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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 190, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2016/190\hfil A waveless free surface flow]
{A waveless free surface flow past a submerged triangular obstacle
in presence of \\ surface tension}
\author[H. Sekhri, F. Guechi, H. Mekias \hfil EJDE-2016/190\hfilneg]
{Hakima Sekhri, Fairouz Guechi, Hocine Mekias}
\address{Hakima Sekhri \newline
Department of Mathematics, Faculty of sciences,
University Setif1.19000, Algeria}
\email{sekhrihakima@yahoo.fr}
\address{Fairouz Guechi \newline
Department of Mathematics, Faculty of sciences,
University Setif1.19000, Algeria}
\email{f\_guechi@yahoo.fr}
\address{Hocine Mekias \newline
Department of Mathematics, Faculty of sciences,
University Setif1.19000, Algeria}
\email{mekho58@gmail.com}
\thanks{Submitted November 5, 2015. Published July 13, 2016.}
\subjclass[2010]{35B40, 35Q35, 76B07, 76D45, 76M40}
\keywords{Free surface; potential flow; Weber number;
\hfill\break\indent
surface tension; nonlinear boundary condition}
\begin{abstract}
We consider the Free surface flows passing a submerged triangular obstacle
at the bottom of a channel. The problem is characterized
by a nonlinear boundary condition on the surface of unknown
configuration. The analytical exact solutions for these problems
are not known. Following Dias and Vanden Broeck \cite{d1},
we computed numerically the solutions via a series truncation method.
These solutions depend on two parameters: the Weber number $\alpha$
characterizing the strength of the surface tension and the angle
$\beta$ at the base characterizing the shape of the apex.
Although free surface flows with surface tension admit capillary waves,
it is found that solution exist only for values of the Weber number greater
than $\alpha_0$ for different configurations of the triangular obstacle.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{remark}[theorem]{Remark}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
We consider the steady two dimensional flow of an inviscid incompressible
fluid passing a submerged triangular obstacle at the bottom of a channel
(See \ref{fig1}), as we shall see the problem is characterized by the
Weber number. Free surface flows around submerged bodies have been
studied by many authors and researchers, for long years.
They modeled their problems by considering bodies of regular shapes:
flows around cylinder \cite{h1,t1},
semi-circle \cite{f1,f2,f3,v1}, triangles \cite{d1}, and
finite flat plates \cite{v2}.
Choi \cite{c2,c3}) carried out an analytical asymptotic calculation
over a small depression in a channel with a shallow water flow,
taking into consideration gravity and neglecting surface tension.
Dias and Vanden Broeck \cite{d1} considering the effect of gravity
and neglected the surface tension, they computed the problem via a
series truncation method solution, for different values of the Froude number.
We used the same method to solve our problem considering the effect of
surface tension and neglecting gravity. For very large values of the Weber
number $\alpha \to \infty$, solutions are approximately the same and the
free surface profiles coincide with the free streamline solution,
in the absence of gravity and surface tension.
It is observed that there is a value $\alpha_0$, $0<\alpha_0<1$,
of the Weber number for which there is no solution, if
$\alpha<\alpha_0$, and a unique negative solitary-wave-like solution
if $\alpha>\alpha_0$, Vanden Broeck \cite{c1} showed that, in presence
of surface tension, capillary waves are exponentially small to all orders.
This may explain the limiting value $\alpha_0$ of the Weber number below
which our procedure fails to describe a waveless solution of the problem.
\section{Formulation of the problem}
We consider the steady two-dimensional flow of a fluid over a triangular
obstacle (See \ref{fig1}). The fluid is assumed to be inviscid, incompressible
and the flow is irrotational. We neglect the effect of the gravity but we
take into account the effect of surface tension. Far upstream and downstream
``far from the triangular $BCD$'', the flow is uniform with a constant velocity
$U$ and a constant depth $L$. As we shall see, the flow is characterized
by two-parameters: the angle $\beta$ at the base characterizing the shape
of the apex and the Weber number $\alpha$ characterizing the strength of the
surface tension and is defined by
\begin{equation}
\alpha=\frac{{\rho}U^{2}L}{T}
\label{2-1}
\end{equation}
where $T$ is the surface tension and $\rho$ is the density of the fluid.
When the effects of surface tension and gravity $g$ are neglected, the
classical exact solution can be found via the hodograph transformation
Birkhoff\cite{b2}. If the effects of surface tension or gravity are
considered, the boundary condition at the free surface is nonlinear
and no exact analytical solution is known. Different combinations and
some varieties of this problem have been considered.
Considering the effect of the surface tension, our results confirm that
there is a solution for different Weber number $\alpha>0$, and for
triangles of arbitrary size by varying the angle $\beta$.
A system of cartesian coordinates is defined, with the $x$-axis along the
horizontal bottom AB, DE and the $y$-axis going through the apex $C$
of the triangle BCD.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1}
\end{center}
\caption{Sketch of the flow and of the system of coordinates.}
\label{fig1}
\end{figure}
We define dimensionless variables by taking $U$ as the unit velocity and
$L$ as the unit length. We denote by $u$ and $v$ the components of the velocity
in the $x$ and $y$ directions, respectively. Since the flow is potential,
it can be described by two functions: a potential function $\phi$ and a stream
function $\psi$. Without loss of generality, we choose $\phi=0$ at $C$ and
$\psi=0$ on the stream line ABCDE. It follows from the choice of the
dimensionless variables that $\psi=1$ on the free stream line FGHJ.
\begin{itemize}
\item[(1)] In the far field (as $|x|\to\infty$), the flow is supposed
to be uniform, hence
\begin{equation}
\phi{(x,y)}=Ux\quad\text{as } |x|\to\infty \label{2-2}
\end{equation}
\item[(2)] On the rigid boundary ($ABCDE$), the normal velocity has to vanish,
that is:
\begin{equation}
\frac{\partial{\phi}}{\partial{\overrightarrow\eta}}=0,
\label{2-3}
\end{equation}
where $\overrightarrow\eta$ is the unit normal vector on the boundary ($ABCDE$).
\item[(3)] On the free surface, the atmospheric pressure $P_0$ is constant,
hence the Bernoulli equation yields:
\begin{equation}
\overline {p}+\frac{1}{2}\rho\overline {q}^{2}=\overline {p}_0
+\frac{1}{2}\rho {U}^{2} \quad\text{on FHJ } \psi=1
\label{2-4}
\end{equation}
\end{itemize}
Here $\overline {p}$ and $\overline {q}$ are the fluid pressure and the
speed just inside the free surface, respectively. The right-hand side of
\eqref{2-4} is evaluated from the condition in the far field.
A relationship between $\overline {p}$ and $\overline {p}_0 $ is given by
Laplace's capillarity formula \begin{equation}
\overline {p}-\overline {p}_0=TK
\label{2-5}\end{equation}Here $K$ is the curvature of the free surface and $T$ is the surface tension. \\If we substitute \eqref{2-5} into \eqref{2-4}, and in dimensionless variables, \eqref{2-4} becomes \begin{equation}
\frac{1}{2} q^{2}-\frac{1}{\alpha}K=\frac{1}{2}\quad\text{on } FHJ, \label{2-6}
\end{equation}
where $\alpha$ is the Weber number defined by \eqref{2-1}.
The physical flow problem as described above can be formulated as a boundary
value problem in the potential function $\phi{(x,y)}$:
\begin{equation}
\begin{gathered}
\Delta \phi=0\quad\text{in the flow domain}, \quad
\phi{(x,y)}=x,\quad |x|\to\infty \\
\frac{\partial{\phi}}{\partial{\overrightarrow\eta}}=0
\quad\text{on the rigid boundary ABCDE }\\
|\nabla\phi|^{2}-\frac{2}{\alpha}K=1 \quad\text{on the free surface.}
\end{gathered}
\label{2-7}
\end{equation}
Solving the problem in this form is very difficult especially that the
nonlinear boundary condition is specified on an unknown boundary
(the free surface). Instead of solving the problem in its partial differential
equation form in $\phi$, we take advantage of the property that for the
bidimensional potential flow (as is in our problem) and if the plane
in which the flow is embedded is identified to the complex plane,
the complex velocity $\xi=u-iv$ and the complex potential function
$f=\phi+i\psi$ are analytic functions of the complex variable $z=x+iy$.
Hence, we use all the necessary properties of analytic functions of a
complex variable: integral formulation, series formulation, conformal mapping,
etc.. Therefore, in the $f$-plane, the flow is the strip $0<\psi<1$
(See \ref{fig2}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig2}
\end{center}
\caption{The flow configuration in the complex potential plane.}
\label{fig2}
\end{figure}
The free surface, the bottom channel and the triangle are parts of a streamline,
are mapped onto the straight lines $\psi=1$ and $\psi=0$, respectively.
In order that the curvature be well defined, we introduce the function
$\tau-i\theta$ as \begin{equation}
\xi=\frac{df}{dz}=u-iv=e^{\tau-i\theta},
\label{2-8}
\end{equation}
where $e^\tau$ represents the strength of the velocity,
$e^\tau=\sqrt{u^{2}+v^{2}}$ and $\theta$ is the angle between the $x$-axis
and the vector velocity. In these new variables, the Bernoulli equation
\eqref{2-6} becomes
\begin{equation}
e^{2\tau}-\frac{2}{\alpha}|\frac{\partial{\theta}}{\partial{\phi}}|e^{\tau}=1
\quad \text{on FHJ } (\psi=1)
\label{2-9}
\end{equation}
The kinematic condition is expressed as
\begin{gather}
\beta=0 \quad\text{ on AB and DE} , \label{2-10}\\
\begin{gathered}
\theta=\beta \quad\text{on BC}, \\
\theta=\beta_2 \quad\text{on CD},
\end{gathered}\label{2-11}
\end{gather}
We shall seek $\tau-i\theta$ as an analytic function of $f=\phi+i\psi$
in the strip $0<\psi<1$, satisfying the conditions \eqref{2-9}, \eqref{2-10}
and \eqref{2-11}.
\section{Numerical procedure}
We define a new variable $t$ by the relation
\begin{equation}
f=\frac{2}{\pi} \log(\frac{1+t}{1-t}) \label{3-1}
\end{equation}
This transformation maps the flow domain into the upper half of
the unit disc in the complex $t$ plane (See \ref{fig3}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3}
\end{center}
\caption{The flow domain in the t-plane.}
\label{fig3}
\end{figure}
The free surface is mapped onto the upper half unit circle and the rigid
bottom is mapped onto the diameter. The apex C of the triangle is
mapped into the origin, the apex B into a point $t_B$,
$-11$.
For example with error less than a ${10^{-8}}$. When $\alpha \to0$,
the algorithm converges less rapidly and ceases to converge, when $\alpha<\alpha_0$,
for some critical values $0<\alpha_0<1$. The critical value $\alpha_0$ depends
on the angle $\beta$. The existence of this critical value of the Weber number
can be explained from the procedure used in this article itself.
The procedure used relies on the series expansion \eqref{3-6} of the analytic
complex velocity $\xi=u-iv$, which does not take into account capillary waves.
In this article, Chapman (See \cite{c1}) showed that capillary waves are
exponentially small to all order. Hence, the capillary waves are not dominant
and the expansion \eqref{3-6} describes the flow very well unless the Weber
number is sufficiently small. For all the values of the Weber number
$\alpha>\alpha_0$ and for the angle $\frac{\pi}{2}<\beta<\pi$,
the free surface profile looks like a symmetric negative solitary wave with
the maximum crest is just above the apex $C$ of the triangular.
In (See \ref{fig5}), we showed different free surface profiles for
$\beta=\frac{3\pi}{4}$ and different values of the Weber number.
It is observed that the maximum crest is obtained for $\alpha \to \infty$
and decreases as $\alpha \to 0$.
For $\alpha\geq300$, all free surface profiles for different values of
$\alpha$ are the same within graphical accuracy and coincide with the
graph of the exact solution without surface tension.
This suggests that the surface tension can be neglected if $\alpha\geq300$.
To obtain different configuration of the triangular, we varied the angle
$\beta$, $\frac{\pi}{2}<\beta<\pi$ and fixed the Weber number $\alpha$.
Profiles of the free surface for different values of the angle $\beta$
and $\alpha=5$ is shown in (See \ref{fig6}).
We remark that when the angle $\beta$ increases $\beta\to\pi$,
the profiles of the free surface take the form of a uniform flow
over a horizontal plan.
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\end{document}