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3-4press-grad.tex
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\section{The effects of pressure gradient}
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Let us follow the incoming flow from infinity A towards the    stagnation point O, and then along the body.    The flow first decelerates from A to O, accelrates from O to E, then decelerates again  from E to F. In  the inviscid flow outside the boundary layer, the pressure variation can be estimated by Bernoulli's theorem. The pressure must increase from A to O,  decrease from O to E and decrease from E to F again as shown in 
Figure \ref{fig:bluntbody}.
\begin{figure} [h]
\begin{center}
\includegraphics[scale=1]{3-5-1.eps}
\end{center}
\caption{Pressure variation along the surface of a a blunt body}. 
\label{fig:bluntbody}
\end{figure}
Inside the boundary layer the pressure variation must be the same.

 Let us denote the tangential coordinate by $x$ and the transverse coordinate by $y$, and recall the boundary layer equation
\be uu_x+vu_y=UU_x+\nu u_{yy}\label{3-BLeqn}\ee
On the body surface, the no slip condition leads to 
\be UU_x+\nu u_{yy} =0\ee
which  implies that
\be   u_{yy} >0 , ~~\mbox{if} ~~U_x<0, ~~~\mbox{deceleration}\label{decel}\ee
\be   u_{yy} =0 , ~~\mbox{if} ~~U_x=0, ~~~\mbox{no acceleration}\label{no-accel}\ee
\be  u_{yy} <0 , ~~\mbox{if} ~~U_x>0, ~~~\mbox{acceleration}\label{accel}\ee
Let us differentiate   (\ref{3-BLeqn}), 
\[ uu_{xy}+vu_{yy}+u_y(U_x+v_y)= \nu u_{yyy}\]
On the body surface the no slip condition  and continuity require  that
\[ u_{yyy}=0\]
therefore the curvature of the velocity profile $u_{yy}$ is an  extremum.
Now let us examine the implied profiles of shear and velocity by integration, sketched in  
Figure \ref{fig:press-grad}. 
In the stage of acceleration,  $u_{yy}<0 $, hence $u_y$ (shear) decreases in $y$. 
At  the point of no acceleration,  $u_{yy}=0 $, hence $u_y$ (shear) must have a negative maxmum at some height $y>0$. 
In the stage of deceleration,  $u_{yy}>0 $, there must be a point where $u_{yy}=0$ at some ehight $y=0$ where    shear  is the greatest,  $u$ has a point of inflection. 

If the flow external flow further decreases, the shear stress at the boundary vanishes,  the velocity profile becomes  tangential to the y axis. Still further deceleration cause flow reversal. The flow separates!!
   \begin{figure} [h]
\begin{center} \includegraphics[scale=1]{press-grad-BL.eps}\end{center}
\caption{Effects of pressure gradient on velocity shear.}. 
\label{fig:press-grad}
\end{figure}

\noindent{\bf Remark: Pressure gradient and vorticity}:
 As another  physical insight, let us consider the same  flow arond the blunt body, see 
Figure \ref{fig:vort-blunt},
 \begin{figure} [h]
\begin{center} \includegraphics[scale=1]{vort-blunt.eps}\end{center}
\caption{Vorticity and pressure gradient }. 
\label{fig:vort-blunt}
\end{figure}
Since the vorticity in the boundary layer is dominated by $\zeta=u_y$, the total vorticity in  the boundary  layer is
\[ \int_0^\delta \zeta dy = \int_0^\delta u_y\,  dy= U(y=\delta) \]
which is the inviscid velocity at the outer edge of the boundary layer. 
Now the total rate of flux of   vorticity is 
\[ \int_0^\delta u \zeta dy=\int_0^\delta uu_y\,  dy= \f{U^2}{2} \]
hence the mean speed of vorticity transport is $U/2$. The spatial variation of vorticity transport is 
\[\f{d}{dx}\lp \f{U^2}{2}\rp=UU_x=-\f{1}{\rho}p_x\]
This vorticity increase  or decrease, which is forced by the external pressure gradient, must come from the body surface. Thus $UU_x$ is the vorticity source strength per unit length of the surface. From O to A, $U_x>0$ vorticity is generated. From A to B, $U_x<0$, vorticity is destroyed. 

 
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