commit 8ce911ece70e6d388d9cd789443ae126f7eddedd
parent c77d1533d1c61411c2bf6ca72e6c8636357994fa
Author: miksa <milutin@popovic.xyz>
Date: Sat, 4 Jun 2022 19:10:50 +0200
new chapters
Diffstat:
2 files changed, 387 insertions(+), 3 deletions(-)
diff --git a/app_pde/basics_fluids.tex b/app_pde/basics_fluids.tex
@@ -90,6 +90,7 @@ neither created nor destroyed anywhere in the fluid, leading us to
\begin{align}
\frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = 0.
\end{align}
+\textbf{NOT SURE HERE YET!!!!!!!!!!!, CHECK LEIBINZ FORMULA}
To get more information we simply ''differentiate under the integral
sign``, also known as the Leibniz Rule of Integration, see appendix
\ref{appendix:leibniz}, the integral equation representing the rate of change
@@ -99,6 +100,7 @@ of mass reads
+\int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u}\cdot\mathbf{n}\ dS
= 0.
\end{align}
+\textbf{----------------------}
The above equation in \ref{eq:mass balance} is an underlying equation, describing that the rate of
change of mass in V is brought about, only by the rate of mass flowing into
V across S, and thus the mass does not change.
@@ -394,7 +396,7 @@ components
Integrating the above expression from bottom to surface, i.e. from
$z=b(\mathbf{x}_\perp,t)$ to $z = h (\mathbf{x},t)$ gives
\begin{align}
- \int_b^h \nabla_\perp \mathbf{u}_\perp\ dz w\bigg|_{z=b}^{z=h} = 0,
+ \int_b^h \nabla_\perp \mathbf{u}_\perp\ dz + w\bigg|_{z=b}^{z=h} = 0,
\end{align}
where we insert the conditions on the free surface and on the bottom surface
\begin{align}
@@ -423,7 +425,7 @@ As a consequence the \textbf{Integrated Mass Condition} is given by
b_t}_{=d_t} = 0.
\end{align}
\subsection{Energy Equation}
-To derive the energy equation we start of with Euler's Equation of Motion
+To derive the energy equation we start off with Euler's Equation of Motion
\begin{align}
\mathbf{u} _t + \nabla
(\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega) = \mathbf{u}\times
@@ -478,7 +480,7 @@ additionally adding $\frac{\partial \Omega}{\partial t} =0$ leads us to
\end{align}
This is called the \textbf{energy equation} and is a general result for a
inviscid and incompressible fluids, which we can apply to study water waves.
-We start of with replacing $\nabla = \nabla_\perp + \frac{\partial }{\partial
+We start off with replacing $\nabla = \nabla_\perp + \frac{\partial }{\partial
z} $ and $\Omega = g z$ and multiplying by $\rho$, then our energy equation
in \ref{eq:energy} becomes
\begin{align}
@@ -551,6 +553,388 @@ can set $P_s=0$, such that $\mathcal{P} =0$ leaving us with the equation
We note that the assumption $P_s=0$ is only possible if the coefficient of
surface tension is set to 0, which usually is not the case.
\section{Dimensional Analysis}
+Our derived model of fluid dynamics yields formal connections between
+physical quantities. These quantities bear units, e.g. the velocity of fluid
+particles $\mathbf{u}$ has the ``SI'' unites of $\frac{m}{s}$, meters per
+second. The idea is the make use of these scales and formulate a model, where
+the quantities are nondimensionalized, i.e. to get rid of physical units by
+scaling each quantity appropriately. The appropriate length scales are that
+of the typical water depth $h_0$ and the typical wavelength $\lambda$ of a
+surface wave.
+
+\subsection{Nondimensionalisation}
+In summary we use these adaptations
+\begin{itemize}
+ \item $h_0$ for the typical water depth
+ \item $\lambda$ for the typical wavelength
+ \item $\frac{\lambda}{\sqrt{g h_0}}$ time scale of wave propagation
+ \item $\sqrt{g h_0}$ velocity scale of waves in $(x, y)$
+ \item $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in the $z$
+ direction.
+\end{itemize}
+$(x, z, t)$, then
+\begin{align}
+ u = \psi _z, \qquad w = - \psi_x;
+\end{align}
+and the scale of $\psi$ must be $h_0\sqrt{g h_0}$. Additionally we write the
+boundary condition on the free surface as follows
+\begin{align}
+ h = h_0 + a \eta (\mathbf{x}_\perp, t) = z,
+\end{align}
+where $a$ is the typical amplitude and $\eta$ nondimensional function. All in
+all we have the following scaling for the physical quantities of our context
+\begin{align}
+ &x \rightarrow\ \lambda x, \quad u \rightarrow \sqrt{gh_0} u, \\
+ &y \rightarrow\ \lambda y, \quad v \rightarrow \sqrt{gh_0} v, \qquad
+ t\rightarrow \frac{\lambda}{\sqrt{gh_0}}t,\\
+ &z \rightarrow\ h_0 z, \quad w \rightarrow
+ \frac{h_0\sqrt{gh_0}}{\lambda} w.
+\end{align}
+with
+\begin{align}
+ h = h_0 + a \eta, \qquad b \rightarrow h_0 b.
+\end{align}
+The pressure is also rewritten into
+\begin{align}
+ P = P_a + \rho g(h_0 -z) + \rho g h_0 p,
+\end{align}
+where $P_a$ is the atmospheric pressure, the term $h_0-z$ represent the
+hydrostatic pressure distribution, i.e. pressure at depth and the term with the pressure
+variable $p$ measures the deviation from the hydrostatic pressure
+distribution. Indeed $p\neq 0 $ for wave propagation. Now we can perform a
+rescaling of the Euler's Equation of Motion, we introduce the notation
+\begin{align}
+ &t = \frac{\lambda}{\sqrt{gh_0}}\tau,\quad x = \lambda \xi,\quad u =
+ \sqrt{gh_0} \tilde{u}\\
+ &y = \lambda \chi,\quad v = \sqrt{gh_0} \tilde{v}\\
+ &z = h_0 \zeta, \quad w = \frac{h_0\sqrt{gh_0} }{\lambda}\tilde{w}.
+\end{align}
+We start off with the $x$ coordinate, substitute and apply the chain rule
+leading us to
+\begin{align}
+ \frac{Du}{Dt}
+ &= \frac{\partial u}{\partial t} +u \frac{\partial
+ u}{\partial x} \\
+ &= \sqrt{gh_{0}}\frac{\partial \tilde{u}}{\partial \tau} \frac{\partial
+ \tau}{\partial t} +gh_0 \tilde{u} \frac{\partial \tilde{u}}{\partial \xi}
+ \frac{\partial \xi}{\partial x} \\
+ &= \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
+ \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right),
+\end{align}
+on the other hand
+\begin{align}
+ \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
+ \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right)
+ &=-\frac{1}{\rho}\frac{1}{\lambda}\frac{\partial P}{\partial x} \\
+ &=-\frac{ g h_0 }{\lambda}\rho \frac{\partial p}{\partial \xi}.
+\end{align}
+Thereby the rescaling evolves to
+\begin{align}
+ \frac{D \tilde{u}}{D\tau} = -\frac{\partial p}{\partial \xi}.
+\end{align}
+Because of the same scaling in $y$ we get the same result as in $x$, that is
+\begin{align}
+ \frac{D \tilde{v}}{D\tau} = -\frac{\partial p}{\partial \chi}.
+\end{align}
+In the $z$ coordinate we have
+\begin{align}
+ \frac{Dw}{Dt}
+ &= \frac{\partial w}{\partial t} +w \frac{\partial
+ w}{\partial \zeta} \\
+ &= \frac{h_0\sqrt{gh_0}}{\lambda} \frac{\sqrt{gh_0}}{\lambda}
+ \frac{\partial \tilde{w}}{\partial \tau} + \frac{1}{h_0}
+ \frac{h_0\sqrt{gh_0} }{\lambda} \frac{h_0\sqrt{gh_0}}{\lambda}
+ \tilde{w}\frac{\partial \tilde{v}}{\partial \zeta}\\
+ &= \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
+ + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right) .
+\end{align}
+On the other side we have
+\begin{align}
+ \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
+ + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right)
+ &=
+ -\frac{1}{h_0\rho} \frac{\partial P}{\partial z} +g \\
+ &=-\frac{1}{h_0\rho}(-\rho gh_0 \frac{\partial \zeta}{\partial \zeta}
+ \rho gh_0
+ \frac{\partial p}{\partial \zeta} ) + g \\
+ &= -g \frac{\partial p}{\partial z}.
+\end{align}
+In total for the $z$ direction we get
+\begin{align}
+ \underbrace{\left( \frac{h_0}{\lambda} \right)^2}_{=: \delta^2}
+ \frac{Dw}{Dt} = -\frac{\partial p}{\partial z},
+\end{align}
+where $\delta$ is the \textbf{long wavelength} or \textbf{shallowness}
+parameter, a very important constant for developing model hierarchies. For
+clarity we resubstitute for $x, y, z, t, u, v$ and $w$, and for completeness
+the we display the equations again, which are
+\begin{align}\label{eq:nondim-motion}
+ \frac{Du}{Dt} = - \frac{\partial p}{\partial x}&, \quad
+ \frac{Dv}{Dt} = - \frac{\partial p}{\partial y}, \quad
+ \delta^2\frac{Dw}{Dt} = - \frac{\partial p}{\partial z}, \\
+ &\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
+ +\frac{\partial w}{\partial z} = 0.
+\end{align}
+We can now turn our attention to the boundary conditions, on both free
+surface $z=h$ and the bottom $z=b$ we have $z \Rightarrow h_0 z$ and thereby
+\begin{align}
+ z = 1+
+ \underbrace{\frac{a}{h_0}}_{:=\varepsilon}\eta(\mathbf{x}_\perp,t) \quad
+ \text{and}\quad z= b,
+\end{align}
+where we arrive at our second very important parameter $\varepsilon$ called
+the \textbf{amplitude} parameter. As for the kinematic condition, we
+substitute the free surface $z=h = 1+\varepsilon \eta$ and get
+\begin{align}
+ \frac{Dz}{Dt} = \varepsilon\left(\eta_t + (\mathbf{u}_\perp
+ \nabla_\perp)\eta\right) \qquad \text{on}\;\; z= 1+\varepsilon \eta.
+\end{align}
+Respectively the bottom condition is not changed
+\begin{align}
+ w = b_t + (\mathbf{u}_\perp \nabla_\perp) b \quad \text{on}\;\; z= b.
+\end{align}
+The general dynamic condition for $h = h(x, y, t)$ yields a rescaling of the
+curvature in terms of
+\begin{align}
+ \frac{1}{R}
+ &= \frac{(1+h_y^2)h_{x x} + (1+h_x^2)h_yy - 2h_xh_yh_{xy}
+ }{\left(h_x^2+h_y^2 +1 \right)^{\frac{3}{2}} } \\
+ &= -\frac{\varepsilon h_0}{\lambda^2} \frac{(
+ 1+\varepsilon^2\delta^2\eta_y^2 )\eta_{x x}+
+ (1+\varepsilon^2\delta^2\eta_x^2)\eta_{yy} -
+ 2\varepsilon^2\delta^2\eta_x\eta_y\eta_{xy}}{\left(
+ 1+\varepsilon^2\delta^2\eta_x^2+\varepsilon^2\delta^2\eta_y^2
+ \right)^{\frac{3}{2}} },
+\end{align}
+together with the pressure difference
+\begin{align}
+ \Delta P = \rho g h_0(p - \varepsilon \eta) = \frac{\Gamma}{R},
+\end{align}
+leaving us ultimately with the dynamic condition
+\begin{align}
+ p-\varepsilon\eta= \varepsilon\left( \frac{\Gamma}{\rho g\lambda^2}
+ \right) \left(\frac{\lambda^2}{\varepsilon h_0}\frac{1}{R}\right),
+\end{align}
+where $W_e = \frac{\Gamma}{\rho g h_0^2}$ is the \textbf{Weber number}. This
+dimensionless parameter can be considered as a measure of the fluid's inertia
+compered to its surface tension, which satisfies the relation
+\begin{align}
+ \delta^2 W_e = \frac{\Gamma}{\rho g \lambda^2}.
+\end{align}
+\subsection{Scaling of Variables}
+Admits a simple observation of the governing equations in the last chapter we
+notice that $w$ and $p$ on the free surface $z = 1 + \varepsilon\eta$ are
+directly proportional to $\varepsilon$. Hence we want to ''scale this way``
+by introducing the following transformation
+\begin{align}
+ p \rightarrow \varepsilon p, \quad w \rightarrow \varepsilon w, \quad
+ \mathbf{u}_\perp \rightarrow \varepsilon \mathbf{u}_\perp.
+\end{align}
+Because of this scaling our material derivative changes slightly to
+\begin{align}\label{eq:mod-material}
+ \frac{D}{Dt} = \frac{\partial }{\partial t} + \varepsilon\left(u
+ \frac{\partial }{\partial x} + v \frac{\partial }{\partial y} + w
+ \frac{\partial }{\partial z} \right)
+\end{align}
+A simple recalculation yields the rescaled, nondimensionalized Euler's
+Equation of motion are the same as in equations \ref{eq:nondim-motion} with
+the modified material derivative from \ref{eq:mod-material}, and the boundary
+conditions are
+\begin{align}
+ p &= \eta - \frac{\delta^2\varepsilon h_0}{\lambda^2} \frac{W_e}{R}\\
+ w &= \frac{1}{\varepsilon}\eta_t + (\mathbf{u}_\perp \nabla_\perp)\eta
+ \quad \text{on}\;\; z = 1+\varepsilon\eta\\
+ w &=\frac{1}{\varepsilon}b_t + (\mathbf{u}_\perp \nabla_\perp)b \quad
+ \text{on}\;\; z=b
+\end{align}
+\subsection{Model Hierarchies}
+As we have derived a model of fluid dynamics, with small parameters
+$\varepsilon$ and $\delta$, we can conduct a series of classifications and
+perform asymptotic analysis on them. The main hierarchies important in this
+review are derived from the following problem classifications
+\begin{itemize}
+ \item $\varepsilon\rightarrow 0$: linearized problem, small amplitude
+ \item $\delta\rightarrow 0$: shallow Water, long-wave
+ \item$\delta \rightarrow 0;\; \varepsilon~1$: shallow Water, large
+ amplitude
+ \item $\delta\ll 1;\; \varepsilon~\delta$: shallow water, medium
+ amplitude
+ \item $\delta\ll 1;\; \varepsilon~\delta^2$: shallow water, small
+ amplitude
+ \item $\delta \gg 1;\; \varepsilon\delta\ll 1$: deep water, small
+ steepness.
+\end{itemize}
+
+\section{The Solitary Wave and The KdV Equation}
+The solitary wave is a wave of translation, it is stable and can travel long
+distances additionally the speed depends on the size of the wave. An
+interesting feature is that two solitary waves do not merge together to form
+one solitary wave, rather the small wave is overtaken by a larger one. If a
+solitary wave is too big for the depth it splits into two, a big and a small
+one. Solitary waves arise in the region $\varepsilon=O(\delta^2)$.
+
+
+\subsection{Solitary Wave}
+To describe
+a solitary wave we begin with Euler's Equation of Motion, where we assume
+there is no surface tension we set $W_e = 0$ and additionally assume
+irrotational flow $\mathbf{\omega}=\nabla \times \mathbf{u} = 0$. This means
+that there exists a velocity potential $\phi(\mathbf{x},t)$ given
+by$\mathbf{u} = \nabla \phi$ satisfying the Laplace equation. In regard of a
+solitary wave being a plane wave, we rotate our coordinate system such that
+the propagation is in the $x$-direction and a stationary \& fixed bottom
+$b=0$. Ultimately leaving us with the following model
+\begin{align}\label{eq:soliton}
+\begin{drcases}
+ & \phi_{zz} + \delta \phi_{x x } = 0,\\
+ &\text{with the boundary conditions}\\
+ &\begin{drcases}
+ &\phi_z = \delta^2 (\eta_t + \varepsilon \phi_x \eta_x) \\
+ &\phi_t + \eta + \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
+ + \phi_x^2\right) =0
+ \end{drcases}\quad \text{on}\;\; z = 1+\varepsilon\eta,\\
+ &\text{and}\\
+ & \phi_z =0 \quad \text{on}\;\; z = b = 0.
+\end{drcases}
+\end{align}
+Since the model arises $\varepsilon = O(\delta^2)$, for convince we set
+$\varepsilon=1$. The fact of the matter is we are seeking a traveling wave
+solution, thereby we can go into the coordinate system of the traveling wave,
+one in the variable $\xi = x - ct$ for a from left to right traveling wave,
+where $c$ is the nondimensional speed of the wave. Our goal is to find the
+solution for the velocity potential $\phi(\xi, z)$ and the wave profile
+$\eta(\xi)$. The chain rule gives us
+\begin{align}
+ \frac{\partial }{\partial x} &= \frac{\partial \xi}{\partial x}
+ \frac{\partial }{\partial \xi} = \frac{\partial }{\partial \xi}, \\
+ \frac{\partial }{\partial t} &= \frac{\partial \xi}{\partial t}
+ \frac{\partial }{\partial \xi} = -c\frac{\partial }{\partial \xi}.
+\end{align}
+Together with the equations in \ref{eq:soliton} we obtain
+\begin{align}\label{eq:soliton-xi}
+ \begin{drcases}
+ & \phi_{zz} + \delta \phi_{\xi\xi} = 0,\\
+ &\text{with the boundary conditions}\\
+ &\begin{drcases}
+ &\phi_z = \delta^2 (\phi_\xi -c)\eta_\xi \\
+ &-c\phi_\xi + \eta + \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
+ + \phi_\xi^2\right) =0
+ \end{drcases}\quad \text{on}\;\; z = 1+\eta,\\
+ &\text{and}\\
+ & \phi_z =0 \quad \text{on}\;\; z = b = 0.
+ \end{drcases}
+\end{align}
+\subsubsection{Exponential Decay}
+We would like to analyze if the equation in \ref{eq:soliton-xi} gives viable a
+solution that decays exponentially, we make the ansatz
+\begin{align}
+ \eta \simeq a e^{-\alpha |\psi|},\quad \phi \simeq \psi(z)e^{-\alpha
+ |\xi|}, \qquad \mid \xi \mid \rightarrow \infty,
+\end{align}
+where $\alpha>0$ is the exponent. The equations in \ref{eq:soliton-xi}
+transforms to
+\begin{align}
+ \psi'' + \alpha^2 \delta^2\psi = 0.
+\end{align}
+The above equation is a standard well known ordinary differential equation
+reading
+\begin{align}
+ \psi = A \cos(\alpha\delta z),
+\end{align}
+where $A$ is the integration constant. On the free surface $z\simeq 1$ gives
+\begin{align}
+ &-cA\alpha\sin(\alpha\delta) = ca\alpha,\label{eq:sol1}\\
+ &cA\alpha \cos(\alpha\delta) = -a \label{eq:sol2}.
+\end{align}
+Dividing equation \ref{eq:sol1} with equation \ref{eq:sol2} gives
+\begin{align} \label{eq:soliton-dispersion}
+ c^2 = \frac{\tan\left(\alpha\delta \right) }{\alpha\delta}.
+\end{align}
+We conclude that the solution for such a wave exists provided that the
+dispersion relation on the wave propagation speed holds, thereby all solitary
+waves exhibit exponential decay in their tail and satisfy the dispersion
+relation in equation \ref{eq:soliton-dispersion}.
+\subsubsection{Asymptotic Analysis}
+The underlining equations in \ref{eq:soliton} extend from $-\infty$ to
+$\infty$, so the length scale is much greater than any finite depth of
+water. Therefore the classification $\delta \rightarrow 0$ is appropriate for
+a solitary wave, this however goes with the assumption
+$\varepsilon\rightarrow 0$ otherwise we cannot make an appropriate expansion.
+Let us look at the main equation
+\begin{align}\label{eq:sol-laplace}
+ \phi_{zz} + \delta \phi_{x x} = 0.
+\end{align}
+For small $\delta$ we conduct the $\delta^2 = O(\varepsilon)$ standard ansatz
+in asymptotic analysis
+\begin{align}
+ \phi_{\delta}(x, t, z) \simeq \sum_{n=0}^{\infty} \delta^{2n}\phi_n(x, t,
+ z).
+\end{align}
+Substituting $\phi_\delta$ into equation \ref{eq:sol-laplace} we get
+\begin{align}
+ \delta^{2\cdot 0}\left( \phi_{0zz} \right) + \delta^{2\cdot 1}\left(
+ \phi_{1zz}+\phi_{0 x x} \right) + \delta^{2\cdot 2}\left( \phi_{2zz}+
+ \phi_{1 x x} \right) + O(\delta^{2\cdot 3}) = 0.
+\end{align}
+We start off with $O(\delta^{2\cdot0}) $, which gives us an arbitrary function
+$\phi_{0} = \theta(x, t)$. Next we may generalize the results for all
+$O(\delta^{2\cdot n})$ in the means of
+\begin{align}
+ \phi_{n+1zz} = -\phi_{nx x}\qquad \forall n\in \mathbb{N} .
+\end{align}
+Therefore leaving us for $\phi_1$ and $\phi_2$ with
+\begin{align}
+ &\phi_1 = -\frac{1}{2} z^2 \theta_0(x,t) + \theta(x, t),\\
+ \Rightarrow& \phi_2 =
+ \frac{1}{24}z^4\theta_0(x,t)-\frac{1}{2}z^2\theta_1(x,t) + \theta_2(x,t).
+\end{align}
+The boundary condition on the bottom comes around to be
+\begin{align}
+ \phi_{nz} =0 \quad \text{on}\;\; z=0.
+\end{align}
+The free surface boundary condition $z= 1+\varepsilon\eta$ n evolves more calculation, we consider
+only terms up the order of $\delta^2$, initializing with
+\begin{align}
+ &\phi_z = \delta^2(\eta_t + \varepsilon\phi_x \eta_x)\\
+ \Leftrightarrow &\frac{1}{\delta}\phi_z = \eta_t + \varepsilon\phi_x
+ \eta_x,
+\end{align}
+substituting $\phi_\delta$ into the above proceeds to be
+\begin{align}
+ \frac{1}{\delta^2}\underbrace{\phi_{0z}}_{=0} + \phi_{1z}+ \delta^2\phi_{zz}
+ O(\delta^{2\cdot 2})
+ &= -z\theta_{x x} + \delta^2\left( \frac{1}{6}z^3\theta_{0 x x x x} - z
+ \theta_{0x x} \right) + O(\delta^{2\cdot 2})\\
+ &=-(1+\varepsilon\eta)\theta_{0 x x} + \delta^2\left(
+ \frac{1}{6}(1+\varepsilon\eta)^3\theta_{0 x x} -
+(1+\varepsilon\eta)\theta_{0 x x} \right) +O(\delta^{2\cdot 2})\\
+ &= \eta_t + \varepsilon\eta_x \left( \theta_{0x}
+ \delta^2(\theta_{1x}-\frac{1}{2}( 1+ \varepsilon\eta)^2 \theta_{0x x x}
+\right).
+\end{align}
+The second condition is
+\begin{align}
+ \phi_t + \eta + \frac{1}{2}\varepsilon \left( \frac{1}{\delta}\phi^2_z
+ \phi_x^2\right) = 0,
+\end{align}
+becomes after substitution
+\begin{align}
+ \theta_{0t}+ \delta^2\left( -\frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x xt}
+ + \theta_{1t}\right) + \eta + O(\delta^{2\cdot 2})
+ &=-\frac{1}{2}\delta^2\varepsilon(-(1+\varepsilon\eta)\theta_{0 x x
+ })^2\\
+ &-\frac{1}{2}\left( \theta_{0 x} + \delta^2\left( \theta_{1x} -
+ \frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x x x x} \right) \right) ^2
+\end{align}
+The simplest case is $\varepsilon,\delta \rightarrow 0$.
+
+
+
+
+
+
+
+
diff --git a/app_pde/build/basics_fluids.pdf b/app_pde/build/basics_fluids.pdf
Binary files differ.
