Updates from Overleaf

chapters/Introduction.tex

@@ -437,7 +437,7 @@ \section{Outline}
 
 Use cases and a derivation of the model can be found in Chap.~\ref{chapter:first_paper}.
 In this chapter the mandatory modelling assumptions are introduced that allow to match the shallow water system with the thin film equation.
-It further highlights which modifications are made to the shallow water lattice Boltzmann algortihm. 
+It further highlights which modifications are made to the shallow water lattice Boltzmann algorithm. 
 With some emphasis on the numerical implementation of e.g., the computation of gradients and the Laplacian in agreement with Ref~\cite{junkDiscretizationsIncompressibleNavier2000, thampiIsotropicDiscreteLaplacian2013}. 
 After the derivation, numerical experiments are displayed to validate the model. 
 Among those are the Rayleigh-Taylor instability, an instability that occurs when a heavy fluid is on top of a light fluid. 

chapters/Paper_one.tex

@@ -234,7 +234,7 @@ \subsection{The Rayleigh-Taylor instability}
 
 \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/Fig_3_new_RTI_spectra_single_color_same_tau_capkc_correct_rescaled_x_axis_inset.png}
+    \includegraphics[width=0.65\textwidth]{graphics/Fig_3_new_RTI_spectra_single_color_same_tau_capkc_correct_rescaled_x_axis_inset.png}
     \caption{Power spectrum of the height fluctuations versus wave number. The different colors and symbols belong to different values of graviational acceleration; $g=4\cdot 10^{-5}$ is given by blue circles (\textcolor{pyblue}{$\bullet$}), $g=6\cdot 10^{-5}$ by orange triangles (\textcolor{pyorange}{$\blacktriangle$}) and $g=8\cdot 10^{-5}$ is given by green squares (\textcolor{pygreen}{$\blacksquare$}). Same colored lines are taken at different time steps. In the inset we show the growth rate $\sigma(k)$ for the largest value of $g$ (symbols) and the theoretical growth rate according to Eq.~(\ref{eq:RTgrowth}) (solid line).}
     \label{fig:RTI}
 \end{figure}
@@ -339,7 +339,7 @@ \subsection{A spreading droplet}
 
 \begin{figure}
     \centering 
-    \includegraphics[width=0.75\textwidth]{graphics/Fig_5_Indirect_Cox_Voinov_all_data_visually_appealing_slip_2_m_nosci.png}
+    \includegraphics[width=0.65\textwidth]{graphics/Fig_5_Indirect_Cox_Voinov_all_data_visually_appealing_slip_2_m_nosci.png}
     \caption{Difference of cubed instantaneous and equilibrium contact angles, $\theta_{num}^3-\theta_{eq}^3$, vs. capillary number $Ca$ for a spreading droplet; the dashed line shows a linear dependence (consistent with the Cox-Voinov law). The different symbols represent different viscosities, while the dashed line is a linear function of the capillary number.}
     \label{fig:Cox-Voinov}
 \end{figure}
@@ -349,7 +349,7 @@ \subsection{A spreading droplet}
 In Fig.~\ref{fig:Cox-Voinov} we plot $\theta^3(t) - \theta_{eq}^3$ vs $Ca(t)$ from a numerical simulation of a spreading drop: A good linear scaling, in agreement with the Cox-Voinov law, is observed, as highlighted by the dashed line. 
 \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/Fig_6_Tanners_law_slip_2_paper_rescaled_t.png}
+    \includegraphics[width=0.65\textwidth]{graphics/Fig_6_Tanners_law_slip_2_paper_rescaled_t.png}
     \caption{Time evolution of the droplet base radius of a spreading droplet; the dashed red line shows a $\tilde{t}^{1/10}$ power law (consistent with Tanner's law). 
     As in Fig.~\ref{fig:Cox-Voinov} different symbols refer to different viscosities. The radius clearly grows with the predicted power law until it saturates. 
     On rescaling the time with $\tau_{\mbox{\tiny{cap}}}$ the curves of all three viscosities collapse into a single one.}
@@ -382,7 +382,7 @@ \subsection{A sliding droplet}
 $Bo_c$ is the critical Bond number, defined in terms of the critical tilting angle $\alpha_c$.
   \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/Fig_7_Ca_Bo_true_with_pic.png}
+    \includegraphics[width=0.65\textwidth]{graphics/Fig_7_Ca_Bo_true_with_pic.png}
     \caption{$Ca$ vs $Bo$ for a sliding droplet: Notice that a finite minimum forcing (corresponding to $Bo_c$) is needed to actuate the droplet. 
     For $Bo > Bo_c$ a linear relation, $Ca \sim Bo$, is observed. 
     In the insets we show the shape of the droplet for both, the pinned (upper left) as well as the sliding (lower right) case.

chapters/Paper_three.tex

@@ -88,7 +88,7 @@ \section{Results} We first investigate how the rupture times depend on the param
 where $U_{\Theta}$ is the retraction speed $U_{\Theta} = \frac{\gamma \Theta^3}{9\mu}$~\cite{edwardsNotSpreadingReverse2016}, with $\Theta = \max_{\mathbf{x}}\{\theta(\mathbf{x})\}$.
 \begin{figure}
     \centering
-    \includegraphics[width=0.4\textwidth]{graphics/Figure_2.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/Figure_2.pdf}
     \caption{Rupture times $\tau_r$ as a function of the pattern wavelength $\lambda$, for $v_{\theta}=0$ (\textcolor{jlblue}{$\bullet$}) and $v_{\theta}=20 v_0$ (\textcolor{jlorange}{$\star$}).
     The continuous and dashed lines indicate the linear, $\sim \lambda$, and quadratic, $\sim \lambda^2$, scaling laws, respectively.
         }
@@ -97,7 +97,7 @@ \section{Results} We first investigate how the rupture times depend on the param
 We now focus on the long-time dynamics, the characterization of the dewetting morphologies, and how they are affected by the speed of the wettability wave.
 \begin{figure}
     \centering
-    \includegraphics[width=0.4\textwidth]{graphics/Figure_3.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/Figure_3.pdf}
     \caption{Main panel: Time evolution of the height fluctuations $\Delta h(t)$ during the dewetting process on the patterned substrate given by Eq.~(\ref{eq:sinetheta}) with $v_{\theta}= 0$ and $\lambda= 512 h_0$ (\textcolor{jlblue}{$\bullet$}), $\lambda=256 h_0$ (\textcolor{jlorange}{$\blacksquare$}) and $\lambda=170 h_0$ (\textcolor{jlgreen}{$\star$}). 
     Inset: Number of droplets $N(t)$ as a function of time. 
     The three horizontal dashed lines indicate the number of minima of Eq.~(\ref{eq:sinetheta}), which is $2\left(\frac{L}{\lambda}\right)^2$. 
@@ -113,7 +113,7 @@ \section{Results} We first investigate how the rupture times depend on the param
 This represents a measure of the mean droplet height $h_d$ (since droplets are essentially monodisperse), decreasing with the pattern wavelength (as expected, due to a decreasing droplet volume, $V_d = \frac{h_0 \lambda^2}{2}$).
 \begin{figure}
     \centering
-    \includegraphics[width=0.4\textwidth]{graphics/Figure_4.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/Figure_4.pdf}
     \caption{Time evolution of the second-order Minkowski structure metric $q_2(t)$ for different $\Gamma$ values, on a substrate with pattern wavelength $\lambda=256 h_0$.
     The grey-scale insets supply snapshots of the corresponding film thickness fields.}
     \label{fig:msm_q2}
@@ -134,7 +134,7 @@ \section{Results} We first investigate how the rupture times depend on the param
 Notice, though, that the $q_2$ signal for any $v_{\theta} >0$ always stays above the one for the static case, suggesting that even the smallest pattern velocity introduces a sizeable deformation of the spherical cap shape.
 \begin{figure}
     \centering
-    \includegraphics[width=0.4\textwidth]{graphics/Figure_5.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/Figure_5.pdf}
     \caption{Main panel: Rivulet lifetimes $\tau_{\text{riv}}$ for various $\Gamma$.
     The dashed line is a guide to the eye to highlight the logarithmic dependence, in agreement with the theoretical prediction, Eq.~(\ref{eq:rivlt}).
     Inset: Height fluctuations $\Delta h(t)$ vs time, along the rivulet axis, for three different $\Gamma$.
@@ -182,13 +182,12 @@ \section{Conclusions}
 \section{Acknowledgements} We acknowledge financial support from the German Research Foundation (DFG) (priority program SPP2171 / project HA-4382/11 and Project-ID 431791331—CRC1452), and from the Independent Research Fund Denmark (grant 9063-00018B).
 \newpage
 
-\section{Supplemental material}.
-\label{suppmat}
+\section{Supplemental material}\label{suppmat}
 \subsection{Minkowski's structure metric \texorpdfstring{$q_2$}{hmm} in an extended parameter space.}
 \begin{figure}
     \centering
-    \includegraphics[width=0.4\textwidth]{graphics/SupMatFig_1.pdf}
-    \includegraphics[width=0.4\textwidth]{graphics/SupMatFig_2.pdf}
+    \includegraphics[width=0.45\textwidth]{graphics/SupMatFig_1.pdf}
+    \includegraphics[width=0.45\textwidth]{graphics/SupMatFig_2.pdf}
     \caption{LEFT PANEL. Minkowski's structure metric $q_2$ for three different wavelengths $\lambda=64 h_0$, $\lambda=128 h_0$ and $\lambda=256 h_0$ (as in figure 4 of the main text) and for $\Gamma=1.5$ (all other parameters are as in the main text). 
     The time interval during which $q_2 \approx 1$ signals the emergence of rivulets, also for $\lambda = 64 h_0$ which is comparable with the spinodal wavelength $\lambda_s \approx 70 h_0$. 
     RIGHT PANEL. Comparison of the Minkowski's structure metric $q_2$ for $\delta\theta=5^{\circ}$ and $\delta\theta=10^{\circ}$, $\Gamma = 15$ and $\lambda = 256 h_0$.}

chapters/Paper_two.tex

@@ -241,7 +241,7 @@ \section{Results}\label{sec:results_two}
 \subsection{Testing the stochastic term}\label{subsec:validation}
 \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/spectrum_theta_20_nob_fill_pattern.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/spectrum_theta_20_nob_fill_pattern.pdf}
     \caption{Height fluctuations spectra from deterministic (circles) and stochastic (triangles) simulations at $t=0.2 t_0$ (filled symbols) and $t=0.7 t_0$ (empty symbols), on a substrate with $\theta =\pi/9$. 
     The theoretical predictions, Eq.~(\ref{eq:structure_factor_2}), are reported with solid ($\sigma=0$) and dashed ($\sigma = \sigma_0$) lines.}
     \label{fig:theory_simulation_structure_factor}
@@ -259,7 +259,7 @@ \subsection{Testing the stochastic term}\label{subsec:validation}
 \subsection{Droplet size distributions}\label{subsec:morphandrup}
 \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/Droplet_height_['00', '1e-7']_distri_new_35_nodist.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/Droplet_height_['00', '1e-7']_distri_new_35_nodist.pdf}
     \caption{Histograms of the droplet height distributions from the deterministic (blue, line-patterned) and stochastic (orange, dot-patterned) simulations with $\theta = \pi/9$, at $t \approx 20 t_0$; the corresponding Gaussian kernel density estimations are also plotted, as a guide to the eye, with solid blue (dashed orange) curve for the deterministic (stochastic) data.}
     \label{fig:droplet_distribution}
 \end{figure}
@@ -282,7 +282,7 @@ \subsection{Droplet size distributions}\label{subsec:morphandrup}
 These findings are in agreement with what was reported by Nesic \textit{et al.}~\cite{nesicFullyNonlinearDynamics2015}.
 \begin{figure}
     \centering
-    \includegraphics[width = 0.75\textwidth]{graphics/Correct_t0_normed_delta_h_evo_seminar.pdf}
+    \includegraphics[width = 0.65\textwidth]{graphics/Correct_t0_normed_delta_h_evo_seminar.pdf}
     \caption{Time evolution of $\Delta h(t) = \max_x\{h(x,t)\} - \min_x\{h(x,t)\}$ for the athermal (bullets) and fluctuating (triangles) systems with $\theta=\pi/9$.
     The blue dashed-dotted line depicts an exponential fit of the data for the athermal case ($\sigma=0$).
     The vertical lines mark the rupture times for the deteministic (solid) and stochastic (dotted) simulations.}
@@ -351,7 +351,7 @@ \subsection{Rupture times and role of contact angle}\label{subsec:results_contac
 The latter relation tells us that $\chi_{\sigma}(\theta)$, indeed, increases with the temperature (confirming that the rupture times are shorter for the fluctuating systems), but decreases with the contact angle.
 \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/Andrea_model_t0_normed_rupture_times.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/Andrea_model_t0_normed_rupture_times.pdf}
     \caption{Ratio of rupture times from athermal and fluctuating dewetting, $\chi_{\sigma_0}(\theta)$ [Eq.~(\ref{eq:defchi}], as a function of the contact angle $\theta$. 
     The dashed line is Eq.~(\ref{eq:chi}) for $\sigma=\sigma_0$.
     Inset: Rupture times $\tau_r$ normalized by $t_0$ (see Table~\ref{tab:t_0s}) vs $\theta$, for the deterministic (orange bullets \textcolor{pyorange}{$\bullet$}) and stochastic (green triangles \textcolor{pygreen}{$\blacktriangle$}) simulations.
@@ -361,7 +361,7 @@ \subsection{Rupture times and role of contact angle}\label{subsec:results_contac
 Equation~(\ref{eq:chi}) is plotted in Fig.~\ref{fig:rupture_times_semilogy_more_theta} (dashed line), showing a nice agreement with the numerical data.
 \begin{figure}
     \centering
-    \includegraphics[width=0.75\textwidth]{graphics/evolution_qm_with_inset_slip0.pdf}
+    \includegraphics[width=0.65\textwidth]{graphics/evolution_qm_with_inset_slip0.pdf}
     \caption{Time dependence of the maximum wavenumber, $q_m$, at which the spectrum has the global maximum for various contact angles $\theta$ and $\sigma = \sigma_0$, using $\delta=0$.
     The full blue, orange and green lines correspond to theoretical curve $q_m(t)$ derived from Eq.~(\ref{eq:structure_factor_2}) for $\theta = \pi/9, 5\pi/36, \pi/6$.
     The dashed lines show the corresponding $q_0(\theta)$. 
@@ -391,7 +391,7 @@ \subsubsection{Sine wave pattern}\label{subsubsec:sine}
 Indeed, the deterministic dewetting leads to the formation of precisely $L/\lambda_{\theta}$ droplets.
 \begin{figure}
     \centering
-    \includegraphics[width=0.95\textwidth]{graphics/spacedepCA_['sine', '1e7', '10', 25, '9_3', 10000000.0]_v2.png}
+    \includegraphics[width=0.75\textwidth]{graphics/spacedepCA_['sine', '1e7', '10', 25, '9_3', 10000000.0]_v2.png}
     \caption{Space-time plot of the height field $h(x,t)$ evolution over a sinusoidally patterned substrate undergoing athermal 
     (a) and fluctuating (b) dewetting, respectively. In panel (c) we report the contact angle profile $\theta(x)$ [Eq.~(\ref{eq:sine_angle})].} 
     \label{fig:patterned_sine8_difference_20-30}
@@ -407,15 +407,15 @@ \subsubsection{Sine wave pattern}\label{subsubsec:sine}
 Consequently, since $q_0$ decreases with $\theta$, we should expect the actual most unstable wavenumber to be slightly below $q_0$.
 \begin{figure}
     \centering
-    \includegraphics[width=0.95\textwidth]{graphics/psd_spacedepCA_sine_25_10_00_[0, 450000, 1150000]_9_3_rescaled.pdf}
+    \includegraphics[width=0.75\textwidth]{graphics/psd_spacedepCA_sine_25_10_00_[0, 450000, 1150000]_9_3_rescaled.pdf}
     \caption{Height profiles from deterministic dewetting ($\sigma=0$) over the sinusoidally patterned substrate, Eq.~(\ref{eq:sine_angle}) at $t=0.26 t_0$ (a) and $t=0.66 t_0$ (b) and corresponding spectra (c).  
     In panel (c) also the initial spectrum, $S_0(q)$, is reported; the vertical lines indicate the pattern wave mode $q_{\theta}$ (solid) and its multiples $2q_{\theta}$ (dashed) and $3q_{\theta}$ (dashed dotted).
     As explained in the text, the convention $t_0 = t_0(\pi/6)$ and $q_0 = q_0(\pi/6)$ applies.}   
     \label{fig:spectral_analysis_deter_sine8}
 \end{figure}
 \begin{figure}
     \centering
-    \includegraphics[width=0.95\textwidth]{graphics/psd_spacedepCA_sine_25_10_1e-7_[0, 450000, 1150000]_9_3_rescaled.pdf}
+    \includegraphics[width=0.75\textwidth]{graphics/psd_spacedepCA_sine_25_10_1e-7_[0, 450000, 1150000]_9_3_rescaled.pdf}
     \caption{Height profiles from stochastic dewetting ($\sigma=\sigma_0$) over the sinusoidally patterned substrate, Eq.(\ref{eq:sine_angle}) at $t=0.26 t_0$ (a) and $t=0.66 t_0$ (b) and corresponding spectra (c).  
     In panel (c) also the initial spectrum, $S_0(q)$, is reported; the vertical lines indicate the pattern wave mode $q_{\theta}$ (solid) and its multiples $2q_{\theta}$ (dashed) and $3q_{\theta}$ (dashed dotted).}
     \label{fig:spectral_analysis_stoch_sine8}
@@ -435,7 +435,7 @@ \subsubsection{Sine wave pattern}\label{subsubsec:sine}
 \subsubsection{Square wave pattern}\label{subsubsec:square_wave}
 \begin{figure}
     \centering
-    \includegraphics[width=0.95\textwidth]{graphics/spacedepCA_['delta', '1e7', '10', 25, '9_3', 2000000.0]_v2.png}
+    \includegraphics[width=0.75\textwidth]{graphics/spacedepCA_['delta', '1e7', '10', 25, '9_3', 2000000.0]_v2.png}
     \caption{Space-time plot of the height field $h(x,t)$ evolution over a square-wave patterned substrate undergoing athermal (a) and fluctuating (b) dewetting, respectively. 
     In panel (c) we report the contact angle profile $\theta(x)$ [Eq.~(\ref{eq:sharp_contact_angle_spatial})].} 
     \label{fig:patterned_step8_difference_20-30}
@@ -474,7 +474,7 @@ \subsubsection{Square wave pattern}\label{subsubsec:square_wave}
 in excellent agreement with the measured value.
 \begin{figure}
     \centering
-    \includegraphics[width=0.95\textwidth]{graphics/Rupture_events_with_film.pdf}
+    \includegraphics[width=0.75\textwidth]{graphics/Rupture_events_with_film.pdf}
     \caption{(a), (b) Space-time plots showing the evolution of the height field for athermal (a) and fluctuating (b) dewetting over the square-wave patterned substrate, in a neighbourhood of the instant of time at which the first rupture event (in the athermal case) occurred; for the sake of visualization we mark the rupture events with red bullets (\textcolor{red}{$\bullet$}).
     (c) Distribution of times of occurrence of rupture events for the athermal ($\sigma=0$, blue, line-patterned) and fluctuating ($\sigma = \sigma_0$, orange, dot-patterned) dewetting.} 
     \label{fig:rupture_time_distri_square_wave8}
@@ -488,7 +488,7 @@ \subsubsection{Square wave pattern}\label{subsubsec:square_wave}
 
  \begin{figure}
      \centering
-     \includegraphics[width=0.75\textwidth]{graphics/square_wave_det_stoch.pdf}
+     \includegraphics[width=0.65\textwidth]{graphics/square_wave_det_stoch.pdf}
      \caption{Spectra of the dewetting film on the $\theta^{(2)}(x)$ pattern, Eq.(\ref{eq:sharp_contact_angle_spatial}), at $t = 0.1t_0(\pi/6)$, with ($\sigma = \sigma_0$, \textcolor{pyorange}{$\blacktriangle$}) and without ($\sigma =0$, \textcolor{pyblue}{$\bullet$}) thermal fluctuations. 
      The vertical lines indicate the substrate wave mode $q_{\theta}$ (solid) and its multiples $2q_{\theta}$ (dashed), $3q_{\theta}$ (thin dashed-dotted) and $7q_{\theta}$ (thick dashed-dotted).}
      \label{fig:square_wave_both}

others/titlepage.tex

@@ -17,7 +17,7 @@
 \node at ($ (current page.center) + (0, 0.7) $) {
     \parbox{\textwidth}{%
 		\begin{center}
-			\large{Der Technischen Fakul\"at \break der Friedrich-Alexander-Universit\"at \break Erlangen-N\"urnberg \break zur \break Erlangung des Doktorgrades Dr.-Ing. \break vorgelegt von} \\[2ex]
+			\large{Der Technischen Fakult\"at \break der Friedrich-Alexander-Universit\"at \break Erlangen-N\"urnberg \break zur \break Erlangung des Doktorgrades Dr.-Ing. \break vorgelegt von} \\[2ex]
 
 			 \large{Stefan Zitz \break aus Graz, \"Osterreich}