# streamlines fluid mechanics

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#1
i have a potential p(x)=-iLog(z-I)

and need to find the equation for the stream lines through the points 1 and 0

I know streamlines are given by Im(p(x))=constant

so

-ln|z-I|=cosnt

which gives

x^2+(y-1)^2=const

so for the streamlines through 0 do I just say x=0,y=0
so streamline is x^2+(y-1)^2=1

and for x=1,y=0 so streamline x^2+(y-1)^2=2?

also I need the direction of these, how do I find that?

thanks
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#2
Anyone
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3 years ago
#3
What does "through the points 1 and 0" mean? Are you sure the question didn't specify coordinates?
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3 years ago
#4
Surely to find the 'direction' of these you need x and y as functions of time or something...
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3 years ago
#5
(Original post by mathz)
also I need the direction of these, how do I find that?
You also have

dp/dz = u - iv

so from this you can work out the velocity and so the direction along the streamline.
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#6
(Original post by RichE)
You also have

dp/dz = u - iv

so from this you can work out the velocity and so the direction along the streamline.
Ok,that makes sense. Does my working for the streamlines seem ok?

For the other reply,the point 1 is 1+0i in complex plane.
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3 years ago
#7
(Original post by mathz)
Ok,that makes sense. Does my working for the streamlines seem ok?
Yes, seemed fine
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#8
The follow up to this question was to find the circulation of q=dp/dz along line segment from 0 to 1 and flux of q across the line segment from 0 to1

Do I just evaluate the integral of q conjugated at 0 and 1 then take real and imaginary parts for my circulation and flux?
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3 years ago
#9
(Original post by mathz)
The follow up to this question was to find the circulation of q=dp/dz along line segment from 0 to 1 and flux of q across the line segment from 0 to1
What is your definition of circulation? For me this is usually calculated around a closed curve.
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#10
(Original post by RichE)
What is your definition of circulation? For me this is usually calculated around a closed curve.

They are defining it as follows

If g(s) is a smooth path and q is the flow velocity function then for each s the flow velocity q(g(s)) has tangential component qt(s) in direction specified by g'(s)

And circulation of q along g(s) is

Integral between end points of path of qt(s)

They then go to show that if Q is the conjugate of q and C is circulation and F is flux

C+iF=integral along g(s) of Q(z)dz

So I'm thinking for my line I just need to do the integration of Q and evaluate at z=0 and z=1

It's just in book only examples they do are round unit circle so it all involves residue theorem ,I'm worried what I'm doing is to simplistic
1
3 years ago
#11
(Original post by mathz)
So I'm thinking for my line I just need to do the integration of Q and evaluate at z=0 and z=1

It's just in book only examples they do are round unit circle so it all involves residue theorem ,I'm worried what I'm doing is to simplistic
Sounds about right. In any case if it's not a closed curve then you can't apply residue theory.
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