\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 78, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/78\hfil On singular solutions]
{On singular solutions of a magnetohydrodynamic nonlinear boundary
layer equation}

\author[M. Benlahsen, A. Gmira, M. Guedda\hfil EJDE-2007/78\hfilneg]
{Mohammed  Benlahsen, Abdelilah Gmira, Mohammed Guedda}  % in alphabetical order

\address{Mohammed Benlahsen \newline
 LPMC, Universit\'e de Picardie Jules Verne\\
 Facult\'e des Sciences, 33, rue Saint-Leu 80039
Amiens, France}
\email{benlahsen@u-picardie.fr}

\address{Abdelilah Gmira \newline
 Universit\'e Abdelamelk Assaadi,
 Facult\'e des Sciences, Tetouan, Maroc}
\email{gmira@fst.ac.ma or gmira.i@menara.ma}

\address{Mohammed Guedda \newline
LAMFA, CNRS UMR 6140, Universit\'e de Picardie Jules Verne,
 Facult\'e de Math\'ematiques et d'Informatique, 33,
 rue Saint-Leu 80039 Amiens, France}
 \email{guedda@u-picardie.fr  Tel: + 33322827796}

\thanks{Submitted September 22, 2006. Published May 23, 2007.}
\subjclass[2000]{34C11, 76D10}
\keywords{Boundary-layer equation;  MHD flow; blowing-up solutions; 
\hfill\break\indent asymptotic behavior}

\begin{abstract}
 This paper concerns the singular solutions  of the  equation
 $$
 f''' +\kappa ff''-\beta {f'}^2 = 0,
 $$
 where $ \beta < 0 $ and $ \kappa = 0 $ or $ 1$.  This equation
 arises when modelling  heat transfer past a  vertical
 flat plate embedded in a saturated porous medium with an
 applied magnetic field. After suitable normalization,  $ f' $
 represents the velocity parallel to the surface or
 the non-dimensional fluid temperature. Our interest is in
 solutions which develop a singularity at some point (the blow-up
 point).  In particular,  we shall examine in detail
 the  behavior of  $f$  near the blow-up point.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}\label{Sect.1}

 We investigate a one layer model of
magnetohydrodynamic (MHD) flow and heat transfer problems, which
are of considerable practical interest. Such a system is important
in understanding  a
variety of geophysical, astrophysical,  chemical engineering and
metallurgical processes (cooling of continuous strips
or filaments, purification of molten metals, etc.).

Much progress has been made during the previous years in the
development of MHD  nonlinear boundary layer equations. Pavlov
\cite{3Pav} was the first who examined the MHD flow over a stretching
wall in an electrically conducting fluid, with an
uniform  magnetic field.  Further studies  are those of Chakrabarti
and Gupta \cite{3ChG},
Vajravelu
\cite{3Vaj}, Takhar et al. \cite{3TRP,3TAG},  Kumari et al.
\cite{3KTN}, Andersson et al. \cite{3And},
    Watanabe and Pop \cite{3WP} and Sobha and Ramakrishna \cite{SR}.
In particular,  paper \cite{SR} focused mainly on the effect of
magnetic field on  temperature distribution.

Following the work by Sobha and Ramakrishna, the study of similarity
solutions of Prandtl's equation for    the steady two-dimensional
heat transfer past a vertical plate embedded in a porous medium with
an
applied magnetic field leads to the  differential equation (see Appendix)
\begin{equation}\label{eq:MHD}
  f''' +\frac{1+m}{2} ff''-m {f'}^2 = 0,
\end{equation}
subject to  the boundary wall conditions
\begin{equation}\label{eq:boundary1MHD}
f(0) = 0,\quad f'(0) = \omega ,
\end{equation}
and the boundary (free) condition
\begin{equation}\label{eq:boundary2MHD}
\lim_{\eta\to\infty}f'(\eta) =f'(\infty)=0.
\end{equation}
The parameter $ m $ is related to
the temperature distribution prescribed on the plate and $\omega $
is a magnetic parameter.  For a physical consideration $ m $ and
$\omega $ satisfy $ -\frac{1}{3} \leq  m \leq 1 $ and
$ 0 < \omega < 1$. In the absence of the magnetic field  the parameter
$ \omega $
is equal to $1$ (see for example \cite{BBT3}, \cite{ChM1}). However,
for the mathematical analysis we will be concerned with
$ - 1 \leq m < 0 $ and  with every value of $ \omega $.


Problem (\ref{eq:MHD})-(\ref{eq:boundary2MHD}) also  arises in
physically different contexts in fluid mechanics, as   boundary layer
flow on permeable surface with mass transfer parameter $ a\not=0$
\cite{ChMP}, \cite{MK}. In  this case initial conditions
(\ref{eq:boundary1MHD}) take the form
\begin{equation}\label{eq:boundary1}
f(0) = a,\quad f'(0) = 1.
\end{equation}
  The real number $a $ is also referred to as the suction/injection
parameter. The case $ a > 0 $ corresponds to suction and $ a < 0 $ to
injection of the fluid.  With $ m = 0 $ equation (\ref{eq:MHD}) is
called the Blasius equation \cite{Blas}.

Problem  (\ref{eq:MHD}), (\ref{eq:boundary2MHD}),
(\ref{eq:boundary1}) has been the subject of intensive study.
Results concerning problem (\ref{eq:MHD})--(\ref{eq:boundary2MHD})
can be found  in  \cite{ChM1} by Cheng and Minkowycz, for a
different physical problem,
in which the numerical solution has been performed in the case where
$ -\frac{1}{3} < m < 0. $  For  $ m > -\frac{1}{2} $
numerical investigations are given in the works \cite{IngB2} by
Ingham and Brown   and
\cite{B7bis} by Banks.  The mathematical analysis  is also
considered  in \cite{IngB2}. Some analytical results have been
obtained  by Belhachmi et al.
\cite{BBT3}. The authors  showed  non-existence of solutions to
(\ref{eq:MHD})--(\ref{eq:boundary2MHD}) for $
m \leq -\frac{1}{2}$. They also proved  that this problem has an
infinite number of solutions when $
m = -\frac{1}{3} $ and  uniqueness holds for
$  0 \leq m \leq \frac{1}{3}$.

Recently, multiple solutions of  (\ref{eq:MHD}),
(\ref{eq:boundary2MHD}),  (\ref{eq:boundary1})
were obtained by Guedda \cite{Gu}, for different values  $
-\frac{1}{3} < m < 0$. In particular, it is proved
that for any $ \tau > -\frac{m+1}{2}a $  the local solution to
(\ref{eq:MHD}), (\ref{eq:boundary2MHD}) such that $ f''(0) = \tau $
is global and satisfies
\begin{equation}\label{behaviour}
  f'(\infty) = 0,\quad  f(\eta) \sim
L\eta^{\frac{1+m}{1-m}},\end{equation}
   as $\eta \to \infty$, for some $ L > 0$. The case
$ -\frac{1}{2} < m < 0 $ is also  studied provided that $ a \geq
\sqrt{\frac{1}{m+1}}$.

In a recent  paper   \cite{BS} Brighi and Sari  have
conducted  a discussion of the existence and the non-existence of
solutions  to problem   (\ref{eq:MHD}), (\ref{eq:boundary2MHD}),
(\ref{eq:boundary1}), where the parameters $ a $ and $ m $ are taken
on the whole range $(-\infty , \infty)$. Using dynamical
system theories, the authors proved, among other results, that for $
0 \leq m \leq 1 $ and for any $ a \in \mathbb{R} $ problem
(\ref{eq:MHD}), (\ref{eq:boundary2MHD}),  (\ref{eq:boundary1}) has
one and only one solution while for $ m > 1 $ multiple solutions
exist included  one and only one concave solution. For
$ - 1 < m < -1/2$ the authors proved that there exists $a_\star^+ > 0 $
such that the problem has no solution for any $ a < a_\star^+$, while
for $ m < -1 $ there exists $a_\star^-  < 0 $ such that a  solution
exists if and only if  $ a < a_\star^-$.

Very recently,  asymptotic properties  of global unbounded  solutions
to a class of degenerate nonlinear differential boundary layer
equations are obtained by Guedda and Kersner \cite{GuKersner}. In
particular, it is proved that any global solution to  (\ref{eq:MHD}),
where $-1 < m < 0$, such that $  f(\infty) = \infty$, satisfies
(\ref{behaviour}).

Based on these previous results, we may  conclude (see below) that
for $ -1 < m < -1/2$ and $ a < a_\star^+ $ any local
solution to (\ref{eq:MHD}), (\ref{eq:boundary1}) blows up at a finite
point.

  The problem of the blowing--up solutions to boundary layer equations
was first mentioned by Coppel
\cite{Cop11}. The author classified all solutions of the Falkner-Scan
differential equation \cite{2FS}
\begin{equation}\label{eq:Cop}
f''' + ff''  + \beta(1-{f'}^2)= 0,
\end{equation}
where $ 0 \leq \beta < 2$. In particular, it is shown that for $ 0
\leq \beta <1/2$,  any blowing-up solution satisfies $
f'(\eta) \sim
-(2-\beta)f(\eta)^2/6 $ as $ \eta\to \eta_c, $ where $  0 < \eta_c <
\infty $ is the blow--up point of $ f$.

 The  initial value problem,  with $m=0$ or $ \beta = 0$,
\begin{equation}\label{eq:BBT}
\begin{gathered}
f''' + \frac{1}{2}ff'' = 0,\\
f(0) = a, \quad f'(0) = b, \quad f''(0) = \tau,
\end{gathered}
\end{equation}
where $ a\in \mathbb{R}$, $b > 0 $ and $ \tau \leq 0$, has been
considered by Belhachmi et al. \cite{BBTbis}. Among  other results,
it is shown that there
exists
$\tau^\star \leq 0 $ such that
the unique solution to the Blasius problem (\ref{eq:BBT}) is not
global, for any $ \tau < \tau^\star.$\\
Recently,  the absence of  global solutions  to
\begin{equation}\label{eq:BBTbis}
\begin{gathered}
f''' + ff'' = 0,\\
f(0) = 1, \quad f'(0) = 0,\quad  f''(0) = \tau,
\end{gathered}
\end{equation}
  has    been  reconsidered in detail  by  Ishimura and  Matsui
\cite{IM}. By introducing the function $ v $ such that $v(-f) =
{f'}^2$,  the authors
proved that for any $ \tau <0$,  the solution $ f $ to
 (\ref{eq:BBTbis}) blows up at a some point $ \eta_c= \eta_c(\tau)$,
and that the blow-up coordinate $ f'/f^2$  tends to $-1/3$ as $
\eta\to \eta_c$. Then they deduced that
\[
\lim_{\eta\uparrow \eta_c}(\eta_c-\eta)f(\eta) = -3.
\]
In this work we  extend the results of \cite{IM} to equation
(\ref{eq:MHD}). Since the case $ m = 0 $ were investigated, we will
suppose $ -1 \leq m < 0$.
Let us note that if $  m  > -1  $ the new  function
\[
\eta \mapsto \sqrt{\frac{m+1}{2}}f\Big(\sqrt{\frac{2}{ m+1}}\eta\Big)
\]
satisfies
\begin{equation}
f''' + ff'' -\beta{f'}^2 = 0,\end{equation}
where $ \beta = \frac{2m}{m+1}$.
If $ m=-1 $ or $ \kappa = 0 $  equation (\ref{eq:MHD}) reads
\begin{equation}\label{criticalcase}
f''' +{f'}^2 = 0.
\end{equation}
So, we shall be concerned with the ordinary differential equation
\begin{equation}\label{equ:general}
f''' + \kappa ff'' -\beta{f'}^2 = 0,
\end{equation}
where $ \beta < 0 $  and $ \kappa = 1 $ or $ \beta =-1 $ and
$ \kappa = 0$ ($m=-1$).  The initial conditions which we wish to
consider are
\begin{equation}\label{eqi:general}
f(0) =a,\quad f'(0) = \omega ,\quad f''(0) =  \tau,
\end{equation}
where $ a$, $\omega   $ are real numbers, $ \tau < 0 $ and $ \beta < 0$.


\section{Existence of singular solutions for $ \beta < 0$}

In this section we are interested in conditions for non-global
existence of solutions to (\ref{equ:general}), (\ref{eqi:general}).
First we note, according to a
standard theory of ODE, that problem  (\ref{equ:general}),
(\ref{eqi:general}) has  a unique
local solution $ f_\tau $
  defined on the maximal interval $
[0,\eta_c), \eta_c \leq \infty$.
This solution is of class
$ C^\infty $ on $[0,\eta_c)$ and satisfies
\begin{equation}\label{identity}
f_\tau''(\eta) + \kappa f_\tau(\eta)f_\tau'(\eta) = \tau + \kappa
a\omega  + (\kappa+\beta)\int_0^\eta f_\tau'(s)^2ds,
\end{equation}
for all $ \eta < \eta_c$. If, in addition, $ \eta_c $ is finite
$\lim_{\eta\uparrow \eta_c}\vert
f_\tau(\eta)\vert + \vert f_\tau'(\eta)\vert+ \vert
f_\tau''(\eta)\vert = \infty$.
In fact, following the work
\cite{Cop11},  the existence time $\eta_c$ is
characterized by the following result.

\begin{proposition} \label{prop2.1}
 Let $ f_\tau $ be the unique local solution to
\eqref{equ:general}, \eqref{eqi:general}, where $ \kappa \in
\{0,1\}, \beta < 0 $ and $ \tau \in \mathbb{R}$. Assume
$\eta_c < \infty$. Then
$$
\lim_{\eta\uparrow \eta_c}  f_\tau(\eta) = -\infty.
$$
\end{proposition}

\begin{proof}
First we show that $ \sup_{[0,\eta_c)}\vert f_\tau(\eta)
\vert = \infty$. We adapt an idea due to \cite{Cop11}.
 Suppose that this is not the case.  Assume that
$ \kappa\not=-\beta$. From (\ref{identity}) we deduce
\begin{equation}\label{Cop}
(\kappa+\beta)\big[f_\tau'(\eta) +
\frac{\kappa}{2}f_\tau^{2}(\eta) -(\tau + \kappa a\omega )\eta
- \omega -\frac{\kappa}{2}a^2\big]  =
(\kappa+\beta)^2\int_0^\eta\int_0^t{f_\tau'}(s)^2\,ds\,dt,
\end{equation}
for all $ \eta < \eta_c$.
Because the right--hand side of (\ref{Cop}) is positive and  monotonic the
left--hand side of (\ref{Cop}), and (therefore)  $
(\kappa+\beta)f_\tau'(\eta) $ tends to $ \infty
$  as $ \eta\to \eta_c.
$  Otherwise, $ f_\tau' $  is bounded and   by (\ref{identity}) $
f_\tau'' $ is also bounded, which is absurd. Consequently,
the function
\[
 v(\eta) = \int_0^\eta\int_0^t{f_\tau'}^2(s)\,ds\,dt
\]
goes to $ \infty $ as $ \eta \to  \eta_c $ and satisfies
\[
\lim_{\eta\to \eta_c}v''(\eta) = \infty
\]
and
\[
v'' \leq 2(\kappa+\beta)^2 v^2
\]
on $(\eta_c-\varepsilon,\eta_c), \varepsilon > 0 $ small. The last
differential inequality yields, for some  constant $ C_1 > 0$,
\[
v(\eta) \geq C_1(\eta_c-\eta)^{-2}
\]
as $ \eta\to \eta_c$.
Returning to (\ref{Cop}) we deduce
\[
(\kappa+\beta)f_\tau'(\eta) \geq C_2(\eta_c-\eta)^{-2},\quad
C_2= \text{const.} > 0,
\]
and this implies, after integration, that $ (\kappa+\beta)f_\tau(\eta)
$ is not bounded as
$ \eta\to \eta_c$, a contradiction.
Next we use  the equation of $ f_\tau $ to deduce
\begin{equation}\label{monotone}
(f_\tau''e^{\kappa F})' = \beta
e^{\kappa F} {f_\tau'}^2,
\end{equation}
where $  F(\eta) = \int_0^\eta f(s)ds, $ and (then)
$ f_\tau'' $
has at most one zero. Therefore, $ f_\tau $
  is monotonic on $ (\eta_c-\varepsilon,\eta_c),\ \varepsilon >  0 $
small enough, and then
  $ \vert f_\tau(\eta)\vert \to \infty $ as
$ \eta\to \eta_c$.

For  $ \kappa = -\beta$ and then $ \kappa=1$, we infer
\begin{gather*}
f_\tau'' +  f_\tau f_\tau' = \tau + a \omega , \\
f_\tau' + \frac{1}{2}f_\tau^2= (\tau + a \omega )\eta + \omega  +
\frac{1}{2}a^2.
\end{gather*}
Hence, if $ f_\tau $ is bounded we deduce that $ f_\tau' $ and
$f_\tau'' $ are bounded, a contradiction.

It remains to prove that $ f_\tau(\eta) $ approaches  $-\infty$ as $ \eta $
approaches $ \eta_c$. Because $ f_\tau $ is  monotonic  on some
$(\eta_0,\eta_c)$
we assume that $ f_\tau'(\eta) $ is nonnegative for all
$ \eta_0 < \eta < \eta_c $ and  $ f_\tau(\eta) \to \infty $ as
$ \eta\to\eta_c$.
Define the energy-type function
\begin{equation}\label{energyfunction}
  E = \frac{1}{2}{f_\tau''}^2 - \frac{\beta}{3}{f_\tau'}^3,
\end{equation}
which satisfies
\[ E' = -\kappa f_\tau {f_\tau''}^2.\]
Thus $ f_\tau' $ is bounded and then $  f_\tau  $ is also bounded
on $(\eta_0,\eta_c)$, which is impossible. Consequently, $ f_\tau(\eta)
$ goes to $-\infty $
as $ \eta \to \eta_c$.
\end{proof}

  The following result indicates that $ f_\tau $ has a singularity  for
any  $ \tau < 0$.

\begin{theorem} \label{thm2.1}
 Let $ \omega  \leq 0, a \in \mathbb{R}$. Assume that
$\kappa\in\{0,1\},  \beta < 0$.
For any $ \tau < 0 $
$ \eta_c  $ is finite  and the function $ f_\tau $ satisfies
\[
\lim_{\eta \uparrow \eta_c}f_\tau(\eta)=-\infty.
\]
\end{theorem}

\begin{proof}
First we assume that $ \omega  < 0$. We suppose for
the sake of contradiction that
$ f_\tau $ is global; that is $ \eta_c = \infty$. Because
$f'_\gamma(\eta)< \omega$,
 for all $  \eta > 0$, $ f_\tau(\eta) < a + \omega \eta  $  and
tends to $ -\infty $ as $ \eta \to\infty$.  Together with
(\ref{equ:general}) the
energy-type function $ E $ defined by (\ref{energyfunction}) is
monotonic increasing on $(\eta_0,\infty)$, where
$ \eta_0= \max\{0,-a/\omega \} $ and this infers
\[
f_\tau''(t)^2 \geq
\frac{2\beta}{3}\big(f_\tau'(\eta)^3 -{f'}(\eta_0)^3\big) + f_\tau''(\eta_0)^2,
\]
for all $ \eta \geq \eta_0$.  One readily verifies that
$f'_\gamma(\eta) $ tends to $ -\infty$. Now,  the function
$ g =-f_\tau $ is positive on $ (\eta_0,\infty)$,
monotonic increasing, goes to $ \infty $  with $ \eta $ and satisfies
\[
g''(\eta) \geq
\sqrt{\frac{\vert \beta\vert}{3}}g'(\eta)^{3/2},
\]
for large $ \eta$. A simple analysis of this inequality  implies that
$g' $ is not global. A
contradiction. Next assume that
$ \omega =0$.
  Because $ \tau < 0$ there exists a (small) real number $ \eta_0 > 0 $
such that     $ f_\tau'(\eta_0) $ and $f_\tau''(\eta_0) $  are negative.
The new function $ {\overline f}(\eta) =  f_\tau(\eta+\eta_0) $
is a solution to (\ref{equ:general})
which satisfies
${\overline f}'(0) < 0$ and $ {\overline f}''(0) < 0$.
   Hence  $ f_\tau $ is not global.
\end{proof}

The next result considers the case $ \omega  > 0$ and $ \beta < -2$
for $ \kappa = 1$. The case $ \kappa = 0 $ will be treated  in
detail in the next section. The condition $ \beta < -\frac{1}{2} $ is
plainly satisfied for $ -1 < m < -\frac{1}{2}$. According to
\cite{BS} there exists $ a_\star^+ > 0 $ such that  the problem
\begin{equation}\label{brighi}
\begin{gathered}
f''' + ff'' -\beta {f'}^2 = 0,\\
f(0) = a,\quad f'(0) = 1, \quad f'(\infty) = 0,
\end{gathered}
\end{equation}
has no solution for any $ a < a_\star^+$. On the other hand, it
should be noticed that if $ f $ is a solution to (\ref{equ:general})
then it is for  the function $ \eta \mapsto \gamma f(\gamma\eta)$,
for any $ \gamma > 0$. Consequently, problem (\ref{brighi}), with
$f'(0) = \omega  $ instead of $ f'(0) = 1 $  has no
solution for any $ a < \sqrt{\omega }a_\star^+$. Clearly, this
deduction and the results of \cite{GuKersner}  lead to  the following
result.

\begin{theorem} \label{thm2.2}
Let $ \omega  >  0$. Assume that $\beta <-2$ and
$ a <\sqrt{\omega }a_\star^+$.
For any $ \tau  $ the local solution $ f_\tau $ is not global
($\eta_c <\infty$) and satisfies
\[
\lim_{\eta \uparrow \eta_c}f_\tau(\eta)=-\infty.
 \]
\end{theorem}

Having proved that $ f_\tau $ blows up at a finite point (under
favorable conditions), we determine its precise asymptotic behavior,
closely following the analysis of \cite{IM}.

\section{Asymptotic behavior of blowing-up solutions}

The purpose  of this section is  to study  the asymptotic behavior of
any possible blowing-up solution to (\ref{equ:general}),  where
$\kappa \in \{0, 1\} $ and $ \beta < 0$
  are mainly assumed.

 \subsection{The limit case $ \beta = -\infty $ ($\kappa = 0$)}
  In this short subsection we examine the structure of solutions to
problem   (\ref{criticalcase}), (\ref{eqi:general}) for different
$\tau $ and $ \omega $. Solving  this problem is equivalent to finding a
solution  $ g\, (= f_\tau') $  to the following ODE
  \begin{equation}\label{kappazero}
  g'' + {g}^2 = 0,
  \end{equation}
  accompanied with the initial conditions
  \begin{equation}\label{kappazeroIC}
   g(0) = \omega ,\quad g'(0) = \tau.
  \end{equation}
  In this subsection, we assume that  the real numbers  $ \omega  $
and $ \tau $ take place on the whole $ \mathbb{R}$.
  In the phase plane $(g,g') $ the curve of the above problem are given by
  \begin{equation}\label{phaseplane}
  {g'}^2 +\frac{2}{3}g^3 =\gamma(\tau,\omega ),
\end{equation}
  where $\gamma(\tau,\omega ) =\tau^2 + \frac{2}{3}\omega ^3$, or
  \begin{equation}\label{phaseplanebis}
  g' = \pm\sqrt{ \gamma(\tau,\omega )-\frac{2}{3}g^3},
\end{equation}
as soon as  $ \gamma(\tau,\omega )\geq\frac{2}{3}g^3$.
  If $\gamma(\tau,\omega ) = 0$ the problem can be solved explicitly. In
this case $ \omega  \leq 0$. If $ \omega  = 0 $
we get $ g \equiv 0 $ and  for $  \omega  <  0 $  we deduce  from
(\ref{phaseplanebis}) that
\begin{equation}\label{explicitsolution}
g(\eta) = -\frac{6}{(\eta_c-\eta)^2},
\end{equation}
where
\[
\eta_c^{2} = \frac{6}{\vert\omega \vert}.
\]
Consequently, if $ \eta_c < 0$ ($\tau > 0$) the solution $ g $ is
global and tends to 0 as $ \eta $ goes to infinity and for
$ \eta_c >0$  ($\tau < 0$) the solution $ g $ is not global and tends
to $-\infty$  as $ \eta $ approaches  $\eta_c$.
For  $ \gamma(\tau,\omega )\not=0$, we assume first that  $ g $ is
global and that $ g' $ is positive for large $ \eta$. Recall
that $ g' $ is monotonic decreasing (see (\ref{kappazero})).
Since $ g $ is monotonic increasing
  we conclude that  $ g $ and $ g' $ are bounded,  there exists
a real number $ g_\infty $ such that $ g(\eta) $ tends to
$ g_\infty$ as $ \eta $ tends to infinity and  $ g'(\eta) $ tends to zero
as $ \eta $ tends to infinity and then $ g_\infty = 0 $ by using
again equation (\ref{kappazero}). This leads to
$ \gamma(\tau,\omega )= 0$, thanks to (\ref{phaseplane}). A contradiction.
Therefore, there exists $ \eta_0 \geq 0 $ such that $ g'(\eta) <0$ on
$(\eta_0,\infty)$. So, we may assume without lost of generality that
$ g'(\eta) < 0$ for all $\eta \geq 0$  ($\tau < 0$),  and
consider  equation (\ref{phaseplanebis}) with minus  instead of
$\pm$, which gives
\[
\int_{g(\eta)}^{\omega } \frac{ds}{\sqrt{\gamma(\tau,\omega
)-\frac{2}{3}s^3}}=\eta.
\]
A simple analysis of the above integral shows that $ g $ cannot be
global. This means that there exists  a real number
$ \eta_c =\eta_c(\tau,\omega )$, such that
$ \lim_{\eta \to \eta_c} g(\eta) = -\infty$. Moreover, the blow-up
point $ \eta_c $  is given by
\begin{equation}\label{etac}
  \int_{-\infty}^{\omega } \frac{ds}{\sqrt{\gamma(\tau,\omega
)-\frac{2}{3}s^3}}=\eta_c.\end{equation}
Next, one sees from (\ref{phaseplane}) that
\[
\lim_{\eta\to \eta_c} g'(\eta)^2g^{-3}(\eta) = -\frac{2}{3},
\]
and then
\[
\lim_{\eta\to \eta_c} (\eta_c-\eta)^2g(\eta) = -6.
\]
Returning to the original function $ f_\tau $ we summarize  the main
result of the present subsection  in the following.

(1) If $ \tau^2 = -\frac{2}{3}\omega ^3 $ and $ \tau > 0 $ the
solution $  f_\tau $ is global and given by
\[
f_\tau(\eta) = 6\frac{1}{\eta_c+\eta} + a - \frac{6}{\eta_c},
\]
where  $ \eta_c = \sqrt{6/\vert\omega \vert}$.

(2) If $ \tau^2 = -\frac{2}{3}\omega ^3 $ and $ \tau < 0$ the
solution $  f_\tau $ is  not global and given by
\begin{equation}\label{explicit}
  f_\tau(\eta) = -6\frac{1}{\eta_c-\eta} + a + \frac{6}{  \eta_c},\quad
\eta_c = \sqrt{6/\vert\omega \vert}.
\end{equation}

(3) If $ \tau^2 \not= -\frac{2}{3}\omega ^3 $ the solution
$f_\tau $ is  not global and satisfies
\begin{equation}\label{eq:3.8}
  \lim_{\eta \to \eta_c}(\eta_c-\eta)f_\tau(t)=-6, \end{equation}
where $ \eta_c > 0 $ is the blow-up point, which is  given by
(\ref{etac}), for $ \tau < 0$.

The above  results show, in particular, that  problem
(\ref{eq:MHD})-(\ref{eq:boundary2MHD})) has no non trivial
(similarity) solution for any  $ \omega  \geq 0$, if $ m = -1 $ even
if $ f(0)\not = 0$ \cite[pp. 244--246]{3Rosenhead}, \cite{3MKP}.

\subsection{The case $ \kappa = 1$}
We shall be concerned with  problem (\ref{equ:general}),
(\ref{eqi:general}), where $ \kappa = 1$ and $ \beta < 0$. Our
interest is in solutions which   develop a singularity. In fact, the
aim of the present subsection is to establish the asymptotic behavior
of any possible singular solution at its   blow-up point.
Let us note that if we look for a singular solution to
(\ref{equ:general}), where $ \kappa \in \{0, 1\} $ and
$\beta < 0$, under the form
\[
f^\star(\eta) = A(\eta_c-\eta)^{-\gamma},
\]
where $ A \not = 0, \eta_c > 0 $ and  $ \gamma > 0$,
we find that
$\gamma = 1$ and $A = 6/(\beta-2\kappa)$,

The main result  is as follows.

\begin{theorem} \label{thm3.1}
Let $ f $ be a solution to \eqref{equ:general},
where $ \kappa = 1 $ and $ \beta < 0$.
Assume that there exists a real number $ \eta_c $  such that $
\lim_{\eta \uparrow \eta_c}f(\eta) = -\infty$. Then
\begin{equation}\label{behavior}
f(\eta)\sim\frac{6}{\beta-2}\frac{1}{\eta_c-\eta},
\quad \text{as } \eta \uparrow \eta_c.
\end{equation}
\end{theorem}

The case $\beta=-1 $ is easy to analyze. Setting $ h  = f' +
\frac{1}{2}f^2 $ one sees, from (\ref{equ:general}),
$ h'' = (1+\beta){f'}^2$.
In such situation  the solution $ f $ satisfies
  the  Riccati equation
  \begin{equation}\label{Riccati}
  f'(\eta) + \frac{1}{2}f(\eta)^2 = \lambda \eta + \delta, \end{equation}
  where
  $ \lambda = f''(0) +  f(0)f'(0) $ and
$\delta=f'(0) + \frac{1}{2}f^2(0)$. Since $ f $ is not global we infer
\begin{equation}\label{limf}
\lim_{\eta\uparrow \eta_c} \frac{f'(\eta)}{f(\eta)^2}=-\frac{1}{2}.
\end{equation}
Finally, a simple integration of the above yields  (\ref{behavior})
with $ \beta =-1$.


To prove Theorem \ref{thm3.1} we use some modification and adaptation  of   an
idea used in \cite{IM} for the Blasius equation and introduced by
Toland \cite{To}.  To obtain (\ref{behavior}), equation (\ref{equ:general})
will be    reduced   to a  second order equation in which $ f $ is
regarded as an independent variable. Since
  $ f $ and $ f' $ are monotonic decreasing and tend to
$-\infty $ as $ \eta $ approaches $\eta_c$, there exists a real
number $  0 \leq \eta_0 < \eta_c $ such
that  $ f $ and $ f' $ are negative on  $ (\eta_0, \eta_c)$.
Without loss of generality we may assume that
$ f(\eta_0)= 0, f''(\eta_0) < 0$. Defining
   \[
x=-f,\quad v(x)= f'(\eta(x))^2.
\]
and using (\ref{equ:general}) we arrive at the second order
differential equation
\begin{equation}\label{eq:v}
v''(x) = -2\beta\sqrt{v(x)} +  x\frac{v'(x)}{\sqrt{v(x)}},\quad x > 0.
\end{equation}
The initial condition is given by
\begin{equation}\label{eq:cv}
  v(0) =f'(\eta_0)^2 > 0,\  v'(0) = -2f''(\eta_0) > 0.
\end{equation}
Setting
\[
w(s) = \frac{v(x)}{x^4},\quad x = e^s,\, x \geq x_0,
\]
for $ x_0$ large, equation (\ref{eq:v}) becomes
\begin{equation}\label{eq:w}
w'' + 7 w' + 12 w -2(2-\beta)\sqrt{w} -  w' w^{-1/2}=0.
\end{equation}
Therefore, in the remainder of this section we   study  equations
(\ref{eq:v}) and (\ref{eq:w}). We shall see that  the solution $ v $
to
(\ref{eq:v}),(\ref{eq:cv}) is global, equation (\ref{eq:w})  is
satisfied on some $(s_0,\infty) $ and
$ w(s)$ goes to $\frac{(2-\beta)^2}{36}$ as
$ s$ tends to infinity, which leads to
\begin{equation}\label{limit}
\lim_{\eta\to \eta_c}\frac{f'(\eta)}{f(\eta)^2} =
\frac{2-\beta}{6},\end{equation}
  and then  (\ref{behavior}) is satisfied. We start with  simple results
which are crucial for the proof. We distinguish between the cases $ 1
+ \beta\geq  0 $ and $1 + \beta <  0$.

\begin{lemma} \label{lem3.1}
Let $ v $ be the solution to {\rm (\ref{eq:v}),
(\ref{eq:cv})} where $ -1 \leq \beta <   0$. Then $ v $ is global,
increasing  and tends to infinity with $x$. Moreover, there exists
$x_1 > 0$ (large) such that the following
\begin{equation}\label{eq:estimate2.9}
\sqrt{v(x)} \leq \frac{3}{2} x^2,
\end{equation}
holds for all $ x $ in $ (x_1,\infty)$.
\end{lemma}

\begin{proof}
Since $ v(0) $ and $ v'(0) $ are positive (see
(\ref{eq:cv})), there exists  an $ \varepsilon > 0 $ such that
$v(x) > 0, v'(x) > 0 $ for all
  $ 0\leq  x \leq \varepsilon$. Assume that there exists
$\varepsilon_0 > \varepsilon $ such that $ v' > 0 $ on
$[0,\varepsilon_0) $ and $ v'(\varepsilon_0) = 0$.
Hence $ v $ is positive on $[0,\varepsilon_0]$ and, using (\ref{eq:v}),
 $v''(x) > 0$, for all $x \in [0,\varepsilon_0]$. Therefore,
$v' $ is monotonic increasing on $ [0,\varepsilon_0]$  and
positive on $ [0,\varepsilon_0)$, which contradicts the hypothesis
$v'(\varepsilon_0) = 0$. Hence,
\[
v > 0,\quad v' > 0,\quad v'' > 0,
\]
as long as $ v $ exists. Note that if $ v $ is global,  necessarily
$v(x) $ tends to infinity with $x$.

To show that $ v $ is global and satisfies  (\ref{eq:estimate2.9}) we
put
\[
H(x) = v'(x) - 2 x\sqrt{v(x)},\quad \forall\ x \in [0,x_c).
\]
Hence
\[
H'(x) = -2(1 + \beta)\sqrt{v(x)} \leq 0,
\]
and then
\begin{equation}\label{estimatevprime}
\begin{gathered}
  v'(x) \leq 2 x\sqrt{v(x)} + v'(0), \\
(\sqrt{v(x)})'  \leq  x + \frac{v'(0)}{2\sqrt{v(0)}},
\end{gathered}
\end{equation}
since $ v(x) \geq v(0)$. Integrating the last inequality over $(0,x) $
leads to
\[
\sqrt{v(x)} \leq  \frac{1}{2}x^2 + \frac{xv'(0)}{2\sqrt{v(0)}} + \sqrt{v(0)}.
\]
Using this and (\ref{estimatevprime}) we deduce that $ v $ is global
and estimate (\ref{eq:estimate2.9}) is satisfied.
\end{proof}

The following result gives the lower bound of $ \sqrt{v}/x^2 $ for $ x $ large.

\begin{lemma} \label{lem3.2}
 Let $ v $ be the solution to \eqref{eq:v},
\eqref{eq:cv} where $ -1\leq \beta < 0$. Then  there exists
$ x_2 >0$, large,
such that
\begin{equation}\label{eq:estimate}
\sqrt{v(x)} \geq \frac{1}{12}(2-3\beta)x^2,
\end{equation}
holds for all $ x $ in $ (x_2,\infty)$.
\end{lemma}

\begin{proof}
Let $G(x)=5v(x)-3xv'(x)$, for $x \geq x_1$,
where $ x_1 $ is given by Lemma \ref{lem3.1}.
We have
\[
G'(x) = 2v'(x)\big[1-\frac{3}{2}\frac{x^2}{\sqrt{v(x)}}\big] +
6\beta x\sqrt{v(x)},
\]
thanks to (\ref{eq:v}). It follows from Lemma \ref{lem3.1} that
$ G'(x) \leq 0 $ for all $ x \geq x_1$, and then
\begin{gather*}
5v(x) -3xv'(x) \leq  5v(x_1) -3x_1v'(x_1), \quad \forall  x \geq x_1,
\\
4v(x) -3xv'(x) \leq  5v(x_1) -3x_1v'(x_1)-v(x), \quad
\forall  x \geq x_1.
\end{gather*}
Since $ v(x) $ tends to infinity with $ x $ we deduce that there
exists $ x_3$, large such that
\begin{equation}\label{eq:vvprime}
4v(x) \leq 3xv'(x),\quad \forall  x  \geq x_3.
\end{equation}
On the other hand, the new function
\[
V(x) = v'(x) - \frac{2}{5}(2-3\beta)x\sqrt{v(x)}
\]
satisfies
\[
V'(x) =
2\frac{1+\beta}{5}\big[\frac{3}{2}x\frac{v'(x)}{\sqrt{v(x)}}-2\sqrt{v(x)}\big].
\]
  Due to  (\ref{eq:vvprime}) we have $ V'(x) \geq 0 $ for all
$x \geq x_3$. Hence
\[
v'(x) - \frac{2}{5}(2-3\beta)x\sqrt{v(x)} \geq v'(x_3)
- \frac{2}{5}(2-3\beta)x_2\sqrt{v(x_3)},
\]
which leads to (\ref{eq:estimate}).
\end{proof}

Next we consider the case $ 1+\beta < 0$.

\begin{lemma} \label{lem3.3}
Let $ v $ be the solution to \eqref{eq:v}),
\eqref{eq:cv} where  $ 1 + \beta < 0$. Then $ v $ is global,
increasing  and tends to infinity with $x. $ Moreover, the following
\begin{equation}\label{eq:estimat}
\sqrt{v(x)} \geq \frac{1}{2}x^2,
\end{equation}
holds for all $ x\geq 0$.
\end{lemma}

\begin{proof}
Arguing as in the proof of Lemma \ref{lem3.2} the function  $ v$ is increasing
and tends to infinity with $ x $ if $ x_c = \infty$.
To prove that $v $ is global we show first that estimate (\ref{eq:estimat})
holds on $ (0,x_c)$. Using again the function
$ H(x) = v'(x) - 2 x\sqrt{v(x)} $ and
the ODE satisfied by  $ v $ to deduce that $ H'(x) > 0 $ for
all $ 0 \leq x < x_c$. Hence
\begin{equation}\label{eq:estimatbis}
v'(x) - 2 x\sqrt{v(x)} \geq  v'(0) \geq 0,
\end{equation}
and then
$(\sqrt{v(x)})' \geq  x$,
which leads to (\ref{eq:estimat}). It remains to prove that $ v $ is
global. To this end we use the equation of $ v $  and estimates
(\ref{eq:estimat}), (\ref{eq:estimatbis}) to get
\begin{gather*}
\frac{v''(x)}{v'(x)}= -2\beta
\frac{\sqrt{v(x)}}{v'(x)} + \frac{x}{\sqrt{v(x)}},
\\
 \frac{v''(x)}{v'(x)} \leq  (\vert \beta\vert +
2)\frac{1}{x},
\end{gather*}
for all $ x \in (0,x_c)$. Integrating the above inequality over
$(x_0,x)$, $x_0 > 0$, yields
\[
v'(x) \leq v'(x_0)\left(\frac{x}{x_0}\right)^{\vert
\beta\vert+2}.
\]
  Hence $ v $ is global.
\end{proof}


\begin{remark} \label{rmk3.2} \rm
Lemmas \ref{lem3.1} and \ref{lem3.2} indicate, in particular, that if
$ 1+\beta \geq 0$  the function
$ s \to w(s) $ is uniformly bounded on
$ (s_1,\infty) $ for some $ s_1  $ large.
\end{remark}

\begin{corollary} \label{coro3.1}
  Set
$\Gamma = \inf\{\frac{1}{2},\frac{2-3\beta}{12}\}$.
Let $ v $ be the  global solution to \eqref{eq:v},
\eqref{eq:cv} where $  \beta < 0$. Then there exists
$ x_0 > 0 $ large such that
\begin{equation}\label{finalestimate}
\frac{\sqrt{v(x)}}{x^2} \geq \Gamma,
\end{equation}
for all $ x \geq x_0$.
\end{corollary}

\begin{remark} \label{rmk3.3} \rm
Estimate  \eqref{finalestimate} can also be used for
proving that the existence interval $(0,\eta_c)$ of \eqref{equ:general}
is bounded. In view of
$ \eta'(x) = \frac{1}{\sqrt{v(x)}} $ we have
\begin{gather*}
\eta_c = \eta(x_0) + \int_{x_0}^\infty \frac{dx}{\sqrt{v(x)}},\\
\eta_c \leq \eta(x_0) + \frac{1}{\Gamma}\int_{x_0}^\infty \frac{dx}{x^2};
\end{gather*}
therefore, $  \eta_c $ is bounded.
\end{remark}

Next, we examine the limit of $ w(s)$ as $ s\to \infty$. We note first that
\begin{equation}\label{lew}
w(s) \geq \Gamma^2,\quad s \geq s_0.
\end{equation}
The proof of Theorem \ref{thm3.1} is an immediate consequence of the following
lemma which is our final result.

\begin{lemma} \label{lem3.4}
Let $ v $ be the global solution to \eqref{eq:v}),
\eqref{eq:cv}, where $ \beta < 0$.  Then
\[
\lim_{x\to\infty} \frac{\sqrt{v(x)}}{x^2} = \frac{2-\beta}{6}.
\]
\end{lemma}

\begin{proof} The proof of this lemma will amount to proving that
\begin{equation}\label{limw}
  \lim_{s\to\infty} w(s) = \frac{(2-\beta)^2}{36}.
\end{equation}
The proof of (\ref{limw}) is short and different from the one given
in \cite{IM}. By (\ref{eq:w})  the function
\[
I(s) = \frac{1}{2}w'(s)^2 +6w(s)^2-\frac{4(2-\beta)}{3}w(s)^{3/2}
\]
satisfies
\[
I'(s) = -7w'(s)^2w(s)^{-1/2}\big[\sqrt{w(s)}-\frac{1}{7}\big].
\]
Therefore, $ I'(s) \leq 0 $, for all $s \geq s_0$, thanks to
(\ref{lew}) and the definition of $\Gamma$. It follows from this
that $ w $ and then $ w' $ are  bounded.
Because
\[
0 \leq { M} w'(s)^2 \leq -I'(s),
\]
where $ M = 7[1-\frac{1}{\Gamma}\frac{1}{7}] > 0$, we deduce that
$w' $ is square integrable. Using again equation (\ref{eq:w})
one sees that $ w'' $ is also bounded. Now, we use the identity
\[
w'(s)^3 = w'(s_0)^3 + 3\int_{s_0}^s w'(\tau)^2w''(\tau)d\tau,
\]
to show that $ w'(s) $ has a finite limit as $ s\to \infty $
and this limit is zero, since $ w' $ is square
integrable.
Next we get, by differentiating (\ref{eq:w}),
\[
w'''+7w'' + 12w' - (2-\beta)w' w^{-1/2} -
w'' w^{-1/2} + \frac{1}{2}{w'}^2w^{-3/2}=0.
\]
Hence $ w''' $ is  bounded and we have
\begin{align*}
\int_{s_0}^s w'''(r)w'(r)dr
&=-\frac{7}{2}\left(w'(s)^2-w'(s_0)^2\right)
-\int_{s_0}^sw'(r)^2\left(12-(2-\beta)w(r)^{-1/2}\right)dr\\
&\quad +\int_{s_0}^s{w''(r)w'(r)}{w(r)^{-1/2}}dr-\frac{1}{2}
\int_{s_0}^s{w'(r)^3}{w(r)^{-3/2}}dr.
\end{align*}
Therefore, the integral
$ \int_{s_0}^\infty w'''(r)w'(r)dr  $ is finite,  and then an integration
by parts shows that $ w'' $ is square
integrable. As a consequence, the previous equality implies that
$w'' $  tend to 0 at infinity. Finally, we deduce from (\ref{eq:w})
and (\ref{lew}),
\[
\vert 1-\frac{6}{2-\beta}\sqrt{w}\vert \leq
\frac{1}{2(2-\beta)}\frac{\vert w'\vert}{\Gamma^2}
+\frac{1}{2\Gamma(2-\beta)} \vert w'' + 7w'\vert,
\]
and  get  (\ref{limw}). The proof is completed.
\end{proof}

\section{Appendix: Mathematical modelling}

The materials presented here are based on  many references. For example  the
works
\cite{4NB} by Bejan  and Nield,  \cite{2Woo} by Wooding and \cite{SR}
by Sobha and Ramakrishna.
  The  starting point is the boundary layer system
 \begin{gather}\label{3eq:1.1}
   \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} =
0,\quad u\frac{\partial T}{\partial x} + v
\frac{\partial T}{\partial y} = \alpha\frac{\partial^2 T}{\partial
y^2}, \\
\label{3eq:1.2}
\frac{\partial p}{\partial x}+ g\rho + \mu k^{-1}u + \sigma B_0^2 u =
0,\quad  \frac{\partial p}{\partial y} + \mu k^{-1}v + \sigma B_0^2 v
= 0,
\end{gather}
where $ u,v $  are the velocity components,  describes the
2D stationary heat convection and $ T $ is the temperature of the
fluid. The constants
$ \mu, k, \alpha,g, \sigma $ and $ B_0 $ are, respectively,
viscosity, permeability, thermal
diffusivity, gravitational acceleration, the electric conductivity and
applied magnetic field. The unknown functions $  p $ and $ \rho $
are, respectively, is the pressure and $ \rho $ is  the
$ T-$dependent density, defined from  the Boussinesq approximation \cite{2Woo}
\begin{equation}\label{Boussinesq}
  \rho = \rho_0(1-\beta_1(T - T_0)),
\end{equation}
where $ \rho_0 $ is the density  at a reference temperature $ T_0$,
and $ \beta_1 $ is a constant. Usually, the reference temperature is $
T_\infty;$ the temperature far from the plate and then $ \rho_0 =
\rho_\infty $  is the value of $ \rho $ far from the plate, the
reference density
\cite{2Schl}.

  The wall temperature distribution is assumed to be  a power function
of the distance from the origin;
\[
T_w(x)= T_0 + Ax^m,
\]
where $ A > 0 $ is a constant and $ m $ is a real number. The
boundary conditions are
\begin{equation}\label{3eq:wvboundary}
v(x,0) = 0,\quad u(x,0) = u_wx^m,\quad T(x,0) =T_w(x),\end{equation}
and
\begin{equation}\label{3eq:Tinftyboundary}
T(x,\infty) =T_\infty,\quad u(x,\infty) = 0.
\end{equation}
The above model can be expressed in a simpler form by
introducing the stream function
$\psi (u=\frac{\partial\psi}{\partial y},\,
v=-\frac{\partial\psi}{\partial x})$  and applying boundary
approximations. PDEs (\ref{3eq:1.1}), (\ref{3eq:1.2}) are reduced to
\begin{gather}\label{3eq:1.7}
(\frac{\mu}{k}+ \sigma B_0^2)\frac{\partial^2
\psi}{\partial y^2} = \rho_0 g\beta_1\frac{\partial T}{\partial
y},\\
\label{3eq:1.8}
\alpha\frac{\partial^2 T}{\partial
y^2} =  \frac{\partial T}{\partial x}\frac{\partial \psi}{\partial
y}-\frac{\partial T}{\partial y}\frac{\partial \psi}{\partial x}.
\end{gather}
We then perform the similarity transformations in the usual way,
\begin{gather}\label{3eq:similarityvariable}
\eta = \sqrt{R_{a_x}}\frac{y}{x}, \\
\psi(x,y) = \alpha\sqrt{R_{a_x}}f(\eta),\quad T(x,y) = T_0 +
(T_w-T_0)\theta(\eta),
\end{gather}
where
\begin{equation}\label{3eq:Rnumbers}
R_{a_x}=\frac{\beta_1 g\rho_0 k(T_w-T_0)x}{\alpha\mu},
\end{equation}
is the   modified local Rayleigh number in a porous medium. Equations
(\ref{3eq:1.7}) and (\ref{3eq:1.8}) now reduce to
\begin{gather}\label{3eq:1.11}
f'' =\omega \theta',\quad \omega  =\frac{M^2}{M^2+N^2}, \\
\label{eq:1.13}
\theta'' + \frac{1+m}{2}f\theta' -mf'\theta=0,
\end{gather}
where
$ M^2 = \frac{1}{k}$ and $ N^2 = \frac{\sigma B_0^2}{\mu} $ is the
magnetic parameter.

  The boundary conditions read
\begin{gather*}
 f(0) = 0,\quad f'(0)=\omega ,\quad  \theta(0) = 1,\\
 \theta(\eta)\to 0,\quad f'(\eta)\to 0,
\end{gather*}
as $ \eta \to \infty$.
Therefore we get
\begin{gather}\label{ftheta}
  f' = \omega \theta, \\
\label{eq:1.14}
\begin{gathered}
f''' + \frac{1+m}{2}ff'' -m{f'}^2=0,\\
f(0) =0,\quad f'(0) = \omega ,
\end{gathered} \\
f'(\eta)\to 0,\quad\mbox{as } \eta\to \infty. \label{e4.39}
\end{gather}
Note that equation (\ref{ftheta}) can also be obtained from the
wall condition
\[
u_w = \omega \beta_1 g\rho_0kA,
\]
and condition \eqref{e4.39} can be replaced by
$f''(0) = \tau$,
where  the real number $ \tau $  has a physical meaning, since the
local Nusselt number, $ Nu_x$, is given by
(in the usual dimensionless form)
\[
Nu_x=-\sqrt{Ra_x}\theta'(0) =-\sqrt{Ra_x}\frac{\tau}{\omega }.
\]

\subsection*{Acknowledgments}
 The authors would like to thank  Robert Kersner for his stimulating
 discussions  and to the referees for their careful  reading of the
 original manuscript and for making constructive suggestions,
 which improved the presentation of this article.
 This work was supported  by grant PAI 05/MA/116  from the French-Morroco
 Scientific Cooperation  Project.

\begin{thebibliography}{00}

\bibitem{3And} {Andersson H.I.,} \emph{MHD flow of a viscous fluid past a
stretching surface,} Acta Mechanica, 95 (1992) 227--230.

\bibitem{B7bis}
{Banks W. H. H.,} \emph{Similarity solutions of the boundary layer
equations for a stretching wall,}  J. de
M\'ecan. Th\'eo. et Appl. {2} (1983)  375--392.

\bibitem{BZ4}
{Banks W.H.H.} \& {Zaturska M. B.,} \emph{Eigensolutions in
boundary-layer  flow adjacent to a stretching wall,}
IMA Journal of Appl. Math.  {36} (1986)   375--392.

  \bibitem{4NB}
{Bejan A. \&  Nield D. A.,}  \emph{Convection in Porous Media}, second
ed., Springer, New York, 1999.

\bibitem{BBTbis}
  {Belhachmi Z., Brighi B. \& Taous K.,} \emph{On the concave
solutions of the Blasius equation,}  Acta Math. Univ. Comenian 69 (2)
     (2000) 199--214.


\bibitem{BBT3}
{Belhachmi Z., Brighi B. \& Taous K.,} \emph{On a family of
differential equation for boundary layer approximations
in porous media, } Euro. Jnl. Appl. Math. {12} (2001)  513--528.

  \bibitem{Blas}
  {Blasius H.,} \emph{Grenzchichten in Flussigkeiten mit kleiner
Reibung,} Z. math. Phys. 56 (1908)  1--37.

  \bibitem{BS}
  {Brighi B. \& Sari T.,} \emph{Blowing-up coordinates for a
similarity boundary layer equation,}
  Discrete and Continuous Dynamical Systems, vol. 12 (2005) 5  929-948.

  \bibitem{3ChG}
{Chakrabarti A. \& Gupta A.S.,} \emph{Hydromagnetic flow and heat
transfer over a stretching sheet,} Quart. Appl. Math. 37 (1979)
73--78.

\bibitem{ChMP}
{Chaudary M. A., Merkin J. H. \& Pop I.,} \emph{Similarity
solutions in free convection boundary layer adjoint to vertical
permeable surfaces in porous media. I. Prescribed surface
temperature,} Eur. J. Mech. B-Fluids  14 (1995) 217--237.

\bibitem{ChM1}
   {Cheng P. \& Minkowycz W. J.,} \emph{Free--convection about a vertical
flat plate embedded in a porous medium with
application to heat transfer from a dike,} J. Geophys. Res. { 82}
(14)(1977)  2040--2044.

\bibitem{Cop11}
{Coppel W. A.,} \emph{On a differential equation of boundary layer
theory,} Phil. Trans. Roy. Soc. London, Ser. A { 253} (1960)
101--136.

\bibitem{2FS}
{Falkner V. M. } \&  {Skan S. W.,} \emph{Solutions of the
boundary layer equations, } Phil. Mag. 12 (1931) 865--896.


\bibitem{Gu}
{Guedda M., } \emph{Similarity  solutions to differential
equations for boundary--layer approximations in  porous media,}
ZAMP, J. Appl. Math. Phy. 56 (2005) 749--762.

\bibitem{GuKersner}
{Guedda M. \& Kersner R.,} \emph{Asymptotic behavior of  the
unbounded solutions to some degenerate boundary layer equations
revisited,}  Archiv der Mathematik, to appear.

\bibitem{Gupta}
{Gupta P. S. \& Gupta A. S.,} \emph{Heat and mass transfer on a
stretching shet with suction or blowing,} Can. J. Chem. Eng. 55
(1977) 744--746.

\bibitem{IngB2}
{Ingham D. B. \& Brown S. N., } \emph{Flow past a suddenly heated
vertical plate  in a porous medium,}  J.
Proc. R. Soc. Lond. A  {403} (1986)  51--80.

\bibitem{IM}
  {Ishimura N.} \&  {Matsui S.,} \emph{On blowing-up solutions
of the Blasius equation,}  Discrete Contin. Dyn.
     Syst. 9 (2003) no. 4 985--992.

     \bibitem{3KTN}
{Kumari M., Takhar H.S. \& Nath G.,} \emph{MHD flow and heat
transfer over a stretching surface with prescribed wall temperature
or heat flux,} Warm und
Stoffubert, 25 (1990) 331--336.

\bibitem {MK}
{Magyari E.\& Keller B,} \emph{Exact solutions for self-similar
boundary layer flows induced by permeable stretching walls,} Eur. J.
Mech. B-Fluids  19 (2000) 109--122.

\bibitem {3MKP}
{Magyari E., Pop I. \&  Keller B., } \emph{The ``missing"
self-similar free convection boundary-layer flow over vertical
permeable surface in a porous medium,}
Transp. Porous Media 46 (2002) 91--2002.

\bibitem{3Pav}
{Pavlov K. B., } \emph{Magnetohydrodynamic flow of an
incompressible viscous fluid caused by deformation of a surface,}
Magnitnaya Gidrodinamika, 4 (1974) 146--147.

\bibitem{3PN}
{Pop I. \& Na T. Y.,} \emph{A note on MHD flow over a stretching
permeable surface,} Mech. Res. Comm. vol 25 No 3 (1998) 263--269.

\bibitem {3Rosenhead}
{Rosenhead L.} (ed.), \emph{Laminar Boundary Layers,} {Calderon
Press} 1963, Oxford.


\bibitem{2Schl}
{Schlichting H.} \& {Gersten K.,}  \emph{Boundary layer
theory,} 8th Revised and Enlarged Ed., Springer--Verlag Berlin Heidelberg 2000.


\bibitem{SR}
{Sobha V. V. \& Ramakrishna K.,} \emph{Convective heat transfer
past a vertical plate embedded in porous medium with an applied
magnetic field,} I. E.
Journal--MC vol 84 (2003) 131--134.

\bibitem{3TAG}
{Takhar H. S., Ali M. A. \& Gupta A. S.,} \emph{Stability of
magnetohydrodynamic flow over a stretching sheet,} In: Liquid Metal
Hydrodynamics
(Lielpeteris and Moreau eds.), 465--471, Kluwer Academic Publishers,
Dordrecht, 1989.

\bibitem{3TRP}
{Takhar H. S., Raptis A.  \& Perdikis C.,} \emph{MHD  asymmetric
flow past a semi-infinite moving plate,} Acta Mechanica, 65 (1987)
287--290.


\bibitem{To}
{Toland J. F.,}  \emph{Existence and uniqueness of heteroclinic
orbits for the equation $\lambda u'''+u'=f(u)$,} Proc.
     Roy. Soc. Edinburgh Sect. A 109  no. 1-2 (1988) 23--36.

\bibitem{3Vaj}
{Vajravelu K.,} \emph{Hydromagnetic flow and heat transfer over a
continuous moving, porous surface,} Acta Mechanica, 64 (1986)
179--185.

\bibitem{3WP}
{Watanabe T. \& Pop I.,} \emph{Hall effects on magnetohydrodynamic
boundary layer flow over a continuous moving flat plate,} Acta
Mechanica, 108 (1995)
35--47.

\bibitem{3Wong}
{Wong J. S. W.,} \emph{On the generalized Emden--Fowler
equations,} SIAM Rev 17 (1975) 339--360.


\bibitem{2Woo}
{Wooding R.A.,} \emph{Convection in a saturated porous medium at
large Rayleigh number or P\'eclet number,} J. Fluid
Mech. 15 (1963)  527--545.

\end{thebibliography}

\end{document}