Streamline functions for one-component velocity

Question

I would like to plot the streamline function for the two-dimensional flow with only one non-zero velocity component ($v_x, v_y=0$). I have seen the comparable question, which has been asked here, but I want to get a more precise answer provided by accurate calculus.

So, imagine that we have a Couette Flow, the law is $v_x=C y$, where $C$ is a constant, let it be $C=1$ for the sake of simplicity.

Now, supposing that the flow is incompressible, we have: $$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}=0$$ and streamline function is defined as: $$v_x=-\frac{\partial \psi}{\partial y}, v_y=\frac{\partial \psi}{\partial x}$$

Integrating over this two terms, we get:

$$\psi=-\frac{y^2}{2}+f(x), \psi=Const+g(y)+C_1.$$

$C_1$ is an arbitrary constant, and we can set it to zero, so $C_1=0$, now equating it to one another: $$-\frac{y^2}{2}+f(x)=Const+g(y),$$ we get $g(y)=-\frac{y^2}{2}, f(x)=Const$, so if I am not mistaken, the streamfunction is equal to

$$\psi=-\frac{y^2}{2}+Const.$$

My questions are:

1. How to find a $Const$ ?
2. There is no dependency for $x$. How can I plot the streamline function, to see the actual flow?

Any help will be greatly appreciated! Thank you in advance!

Answer 1

First, $Const + C_1$? What is the point of having two constants? Where did the second one come from? Of course, you immediately say you can just set $C_1 = 0$, which is true only because there is a second constant to roll its value into. But, why did you introduce it in the first place?

A more straightforward approach to this is after deriving $\psi = -\frac{y^2}2 + f(x)$, you differentiate that expression for $\psi$ with respect to $x$: $$\frac{\partial \psi}{\partial x} = f'(x)$$ and since you know $\frac{\partial \psi}{\partial x} = v_y = 0$, you get $f'(x) = 0$ and thus $f(x) = Const$.

But still, it leads you to the same place. So as for your questions:

1. The constant is going to be determined by your boundary conditions. Each constant gives you a solution to the general problem without conditions. The additional conditions tell you which solution is appropriate for your specific problem. Edit: RRL is correct. I was mixing up with solving general PDEs.
2. Streamlines are tangent to the flow velocity vector. As such, $\psi$ is constant along streamlines. Indeed, the streamlines are exactly the curves on which $\psi$ has constant value. Now you've noticed that $\psi$ is constant with respect to changes in $x$. What does that tell you about the streamlines? This is exactly what you would expect from the start. Since your velocity vectors all pointed in the $x$ direction, that is the direction streamlines must travel.

Answer 2

The streamfunction is determined only up to an arbitrary constant. The constant arising by integrating the velocity components can safely be chosen as $0$.

The streamfunction has two principal applications for two-dimensional, incompressible flow, where it can be seen that the value of this constant is irrelevant.

(1) In determining the velocity field, the components $v_x$ and $v_y$ are partial derivatives of the streamfunction to which the value of this constant makes no contribution.

(2) Through any contour $C = \{(\alpha(t), \beta(t)) \, : \, 0 \leqslant t \leqslant 1\}$ joining the points $(x_0,y_0)= (\alpha(0), \beta(0))$ and $(x_1,y_1) = (\alpha(1), \beta(1))$, the flux (volumetric rate of flow per unit width in the $z-$direction) is given by the integral,

$$Q = \int_C \mathbf{v} \cdot \mathbf{n}\, dl = \int_Cv_x \, dy - v_y\, dx = -\int_0^1\left[\frac{\partial \psi}{\partial y}(\alpha(t), \beta(t))\alpha'(t)+ \frac{\partial \psi}{\partial x}(\alpha(t), \beta(t))\beta'(t)\right] \, dt \\ = \psi(\alpha(0), \beta(0)) - \psi(\alpha(1), \beta(1)) = \psi(x_0,y_0) - \psi(x_1, y_1)$$

Hence, the flux through any contour joining two points is the difference in the values of the streamfunction at those points and, therefore, independent of any additive constant.

Streamlines are defined as the level curves of the streamfunction, that is curves in the plane where $\psi$ assumes a constant value and it is easy to show that the velocity vector at any point is tangential to a streamline. In this case the streamlines asre horizontal lines parallel to the $x-$axis.