Consider the following quote:
"... The velocity distribution outside the boundary layer can be determined from the spacing of the streamlines. Since there can be no flow across streamlines, we would expect the flow velocity to increase in regions where the spacing between streamlines decreases. Conversely, an increase in streamline spacing implies a decrease in flow velocity."
"... For inviscid flow, an increase in velocity is accompanied by a decrease in pressure; Conversely, a decrease in velocity is accompanied by an increase in pressure." (pgs.36-37 [1])
Now, let me clarify a few definitions: A velocity distribution is in this context, a time-dependent vector field; Inviscid means non-viscous; A boundary layer develops in a neighborhood of a solid surface due to the "no-slip condition"; And a streamline is a time-dependent curve that, at each instant, is everywhere tangent to the velocity distribution.
To put this last one in math language:
$$\color{blue}{\forall t:\forall s:[\partial_s\gamma_{stream,p_0}(s,t) = \partial_t\theta(\gamma(s,t),t) = X_{\gamma(s,t)}(t)]}$$
The parameter-derivative of the streamline through $p_0$ at $\gamma(s,t)$ equals the flow velocity at that point.
*Note that the above is my own notation adapted from differential geometry notation for flows and vector fields. This definition differs from that of say wiki (but is similar). The text I'm working from doesn't articulate.
Question:
Can we prove the statements in the block quote using this definition in blue and the definition of pressure as Force/Area?
[1] Fox, Robert W.; McDonald, Alan T., Introduction to fluid mechanics., New York, NY: Wiley. xvi, 781 p. (1994). ZBL0812.76001.)
This relationship can be observed for internal flows (such as through a pipe) by way of conservation of mass. If a fluid is incompressible, forcing in fluid one end will force fluid out the other in equal amounts. Since mass is related to volume by density and density is assumed constant, we have: $$m_{in}/V_{in}= \rho_{constant} = m_{out}/V_{out}$$ $$\implies V_{in} = V_{out}.$$ So importantly, if we constrict the pipe, we necessarily have to compensate with an elongation to equate volumes. Say for example: $$\pi r^2h = \pi\bigg(\frac{r}{2}\bigg)^2(4h).$$ If we have the same volume per second in as out, with the same height of both cylinders, it must be the case that (in the example) 4 smaller volumes worth of fluid pass through in the same instant. Based on this argument, we could say the outbound fluid has greater velocity.
Since the fluid is constricted by the pipe, naturally streamlines will be pushed together. We may compare initial and final geometries with a homotopy, which in theory would detail streamline distance throughout the transition in laminar flow.
From the reverse viewpoint, one could then quantify how streamline separation implies volume/geometry changes, which as we've seen yields a velocity comparison. To be explicit here would give the desired result for the first statement. I think experimental relations would be more useful and allow for more variables.
Just to state it, sources I have looked up call this volume per second equality the continuity equation.
The second statement is more tricky and everything I do with $P = F/A$ seems to contradict the assertion. In our example, increase in velocity is accompanied by decrease in area, which the equation says leads to increase in pressure. Fellow Stack Members, I defer to you!