I am curious about the common explanation for the vena contracta effect that occurs as a flow moves around a sharp corner, or within a free jet of liquid issuing from a nozzle.
The explanation goes something like this.
Because the streamlines cannot abruptly change direction, when a flow moves around a sharp corner, travels through a contraction in a pipe, or exits through a sharp orifice, the streamlines curve following a smooth path so that the flow converges.
The definition of the streamline is that
at every point within a flow, the streamline is a path that is always tangent to the local velocity vector.
For a velocity vector $\mathbf{u}=\left(u,v,w\right)$ and an element of the arc along the streamline $d\mathbf{s}=\left(dx,dy,dz\right)$ then by taking the tangency from the definition of a streamline, i.e. $\mathbf{u}\times d\mathbf{s}=0$, $$\frac{dx}{u}=\frac{dy}{v}=\frac{dz}{w}$$ can be used to define the equation of a streamline if the velocity components are known.
So here is my question: Why can't a streamline change direction abruptly? I'm looking for a mathematical explanation of this starting from basic principles to understand why streamlines can't abruptly change direction. Physically I understand it this way, at least for the example of a free jet exiting from a pipe.
The pressure in a pipe will adjust itself by adjusting the flow rate such that at the exit of the pipe, the pressure is the same as the ambient pressure around it (for incompressible flow). Otherwise, the jet coming out would either suddenly contract or suddenly expand. When the flow is not incompressible like in supersonic flows, this can happen because pressure cannot be felt upstream when the flow is supersonic. “The flow doesn’t “know” that the air is at a lower or higher pressure until it flows out of the exit plane” quoting one of my professors.
Amendation Regarding 'abrupt change'
As per the suggestion of Calvin in the comments, an abrupt change may be understood to be the case where the change of velocity along the direction of the streamline is not differentiable. So if it is $C^0$ but not $C^1$, it may have an abrupt change.
Amendation regarding other thoughts
The definition of the flow rate $Q$ and incompressibility are as follows.
The conservation of mass is given by $\nabla\cdot\mathbf{u}+\frac{1}{\rho}\frac{D\rho}{Dt}=0$ where $\rho$ is the density. For a steady flow $\frac{D\rho}{Dt}=0$. If the density is independent of pressure $p$, then the flow is considered incompressible. Therefore, if the flow is incompressible $\nabla\cdot\mathbf{u}=0$.
The volume flux through a boundary is $Q=\int_A\mathbf{u}\cdot d\mathbf{A}$ and via the divergence theorem $Q=\int_V\left(\nabla\cdot\mathbf{u}\right)dV$. If the flow rate being considered is through a single inlet/outlet then the cross-sectional flow rate $Q_c=\int_{A_c}\mathbf{u}\cdot d\mathbf{A}_c=V_{avg.} A_c$ where $V_{avg.}$ is the average velocity across the cross-section.
Further, in 2D, the difference between the values of the streamfunctions at any two points is the volume flow rate. $$dQ=u dy – v dx=d\psi$$ $$Q=\int_{\psi_1}^{\psi_2} d\psi=\psi_2-\psi_1$$
If I can relate the streamlines to the pressure in a converging pipe or a 2D flow going around a sharp bend, then there will be no problem with multiple fluids interacting like in the jet.
Best Answer
There can be an abrupt change in direction of streamlines at a stagnation point. For example, consider the two-dimensional biaxial extensional flow with velocity field
$$u = Ex,\quad v = -Ey$$
In this case, the Navier-Stokes equations reduce to
$$E^2x = u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial x}, \\ E^2y = u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial y},$$
where the pressure can be obtained as $p = p_0 -\frac{\rho E^2}{2}(x^2+ y^2)$. Note that this is an exact solution to the full Navier-Stokes equations since the velocity components are linear with respect to the spatial variables and the second-order partial derivatives must vanish.
The velocity vector is tangential to the $x-$axis since the velocity components on that axis are $u=Ex$ and $v = 0$. Similarly the velocity vector is tangential to the $y-$axis. Hence, the $x-$ and $y-$axes are streamlines that meet perpendicularly at the stagnation point where $x=0,y=0$.
Also note that the first-order partial derivatives of the velocity components happen to be continuous throughout the domain. The abrupt change in direction of streamlines at the origin is physically acceptable for a combination of reasons: (1) the fluid comes to rest at that point; (2) the viscous stresses are finite there since the velocity gradient is everywhere constant; and (3) the pressure is a quadratic function where the components of the pressure gradient are also zero there.
The problem with the explanation you cite is not that it is based on a non-rigorous, physically intuitive argument. Rather, the problem is it is an imprecise statement. What is probably meant is that streamlines cannot exhibit a kink at a point where the velocity changes direction in such way that components of the velocity gradient are infinite -- giving rise to physically impossible infinite viscous stress.