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Post by mondo on Sept 7, 2023 23:45:22 GMT
Mondo Yes it does. But you have written it wrong. The plus sign should be minus. Vasco No my point was that with the "+" sign between these terms the streamlines are also a straight lines, right? Even when I think about it intuitively it doesn't make sense to me. The uniform flow with velocity $1$ in the direction of positive real axis is described by $H(z) = 1$ now if I want to rotate it in the counter clockwise direction by some angle $\phi$ I should multiply it by $e^{i\phi}$. Hence every calculation I made for the original calculation should be multiplied by $e^{i\phi}$ for the case of the flow at an angle. |
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Post by Admin on Sept 8, 2023 6:56:09 GMT
Mondo
Look. The velocity is $\overline{H}=e^{i\phi}$ and so $\Omega'=H=e^{-i\phi}$ and so $\Omega=e^{-i\phi}z$.
Vasco
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Post by mondo on Sept 8, 2023 7:37:50 GMT
Hmm this keeps confusing me. Shouldn't that be the other way around? We are given a flow of $e^{i\phi}$ and so for the calculation of potential function we should take the Polya vector of it, namely $e^{-i\phi}$ which would give us $\Omega = e^{-i\phi}z$ and then when we want to recover the original flow, hence calculating the derivative we need to "remember" that to take the conjugate of it?
Theoretically $\overline{H}$ is neither differentiable not integrable as we discussed in some other thread.
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Post by Admin on Sept 8, 2023 7:43:14 GMT
Mondo
When you say flow what exactly do you mean?
Vasco
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Post by Admin on Sept 8, 2023 7:56:55 GMT
Mondo
We did not say that $\overline{H}$ was not differentiable. We said that $\overline{z}$ is not differentiable. Not the same thing at all.
Vasco
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Post by Admin on Sept 8, 2023 8:07:12 GMT
Mondo
I mean not differentiable with respect to $z$ of course.
Vasco
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Post by mondo on Sept 8, 2023 8:14:20 GMT
Mondo When you say flow what exactly do you mean? Vasco For instance a flow of fluid or electric charge. Hmm what if $H(z) = z$. I don't get the message here |
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Post by Admin on Sept 8, 2023 8:22:33 GMT
Mondo
I meant what do you mean mathematically when you say flow?
Vasco
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Post by Admin on Sept 8, 2023 8:31:14 GMT
Mondo
But $H=z$ is a special case when $\overline{H}$ is not differentiable. In general we cannot say that $\overline{H}$ is not differentiable.
For example if $\overline{H}=z$ then it is differentiable.
Vasco
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Post by Admin on Sept 9, 2023 14:28:30 GMT
Mondo
$\Omega$ does not represent the flow, but its level curves are streamlines of the flow $\overline{H}$. The real part of $\Omega$ represents the value of $\Phi$ and the imaginary part represents the value of $\Psi$.
Vasco
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Post by mondo on Sept 9, 2023 18:37:39 GMT
Mondo I meant what do you mean mathematically when you say flow? Vasco A complex function $f(z)$ which argument is a complex number $z$ and image is a vector emanating from that point. Hence it can be represented on a single plane as introduced on page 450. For instance, the example 2 on page 503 talks about uniform, eastward flow - this can be represented by a horizontal vector of length $1$ emanating from each $z$. Mondo But $H=z$ is a special case when $\overline{H}$ is not differentiable. In general we cannot say that $\overline{H}$ is not differentiable. For example if $\overline{H}=z$ then it is differentiable. Vasco I agree that $\overline{H}$ may be a legit function to differentiable. The only thing that confuses me is the fact that for the calculation of $\Omega$ we used $\overline{H}$ but when we differentiate it we are supposed to get $H$ instead. The way I understand it is in general the integral to get complex potential is on $H$ but we use the polya vector field to get that as a "trick" to get it easier. Hence we have to keep in mind that in one direction we use $\overline{H}$ while in the other it is just $H$. Mondo $\Omega$ does not represent the flow, but its level curves are streamlines of the flow $\overline{H}$. The real part of $\Omega$ represents the value of $\Phi$ and the imaginary part represents the value of $\Psi$. Vasco Well that is another confusing part and I also mentioned that in the other thread - I agree that $\Omega$ is not a flow in general but rather a mathematical object describing a streamlines and potentials of some flow. But then why at the very bottom of page 539 author says on $\tilde{\Omega}(w)$ "..the factor of (2/3) signifying the velocity.." velocity is a property of flow so how can a complex potential have a velocity? |
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Post by mondo on Sept 15, 2023 8:04:02 GMT
Vasco, do you understand the velocity $2/3$ of complex potential from my last qiestion in the above post?
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Post by mondo on Sept 15, 2023 8:07:58 GMT
And also why at the bottom of page 542 autjor say $[\phi] = 4v$? This is a mystery to me.
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Post by Admin on Sept 15, 2023 12:22:20 GMT
And also why at the bottom of page 542 autjor say $[\phi] = 4v$? This is a mystery to me. Mondo It's all explained on page 541 in subsection 2. Vasco |
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Post by Admin on Sept 15, 2023 15:40:44 GMT
Mondo
I'd like you to show me where in the book Needham says that a complex potential has a velocity.
You've made that up in your own head. What you have quoted does not say what you claim it says.
Vasco
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