Conformal Transformation: Fluid flow over surface waves

[stedwards]

I would like to obtain the conformal map from a uniform rectilinear fluid flowing in the x-direction, where the field is bounded below by the x-axis, to the flow in the w-plane.

In the w-plane the flow is correspondingly bounded from below by a trochoid. (A trochoid is a continuous waveform shaped something a sine wave but with pointier tops.)

With

z=x+iy
w=u+iv,​

the trochoid boundary is given parametrically as

u=a \theta – b sin \theta
v= -a + b cos \theta​

where a is greater than b.

But how do I map the x-axis to the trochoid? There seem to be an infinity of maps. How do I select the correct one?

This problem seems to be isomorphic to finding the electric field and equipotentials of a charged trochoid shaped conductor with the return conductor at y=\inf.

 

[fzero]

Your parametric equations already give the mapping of the interval $\theta \in [0,2\pi)$ to the trochoid. The corresponding conformal map from the infinite ribbon subregion of the conformal plane would be obtained by replacing $\theta$ by $z=\theta + i t$ in the parametric equations.

 

[stedwards]

But I've already defined z=x+iy. Did you intend a different variable instead?

 

[fzero]

But I've already defined z=x+iy. Did you intend a different variable instead?

Well $\theta = x ~(\text{mod}~ 2\pi)$, but you can set $t\equiv y$, so I could have been clearer in the notation. The variable we want to use to define the conformal transformation is $\theta + i y$ whatever we want to call it ($\zeta$?). Because of the linear term in $\theta$ we can't just substitute $x$ into that formula, so we don't want to use your original $z$.

 

[stedwards]

Now I see what you mean. That's amazing. However did you see that so fast??

Also, for instance, is z=\theta^3+it also a solution?

In order to find the relationship between w and t at other than the boundary, do I apply the Cauchy-Riemann relations?

 

[fzero]

Now I see what you mean. That's amazing. However did you see that so fast??

With $\zeta = \theta + iy$, we are looking for a holomorphic function $f(\zeta) = w$ such that $f(\theta)$ is given by the trochoid equations. Since $f(\zeta)$ must be holomorphic, we are led to the conclusion that we need to replace $\theta$ in our expression by the full complex variable to find the full map.

Also, for instance, is z=\theta^3+it also a solution?

You'd have to give a candidate $f(z)$ to have a well-defined question. I don't believe one exists though.

In order to find the relationship between w and t at other than the boundary, do I apply the Cauchy-Riemann relations?

Once $f(\zeta)$ has been specified you should have all that you need.

 

[stedwards]

For notation

\zeta=\theta+iy and
w=u+iv,​

the solution seems to be

u=a \zeta – b sin \zeta
v= -a + b cos \zeta,​

yes?

What's been bothering me is the upper boundary condition at y \rightarrow \infty

I get \zeta \rightarrow +\infty + i\infty, where I would expect instead, \zeta = \theta + i\infty.

This isn't really a big deal, but just for completeness.

On an aside, with a flow field across surface waves, you could imagine a vortex residing between each crest which would be an alternative solution with the given boundary conditions, only with singular points. In the static electric field model, it would correspond to placing charge sources between each peak.

 

[fzero]

For notation

\zeta=\theta+iy and
w=u+iv,​

the solution seems to be

u=a \zeta – b sin \zeta
v= -a + b cos \zeta,​

yes?

No, $u$ and $v$ are no longer real, so they are not the real and imaginary parts of $w$. Instead we should write

$$ w = a \zeta – b sin \zeta + i ( -a + b cos \zeta),$$

and some manipulation is necessary to determine $u,v$.

What's been bothering me is the upper boundary condition at y \rightarrow \infty

I get \zeta \rightarrow +\infty + i\infty, where I would expect instead, \zeta = \theta + i\infty.

This isn't really a big deal, but just for completeness.

Generally a conformal transformation from some subregion of the complex plane will map to some different region. Anyway, I think if you work out the correct version of $u$, you'll find that it's finite (it looks to contain $\cosh y- \sinh y$).

 

[stedwards]

No, $u$ and $v$ are no longer real, so they are not the real and imaginary parts of $w$. Instead we should write

$$ w = a \zeta – b sin \zeta + i ( -a + b cos \zeta),$$

and some manipulation is necessary to determine $u,v$.

douh! Of course.

w = (a \zeta – b \sin \zeta) + i(- a + b \cos \zeta)

using identities:

\sin(\zeta)= \sin \theta \cosh y + i \cos \theta \sinh y
\cos(\zeta)= \cos \theta \cosh y – i \sin \theta \sinh y​

w = [a (\theta + iy) – b(\sin \theta \cosh y + i \cos \theta \sinh y)] + i[-a + b(\cos \theta \cosh y – i \sin \theta \sinh y)]

w = [a \theta – b(\sin \theta \cosh y) - \sin \theta \sinh y)] + i[ay – a + b(\cos \theta \cosh y – \cos \theta \sinh y) ]

w = [a \theta – b \sin \theta (\cosh y - \sinh y)] + i[ay – a + b \cos \theta (\cosh y – \sinh y) ]

using the identity: \cosh y – \sinh y = e^{-y}

u = a \theta – b \sin \theta \cdot e^{-y}
v = ay – a + b \cos \theta \cdot e^{-y}​

At y=0,

u = a \theta – b \sin \theta
v = ay – a + b \cos \theta
w = (a \theta – b \sin \theta) + (- a + b \cos \theta)i, verifying the lower boundary condition.​

In the limit y \rightarrow +\infty,

u \rightarrow a \theta
v \rightarrow +\infty​

Is u \rightarrow a \theta correct? The scaling factor of $a$ seems odd.

 

[fzero]

It's the same scaling we had for the original curtate trochoid. We start with domain of $\zeta$ as the ribbon between $0$ and $2\pi$ and this is mapped to another ribbon between $0$ and $2\pi a$.

 

[stedwards]

I see. The original trochoid could have been scaled to match theta with w=ζ–(b/a)sinζ+i(−1+(b/a)cosζ).

Thanks for the all the help, fzero! What text on introductory complex analysis do you recommend?

 

[fzero]

I don't really have a strong recommendation for a particular text. My undergrad course used Kreszig's Advanced Engineering Mathmatics text that I wasn't too thrilled with. I probably learned more from the lectures than the text. The rest of what I know was picked up from random sources as needed over years after, so I can't point to a specific text, except maybe Arfken, which isn't really a complex analysis book or particularly introductory, but is a valuable reference.

Kreyszig's Introductory Functional Analysis seems to be well-rated here on PF. Brown and Churchill has also been recomended, as well as the book by Marsden and Hoffman. If you post in the textbook forum you'd probably get better suggestions or stronger recommendations for one of the the books I've listed.

 

[FactChecker]

I am having trouble following your definition of the map and why it is conformal, but this link claims to define a family of conformal maps onto trochoids with various parameters. Maybe your function is the same as one of these. http://userpages.monmouth.com/~chenrich/Trochoids/Conformal.html (I have not verified that this is indeed a family of conformal maps. Please forgive me if you already have the solution and I am just muddying the waters.)

 

[fzero]

I am having trouble following your definition of the map and why it is conformal, but this link claims to define a family of conformal maps onto trochoids with various parameters. Maybe your function is the same as one of these. http://userpages.monmouth.com/~chenrich/Trochoids/Conformal.html (I have not verified that this is indeed a family of conformal maps. Please forgive me if you already have the solution and I am just muddying the waters.)

A function $f(z)$ is a conformal map iff it is holomorphic and its derivative is nonzero everywhere in its domain. By construction, the function above is holomophic and I believe it's not too difficult to show that it's derivative is everywhere nonzero (mainly because cos and sin have their zeros out of phase).

The trochoid on those pages is for a different geometry, namely the interior of a disk. The tangent function maps the ribbon to the unit disk, but the map is such that the vertical boundaries of the ribbon are mapped to the circumference of the disk, not the segment of the $x$-axis. So I don't see immediately how to relate these solutions in a conformal way.

 

[FactChecker]

A function $f(z)$ is a conformal map iff it is holomorphic and its derivative is nonzero everywhere in its domain. By construction, the function above is holomophic and I believe it's not too difficult to show that it's derivative is everywhere nonzero (mainly because cos and sin have their zeros out of phase).

Ok. I couldn't figure out exactly what the function f(z) is. Does θ = arg(z)?

The trochoid on those pages is for a different geometry, namely the interior of a disk. The tangent function maps the ribbon to the unit disk, but the map is such that the vertical boundaries of the ribbon are mapped to the circumference of the disk, not the segment of the $x$-axis. So I don't see immediately how to relate these solutions in a conformal way.

I thought the reference gave a mapping from a straight line to a trochoid. So it would be a conformal mapping from a half plane to the interior of a trochoid. Maybe I misunderstood the reference.

 

[fzero]

Ok. I couldn't figure out exactly what the function f(z) is. Does θ = arg(z)?

Let me write down an expression for the linear trochoid that doesn't use the same letters as that web page. We have

$$f(\zeta) = a(\zeta -i) + i b e^{-i\zeta}.$$

Here $\zeta = \phi + i y$, where $\phi = x~(\text{mod}~2\pi)$. As a function $f:U\rightarrow \mathbb{c}$, the domain is the vertical ribbon-shaped region between $0$ and $2\pi$ on the $x$-axis and $\pm\infty$ on the $y$-axis.

I thought the reference gave a mapping from a straight line to a trochoid. So it would be a conformal mapping from a half plane to the interior of a trochoid. Maybe I misunderstood the reference.

According to http://userpages.monmouth.com/~chenrich/Trochoids/Trochoids.html the trochoid there describes a wheel rotating inside a second larger circle. There is a map from the disk to the upper half-plane, but I don't think it preserves the periodicity of the trochoid. So it's going to be hard to compare these directly.