Graph function help - fluid mechanics/streamlines related

[elle]

Graph function help please - fluid mechanics/streamlines related

I'm not too sure whether I should have posted this in the Physics section or in the maths Please move this topic if its in the wrong place!

this probably seems like a silly question but is it possible to plot x*y + c on a graph where c = a constant? I'm working on a question on fluid mechanics where I have to find the streamfunction and plot the stream lines. However I'm in doubt whether i have done the question correctly in the first place.

Please help!

Homework Statement

For the 2D velocity field u = xi - yj find the streamfunction 'psi' (apologises, I'm not too familiar with Latex)

Homework Equations

given above

The Attempt at a Solution

After partially differentiating and using formulas from my notes, i got psi = x*y + c

 

[HallsofIvy]

I'm not too sure whether I should have posted this in the Physics section or in the maths Please move this topic if its in the wrong place!

this probably seems like a silly question but is it possible to plot x*y + c on a graph where c = a constant? I'm working on a question on fluid mechanics where I have to find the streamfunction and plot the stream lines. However I'm in doubt whether i have done the question correctly in the first place.

What do you mean by "plot x*y+ c". That's neither a function nor an equation. If you mean xy= c or xy+ c= 0, then, yes, of course, you can plot it- it's a family of hyperbolas.

Please help!

Homework Statement

For the 2D velocity field u = xi - yj find the streamfunction 'psi' (apologises, I'm not too familiar with Latex)

Then $\nabla \psi= x\vec{i}- y\vec{j}$, right?
So
$$\frac{\partial \psi}{\partial x}= x$$
and
[tex]\frac{\partial \psi}{\partial y}= -y[/itex]<br /> That should be easy to integrate.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> <h2>Homework Equations</h2><br /> <br /> given above<br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> After partially differentiating and using formulas from my notes, i got psi = x*y + c <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f615.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":confused:" title="Confused :confused:" data-smilie="5"data-shortname=":confused:" /> </div> </div> </blockquote> Perhaps I am misunderstanding your "stream function". I get the orthogonal family to xy= C which would be the potential function.[/tex]

 

[arildno]

You are confusing "stream function" with "velocity potential", HallsofIvy!

A stream function may always be defined if the velocity field is solenoidal, i.e, $$\nabla\cdot{\vec{v}}=0$$
In the 2-D case, we may define: $$\vec{v}=\nabla\times\psi\vec{k}$$, where $\psi$ is the scalar stream function.

The analogue in electro-magnetism is called the magnetic vector potential, I believe..

 

[HallsofIvy]

Very likely- it's been a long time since I have looked at fluid dynamics.

So the equations are
$$\frac{\partial \phi}{\partial x}= y$$
and
$$\frac{\partial \phi}{\partial x}= x$$

Yes, that tells us that $\phi(x,y)= xy+ c$.
Since the stream lines (again, if I remember correctly!) are the lines on which $\phi$ is a constant they are given by xy+ c= c' or xy= C, the hyperbolas that my family, $x^2- y^2= c$ are orthogonal to.

elle, perhaps your only problem was not realizing that you get an equation to graph by setting your solution for $\phi$ equal to a constant.

 

[elle]

Very likely- it's been a long time since I have looked at fluid dynamics.

So the equations are
$$\frac{\partial \phi}{\partial x}= y$$
and
$$\frac{\partial \phi}{\partial x}= x$$

Yes, that tells us that $\phi(x,y)= xy+ c$.
Since the stream lines (again, if I remember correctly!) are the lines on which $\phi$ is a constant they are given by xy+ c= c' or xy= C, the hyperbolas that my family, $x^2- y^2= c$ are orthogonal to.

elle, perhaps your only problem was not realizing that you get an equation to graph by setting your solution for $\phi$ equal to a constant.

ohh yeah! I forgot that its equal to a constant Thanks guys for the help