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August 2, 2007, 05:10 
CFD: Using Taylor series

#1 
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Hi
I've been looking at a fluid dynamics paper and have a specific question. Here's the link to the paper; http://arxiv.org/PS_cache/physics/pd.../0110081v1.pdf "Drop Formation in a OneDimensional Approximation of the NavierStokes Equation" It involves fluid falling from a pipe, and breaking up into drops due to surface tension. It starts with general equations [NS, continuity, BC's], and creates a 1D computational model. So it is CFD, just pretty imple computationally! What im really interested in is their simplication of the navier stokes equations, continuity etc, by the use of a power series. Looking at the paper, i don't understand how they got equations 9, 10 and 11. Looking at Eq. 9, it seems that it is a power series. Now i understand that a power series looks like; f(x)=a + bx + cx^2 + dx^3... So in equation 9, where is v_11*r?! They say by symmetry we get equation 9  what does this mean? Equation 11 seems to follow the same pattern  again, why? And where in earth do they get equation 10 from  what do they differentiate to get it? I understand that this is quite a specific question, but pls, any help is greatly appreciated! If the link doesn't work please tell me! Thanks Charlie 

August 2, 2007, 05:35 
Re: CFD: Using Taylor series

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I don't know where equations 9 and 11 come from, though presumably if they are right they have the same origin. But equation 10 comes from substituting equation 9 into equation 3.


August 2, 2007, 08:49 
Re: CFD: Using Taylor series

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I dont know the full answer but
first assume v_z(z,r) = v_0 + v_1*r + ... Note that v_0(z) = v_z(z,r=0) v_1(z) = d(v_z)/dr(z,r=0) Now v_z has axisymmetry, just draw zaxis downwards, raxis horizontal think how the profile of v_z should look like near r=0. Then you will realize that v_z must have a smooth extremum at r=0, otherwise it will not have slope continuity at r=0. Hence v_1=0. 

August 2, 2007, 09:11 
Re: CFD: Using Taylor series

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It looks to me like the key assumption in establishing the form of (9) is not symmetry but rather the relative sizes in the two dimensions r and z. They specifically state that they are going to focus on very thin columns relative to the elongation, indicating that gradients in r will be much larger than those in z. Then symmetry is used (as noted by Praveen) to further reduce the series. This is just a first impression based on a very cursory look at the article.


August 3, 2007, 04:56 
Re: CFD: Using Taylor series

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Hi
Thanks for your help  very enlightening! Richard i think your right  to get 10 you do just substitute 9 into Eq. 3. So basically they get rid of v_1 by saying that the gradient at at r=0 is zero  i.e. the velocity profile is smooth across the r axis. One other Q  would there be r^3 terms in Eq. (9) and (11)? Thanks very much for your help Charlie 

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