The continuity equation for fluid flow assuming steady-state and constant fluid density, that is: $\rho = \text{constant}$ and $\partial_t=0$ can be written in the two following forms (differential and integral):
- Differential form:
$${\partial u\over\partial x} = 0 \tag{1}$$
Where $u$ is the $x$ component of velocity.
This equation states that $$u = \text{constant} \tag{2}$$ otherwise the change of $u$ in $x$ direction must be balanced by the change of $v$ in the $y$ direction (this follows from the 2D form of the continuity equation ${\partial u \over \partial x} = -{\partial v \over \partial y}$).
- Integral form, aka flow rate conservation (correct me if I am wrong):
$$ Q = u\cdot A = u_1\cdot A_1 = u_2\cdot A_2 = \cdots = \text{constant} \tag{3}$$
Where the numbers 1, 2, … denote sections of streamtube along $x$ axis.
My questions
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Could you please explain me this ambiguity, that is why the first form states that there is no change in $u$ otherwise the flow will be 2D and in the other hand the integral form there is a change in $u$ along $x$ axis but the flow still 1D.
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Is it possible to consider flows in diverging (or converging) nozzles as 1D flows?
Best Answer
Okay now I understand your question, I think.
I'll leave a brief answer because I'm not sure if it will be enough, but I don't have much time now.
You're right, in 1D, there is no $y$, nor $d/dy$, forget about vertical axes.
It is true that $dv/dx=0$, provided that the flow is non-compresible.
And, why does it seem to contradict the other one? Well, you must keep in mind that the divergence is after all sort of a volumetric quantity. It refers to an infinitesimal volume. $div(v)=0$ means that there is no growth of flow from that point, or, in other words: there are as many flow lines coming in as coming out.
That is coherent with the flow conservation.
On the other hand, the integral form has been converted into a surface integral, but not any surface, but the boundary surface. It is now the velocity crossing the surface, not a volume. Of course the flow crossing the entering surface must be the same than the exiting surface.
So, the key is that one: the first one is something related to infinitesimal volumes, whereas the second one is a velocity crossing the surface.
If you want to go deeper, this has to do with Eulerian versus Lagragian points of view. Considering that $v$ is a vector field, versus a stream of particles.