commit b87c4ff6232fd1ed6b64dd3f916739894690e7ff
parent 2c07f08b13fa5a7e325fe0fbb5b2da9c95d3a874
Author: miksa <milutin@popovic.xyz>
Date: Thu, 2 Jun 2022 18:47:22 +0200
boundary conditions and checking
Diffstat:
3 files changed, 180 insertions(+), 56 deletions(-)
diff --git a/app_pde/appendix.tex b/app_pde/appendix.tex
@@ -0,0 +1,60 @@
+\appendix
+\section{Appendix: Mathematical Preliminaries}
+\subsection{Leibniz Rule of Integration}
+\label{appendix:leibniz}
+The Leibniz integral rule for differentiation under the integral sign
+initiates with an integral
+\begin{align}
+ \mathcal{I}(t, x) = \int_{a(t)}^{b(t)} f(t, x) dx = \mathcal{I}(t, a(t,
+ a(t), b(t))).
+\end{align}
+And upon differentiation w.r.t. $t$, utilizes the chain rule on $a(t)$ and
+$b(t)$ respectively, by
+\begin{align}
+ \frac{d\mathcal{I}}{dt} =
+ \frac{\partial \mathcal{I}}{\partial t}+
+ \frac{\partial \mathcal{I}}{\partial a}\frac{\partial a}{\partial t}+
+ \frac{\partial \mathcal{I}}{\partial b}\frac{\partial b}{\partial t}.
+\end{align}
+Which in integral representation reads
+\begin{align}
+ \frac{d\mathcal{I}}{dt} = \int_{a(t)}^{b(t)}\frac{\partial f(t,
+ x)}{\partial t} dx + f(t, b(t)) \frac{\partial b(t)}{\partial t}
+ - f(t, a(t)) \frac{\partial a(t)}{\partial t}
+\end{align}
+
+\subsection{Gaussian Integration Law}
+\label{appendix:gauss integration}
+This should explain the Gaussian integration law
+
+\subsection{Identity for Vorticity}
+We start off with the standard material derivative
+\begin{align}
+ \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
+ (\mathbf{u}\nabla)\mathbf{u}.
+\end{align}
+We will use Einstein's Summation Convention, where we sum over indices that
+both appear at as the bottom as the top index, to rewrite the second part of
+the material derivative $(\mathbf{u}\nabla)\mathbf{u}$ into
+\begin{align}
+ (\mathbf{u}\times (\nabla \times \mathbf{u}))_k
+ &= \varepsilon^{ijk}u_j(\nabla \times \mathbf{u})_k \\
+ &= \varepsilon^{ijk}u_j\varepsilon_{klm}\partial^l u^m\\
+ &=(\delta^i_l\delta^j_m-\delta^i_m\delta^j_l)u_j\partial^l u^m\\
+ &=u_m\partial^i u^m - u_l \partial^l u^i.\label{eq:identity split}
+\end{align}
+Now the first part in equation \ref{eq:identity split} can be rewritten into
+\begin{align}
+ u_m\partial^i u^m =\partial^i (\frac{1}{2}u_mu^m) .
+\end{align}
+Thus we get
+\begin{align}
+ (\mathbf{u}\times (\nabla \times \mathbf{u}))_k
+ = \frac{1}{2}\partial^i(u_m u^m) + u_l \partial^l u^i,
+\end{align}
+which is
+\begin{align}
+ (\mathbf{u}\nabla)\mathbf{u} = \nabla(\frac{1}{2}\mathbf{u}\mathbf{u}) -
+ \left(\mathbf{u}\times (\nabla \times \mathbf{u})\right)
+\end{align}
+
diff --git a/app_pde/basics_fluids.tex b/app_pde/basics_fluids.tex
@@ -6,23 +6,24 @@
\tableofcontents
\section{Governing Equations of Fluid Dynamics}
-We first of start with a fluid with density/mass
+We first start of with a fluid with a density
\begin{align}
\rho(\mathbf{x}, t),
\end{align}
-with a position $\mathbf{x} = (x, y, z)$ in three dimensional space at time
+in three dimensional Cartesian coordinates $\mathbf{x} = (x, y, z)$ at time
$t$. For water-wave applications, we should note that we take
-$\rho=\text{constant}$. The fluid moves in time and space with a velocity field
+$\rho=\text{constant}$, but we will go into this fact later. The fluid moves
+in time and space with a velocity field
\begin{align}
\mathbf{u}(\mathbf{x}, t) = (u, v, w).
\end{align}
-Additionally it is also described by the pressure of the fluid
+Additionally it is also described by its pressure
\begin{align}
P(\mathbf{x}, t),
\end{align}
generally depending on time and position. When thinking of e.g. water the
pressure increases the deeper we go, that is with decreasing or increasing $z$
-coordinate (depending how we set up our system $z$ pointing up or down
+direction (depending how we set up our system $z$ pointing up or down
respectively).
The general assumption in fluid dynamics is the \textbf{Continuum
@@ -30,16 +31,16 @@ Hypothesis}, which assumes continuity of $\textbf{u}, \rho$ and $P$ in
$\mathbf{x}$ and $t$. In other words, we premise that the velocity field,
density and pressure are ''nice enough`` functions of position and time, such
that we can do all the differential operations we desire in the framework of
-fluid dynamics.
+differential analysis.
\subsection{Mass Conservation}
Our aim is to derive a model of the fluid and its dynamics, with respect to
-time and position, in the most general way. This is generally done thinking
-of the density of a given fluid, which is a unit of mass per unit volume,
-intrinsically an integral representation to derive these equations is going
-to be used.
+time and position, in the most general way. This is usually done thinking
+of the density of a given fluid, which is a unit mass per unit volume,
+intrinsically an integral representation to derive these equations suggests
+by itself.
Let us now thing of an arbitrary fluid. Within this fluid we define a fixed
-volume $V$ relative to a chosen inertial frame and bounded by a surface $S$
+volume $V$ relative to a chosen inertial frame and bound it by a surface $S$
within the fluid, such that the fluid motion $\mathbf{u}(\mathbf{x}, t)$ may
cross the surface $S$. The fluid density is given by $\rho(\mathbf{x}, t)$,
thereby the mass of the fluid in the defined Volume $V$ is an integral
@@ -79,22 +80,23 @@ The figure bellow \ref{fig:volume}, expresses the above described picture.
with a surface normal vector $\mathbf{n}$ \label{fig:volume}}
\end{figure}
-Since we want to figure out the fluid dynamics, we can consider now the rate
-of change of mass in of this completely arbitrary $V$, which needs to be
-disappear, i.e. is equal to zero since we cannot lose mass. Matter (mass) is
-neither created nor destroyed anywhere in the fluid
+Since we want to figure out the fluid's dynamics, we can consider the rate
+of change in the completely arbitrary $V$. The rate of change of mass needs to
+disappear, i.e. it is equal to zero since we cannot lose mass. Matter (mass) is
+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}
-We may get more information with simply ''differentiating under the integral
+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 above integral equation reads
+\ref{appendix:leibniz}, the integral equation representing the rate of change
+of mass reads
\begin{align}\label{eq:mass balance}
\frac{dm}{dt} = \int_V \frac{\partial \rho(\mathbf{x}, t)}{\partial t}\ dV
+\int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u}\cdot\mathbf{n}\ dS
= 0.
\end{align}
-The above equation \ref{eq:mass balance} is an underlying equation, describing that the rate of
+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.
@@ -115,8 +117,8 @@ called the \textbf{Continuity Equation}
\partial_t \rho + \nabla(\rho \mathbf{u}) = 0
\end{align}
-In the light of the results of the equation of mass conservation
-\ref{eq:continuity}, an expansion, of the nabla gives
+In light of the results of the equation of mass conservation
+in \ref{eq:continuity}, an product rule gives
\begin{align}
\partial_t \rho + (\nabla \rho)\mathbf{u} + \rho(\nabla \mathbf{u}),
\end{align}
@@ -125,7 +127,7 @@ as follows
\begin{align}
\frac{D}{Dt} = \partial_t + \mathbf{u}\nabla.
\end{align}
-Thus the equation of mass conservation becomes
+With the material derivative the equation of mass conservation reads
\begin{align}
\frac{D\rho}{Dt} + \rho \nabla\mathbf{u} = 0
\end{align}
@@ -153,7 +155,7 @@ element in the fluid. The integral formulation of the force would be
\int_V \rho \mathbf{F}\ dV - \int_S P\mathbf{n}\ dV.
\end{align}
Now applying the Gaussian rule of integration on the second integral over the
-surface, the resluting force in per unit volume is
+surface, the resulting force in per unit volume is
\begin{align}
\int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
\end{align}
@@ -161,54 +163,116 @@ The acceleration of the fluid particles is given by $\frac{D\mathbf{u}}{Dt}$,
and thus the total force per unit volume on the other hand is
\begin{align}
\int_V \rho \frac{D\mathbf{u}}{Dt}\ dV =
- \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV
+ \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
\end{align}
-
-
-
-
-
-
-
-
-\newpage
-\appendix
-\section{Appendix: Mathematical Preliminaries}
-\subsection{Leibniz Rule of Integration}
-\label{appendix:leibniz}
-The Leibniz integral rule for differentiation under the integral sign
-initiates with an integral
+Newton's Second Law for a fluid in an Volume is essentially saying that the
+rate of change of momentum of the fluid in the fixed volume $V$, which is the particle
+acceleration is the resulting force acting on V together with the rate of
+flow of momentum across the surface $S$ into the volume $V$. Hence we arrive
+at the \textbf{Euler's Equation(s) of Motion}
\begin{align}
- \mathcal{I}(t, x) = \int_{a(t)}^{b(t)} f(t, x) dx = \mathcal{I}(t, a(t,
- a(t), b(t))).
+ \frac{D\mathbf{u}}{Dt} = \left(\frac{\partial \mathbf{u}}{\partial t}
+ (\mathbf{u}\nabla)\mathbf{u}\right) =
+ -\frac{1}{\rho}\nabla P + \mathbf{F}.
\end{align}
-And upon differentiation w.r.t. $t$, utilizes the chain rule on $a(t)$ and
-$b(t)$ respectively, by
+As a side note we have mentioned that there is another contribution if the
+fluid is viscid. Indeed there is a tangential force due to the velocity
+gradient, which into introduces the additional term
\begin{align}
- \frac{d\mathcal{I}}{dt} =
- \frac{\partial \mathcal{I}}{\partial t}+
- \frac{\partial \mathcal{I}}{\partial a}\frac{\partial a}{\partial t}+
- \frac{\partial \mathcal{I}}{\partial b}\frac{\partial b}{\partial t}.
+ \mu \nabla^2 \mathbf{u}, \qquad
+ \mu = \text{viscosity of the Fluid}.
\end{align}
-Which in integral representation reads
+Thereby the equations become
\begin{align}
- \frac{d\mathcal{I}}{dt} = \int_{a(t)}^{b(t)}\frac{\partial f(t,
- x)}{\partial t} dx + f(t, b(t)) \frac{\partial b(t)}{\partial t}
- - f(t, a(t)) \frac{\partial a(t)}{\partial t}
+ \rho\frac{D\mathbf{u}}{Dt}
+ = -\nabla P + \rho \mathbf{F} + \mu \nabla^2 \mathbf{u}.
\end{align}
-\subsection{Gaussian Integration Law}
-\label{appendix:gauss integration}
-This should explain the Gaussian integration law
-
-
-
+For now we have separated two simplifications, that define an
+\textbf{idealized/perfect fluid}
+\begin{enumerate}
+ \item \textbf{incompressible} $\qquad \mu=0$
+ \item \textbf{inviscid} $\quad \rho = \text{const.},\ \nabla \mathbf{u}=
+ 0$
+\end{enumerate}
+\subsection{Vorticity and irrotational Flow}
+The curl of the velocity field $\mathbf{\omega} = \nabla \times \mathbf{u}$
+of a fluid (i.e. the vorticity), describes a spinning motion of the fluid
+near a position $\mathbf{x}$ at time $t$. The vorticity is an important
+property of a fluid, flows or regions of flows where $\mathbf{\omega}=0$ are
+\textbf{irrotational}, and thus can be modeled and analyzed following well
+known routine methods. Even though real flows are rarely irrotational
+anywhere (!), in water wave theory wave problems, from the classical aspect
+of vorticity have a minor contribution. Hence we can assume irrotational flow
+modeling water waves. To arrive at the vorticity in the equations of motions
+derived in the last section we resort to a differential identity derived in appendix
+\ref{appendix:diff identity}, which gives for the material derivative
+\begin{align}
+ \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
+ \nabla(\frac{1}{2}\mathbf{u}\mathbf{u)}
+ - \left( \mathbf{u}\times (\nabla \times \mathbf{u} \right).
+\end{align}
+Thus the equations of motion become
+\begin{align}
+ \frac{\partial \mathbf{u}}{\partial t} + \nabla\left(
+ \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right)
+ = \mathbf{u} \times \mathbf{\omega},
+\end{align}
+where $\Omega$ is the force potential per
+unite mass given by $\mathbf{F} = -\nabla \Omega$.
+At this point we may differentiate between \textbf{stead and unsteady flow}.
+For \textbf{Steady Flow} we assume that $\mathbf{u}, P$ and $\Omega$ are time
+independent, thus we get
+\begin{align}
+ \nabla\left( \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega
+ \right) = \mathbf{u} \times \mathbf{\omega}.
+\end{align}
+It is general knowledge that the gradient of a function $\nabla f$ is
+perpendicular the level sets of $f(\mathbf{x})$, where $f(\mathbf{x}) =
+\text{const.}$. Thus $\mathbf{u} \times \mathbf{\omega}$ is orthogonal to
+the surfaces where
+\begin{align} \label{eq:bernoulli}
+ \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega =
+ \text{const.},
+\end{align}
+The above equation is called \textbf{Bernoulli's Equation}.
+Secondly \textbf{Unsteady Flow} but irrotational (+ incompressible), first of
+all gives us the condition for the existence of a velocity potential $\phi$
+in the sense
+\begin{align}
+ \mathbf{\omega} = \nabla \times \mathbf{u} = 0 \quad \Rightarrow \quad
+ \mathbf{u} = \nabla \phi,
+\end{align}
+where $\phi$ needs to satisfy the Laplace equation
+\begin{align}
+ \Delta \phi = 0.
+\end{align}
+According to the Theorem of Schwartz we may exchange $\frac{\partial
+}{\partial t}$ and $\nabla$, giving us an expression for the material
+derivative
+\begin{align}
+ \nabla\left( \frac{\partial \phi}{\partial t} +\frac{1}{2}
+ \mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right) = 0
+\end{align}
+Thus the expression differentiated by the $\nabla$ operator is an arbitrary
+function $f(\mathbf{x}, t)$, writing
+\begin{align}
+ \frac{\partial \phi}{\partial t} +\frac{1}{2}
+ \mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega = f(\mathbf{x}, t).
+\end{align}
+The function $f(\mathbf{x}, t)$ can be removed by gauge transformation of
+$\phi \rightarrow \phi + \int f(\mathbf{x}, t)\ dt$, never the less this is
+not further discussed and left to the reader in the reference.
+\subsection{Boundary Conditions for water waves}
+Surface, Bottom, Pressure
+\newpage
+\include{appendix.tex}
diff --git a/app_pde/build/basics_fluids.pdf b/app_pde/build/basics_fluids.pdf
Binary files differ.
