# Introducing stream function with given velocity equation

Bit information about the problem We are dealing with the slide coating process - where basically a polymer is being put onto a slot, which is moving in the $$x$$-direction with velocity $$v_0$$. The volumetric flow rate $$Q_w=Q/W$$ where $$Q$$ is total volume-rate flow and W is the length in the $$z$$-direction. The polymer is solidifying the height is dropping from $$h_0$$ to $$h_1$$ for $$x[0;L]$$.

It is known: - constant density and viscosity - steady state - newtonian fluid - no gravity forces - lubrication approximation

Help required for

We are given the following expression for velocity (in $$x$$ direction),

$$v_x=(3v_0 - 6\frac{Q_w}{h})(\frac{y}{h})^2+(6*\frac{Q_w}{h}-4v_0)\frac{y}{h}+v_0$$

We want to introduce the following stream functions $$v_x=-\frac{\delta \psi}{\delta y}$$ $$v_y=-\frac{\delta \psi}{\delta x}$$

We are also given that $$\psi(x,h(x))=0$$. We wish to get an expression for $$\psi(x,y)$$ corresponding to $$x[0;L]$$ and $$y[0;h]$$.

My approach So my approach to this problem was something like this.

I integrated the given $$v_x$$ equation to get the following

$$\psi(x,h)=\int v_xdy + \phi(x)=v_x=(3v_0 - 6\frac{Q_w}{h})(\frac{y^3}{3h^2})+(6*\frac{Q_w}{h}-4v_0)\frac{y^2}{2h}+v_0 y + \phi(x)$$

where $$\phi(x)$$ is a function.

Then I said that since we are not given an expression for $$v_y$$ therefore, I can differentiate the above w.r.t x, giving the following,

$$\frac{\delta \psi}{\delta x} = 0+0+0+\frac{\phi(x)}{\delta x} = v_y$$

From there I just assumed that $$v_y = 0$$.

I am pretty sure the last part is definitely wrong. But is there any other way of doing it?

My thoughts: Should I determine an expression for $$v_y$$ from Navier-Stoke equation and only then use the stream function? Or I don't necessarily have to do that.

If additional information is required. Please write in the comment section, so I can provide it.

• It isn't clear to me what you are describing. Is the solidification taking place on the moving surface? Does your analysis take into account the volumetric throughput rate of the solid? – Chet Miller Oct 24 '18 at 0:36

The Integration with respect to variable $$y$$ was correct. Now you have to Substitute $$y=h(x)$$ and use your given condition $$\psi(x,y=h(x))=0$$. From this you will know the x-dependent function $$\phi(x)$$. By differentiating this by $$x$$, you get $$v_y$$.
Remark: I know that the stream function $$\psi$$ is defined as
$$v_x = \frac{\partial \psi}{\partial y}, v_y = - \frac{\partial \psi}{\partial x}$$ (one positive sign and one negative sign). Is the Definition with both negative signs (both -) correct?