepsilon^{-1} \lambda \right) 382 \end{align} 383 \end{ceqn} 384 \item[$\circ$] KdV dynamics is when the waves are ordered with the 385 highest in front following an oscillatory tale 386 \item[$\circ$] This happens because wave amplitude is proportional to 387 wave speed 388 \end{itemize} 389 390 \end{frame} 391 392 \begin{frame} 393 \frametitle{2004 Tsunami: Regime of Validity} 394 \begin{ceqn} 395 \begin{align} 396 \lambda = 100\ \text{km}\qquad a = 1\ \text{m}\nonumber 397 \end{align} 398 \end{ceqn} 399 \begin{columns} 400 \column{0.45\textwidth} 401 Waves propagating westwards to India/Sri Lanka 402 \begin{align} 403 h_0 = 4\ \text{km} \Rightarrow 404 \begin{cases} 405 \varepsilon \simeq 25 \cdot 10^{-5}\\ 406 \delta \simeq 4\cdot 10^{-2} 407 \end{cases}\nonumber 408 \end{align} 409 410 \column{0.45\textwidth} 411 Waves propagating eastwards to Thailand 412 \begin{align} 413 h_0 = 1\ \text{km} \Rightarrow 414 \begin{cases} 415 \varepsilon \simeq 10^{-3}\\ 416 \delta \simeq 10^{-2} 417 \end{cases}\nonumber 418 \end{align} 419 \end{columns} 420 \vspace{0.5cm} 421 \centering 422 $\Rightarrow$ Both enter the regime $\varepsilon = O(\delta^2)$ 423 \end{frame} 424 425 \begin{frame} 426 \frametitle{2004 Tsunami: Regime of Validity} 427 \begin{columns} 428 \column{0.45\textwidth} 429 Waves propagating westwards to India/Sri Lanka ($\simeq 1600\ \text{km}$) 430 \begin{align} 431 \begin{drcases} 432 \varepsilon \simeq 25\cdot 10^{-5}\\ 433 \lambda = 100\text{km} 434 \end{drcases}\Rightarrow\; x \simeq 4\cdot 10^{5}\ \text{km}\nonumber 435 \end{align} 436 437 \column{0.45\textwidth} 438 Waves propagating eastwards to Thailand ($\simeq 700\ \text{km}$) 439 \begin{align} 440 \begin{drcases} 441 \varepsilon \simeq 10^{-3}\\ 442 \lambda = 100\text{km} 443 \end{drcases}\Rightarrow\; x \simeq 10^{5}\ \text{km}\nonumber 444 \end{align} 445 \end{columns} 446 \vspace{0.5cm} 447 \centering 448 $\Rightarrow$ The propagation distance of the tsunami waves in both 449 directions is not enough for KdV dynamics to take place. 450 \end{frame} 451 452 \begin{frame}{Bibliography} 453 \printbibliography 454 \end{frame} 455 \end{document} 456