Exploring Streamlines and Cutoff Points of a Fluid Flow

[mathmari]

Hey!

The velocity field $u=ax^2, v=bxy$ ($a, b=\text{ constant }$) represents a flow field of an incompressible fluid. Construct the streamlines and find the cutoff points of the flow ($\overrightarrow{u}=(u,v)=0$).

From the fact the fluid is incompressible we have that $$div u=0 \Rightarrow \partial_x u+\partial_y v=0 \Rightarrow 2ax+bx=0 \Rightarrow b=-2a$$

To construct the streamlines we do the following:

$$\frac{dx}{u}=\frac{dy}{v}$$

$$\frac{dx}{ax^2}=\frac{dy}{bxy} \Rightarrow \frac{dx}{ax}=-\frac{dy}{2ay} \Rightarrow \frac{dx}{x}=-\frac{dy}{2y} \Rightarrow \ln x=-\frac{1}{2}\ln y +c \Rightarrow \ln x=\ln y^{-\frac{1}{2}} +c \Rightarrow x=cy^{-\frac{1}{2}} \Rightarrow xy^{\frac{1}{2}}=c \Rightarrow x^2y=C$$

Is this correct?? (Wondering)

So, are the streamlines the curves $x^2t=C$ ?? (Wondering) To find the cutoff points of the flow do we have to do the following??

$$\overrightarrow{u}=(u, v)=(0, 0) \Rightarrow ax^2=0 \text{ and } bxy=0 \Rightarrow ax^2=0 \text{ and } -2axy=0$$

when $a>0$ we have that $x=0$ or $x=y=0$

(Wondering)

 

[jvicens]

Hello! It looks like you have correctly found the streamlines for this flow field. The curves $x^2y=C$ are indeed the streamlines, and they represent the paths that a fluid particle would take as it moves through this flow field.

For the cutoff points, you are correct in saying that when $a>0$, the cutoff points would be at $x=0$ or $x=y=0$. This means that the flow would stop at these points, as the velocity is equal to 0. However, when $a<0$, there would be no cutoff points as the velocity field is defined for all values of $x$ and $y$.

Overall, your analysis is correct and you have a good understanding of this flow field. Keep up the good work!