Re: [ESPResSo-users] No conservation of momentum/mass in LBM ??

[Kai Szuttor]

Hi there,

actually in the book of Bruus "Theoretical microfluidics" there is
an expression for the pressure driven flow in rectangular channels:

I attached texified formulas for velocity and flow rate.

Cheers,

Kai

Am 15/03/16 um 14:07 schrieb Ulf D Schiller:
> Did you check the flow rates directly, i.e., the momentum flux per plane? 
> Your argument seems correct, so I can only guess that there's some
> flaw in the calculation of the mean velocity. I think there's an expression 
> for the flux in rectangular channels that one could use.
> 
> Best,
> Ulf 
> 
> Sent from a mobile device.
> 
> 
> -------- Original message --------
> From: "Wink, Markus" <address@hidden>
> Date: 3/15/2016 8:47 AM (GMT-05:00)
> To: 'Ivan Cimrak' <address@hidden>, address@hidden
> Subject: Re: [ESPResSo-users] No conservation of momentum/mass in LBM ??
> 
> Hi Ivan, Hi Florian,
> 
>  
> 
>>/How did you compute the expected maximum velocity? As far as I know, the 
>>poisseuille flow has an exact expression for the velocity in the case
> of channel with circular cross section, and you have a rectangular one.///
> 
> / /
> 
> I know the velocity of the rhomboid. Thus I know the mean velocity of the 
> fluid (assuming it is incompressible). I took that for calculating the
> Reynoldsnumber, pressure gradient and theoretical velocity profile (using the 
> expression in the book  “Viscous Fluid Flow” of Frank M. White).
> 
>  
> 
> /> //The boundaries are momentum sinks. (Florian)/
> 
> /> Now I read the comment of Florian -//does that mean that amount of fluid 
> is decreasing when no-slip is prescribed?/
> 
> I still don’t get it. That the boundaries are momentum sinks, I agree. Due to 
> the present of the walls and the “friction” of the fluid there, I
> achieve the poiseuille profile. But I still hold the opinion, that the mean 
> velocity of the fluid should be the same.
> Imagine the following physical experiment: you have a syringe pump set up 
> with a constant flow rate Q0 connected to a rectangular channel having
> a cross section A=w*h. The fluid in the channel then has a mean velocity of 
> v_mean=Q/A. Assuming an incompressible medium, this means the
> velocity should be the same at every slice normal the direction of transport.
> In my simulation, the mean velocity should be velocity v0 of the rhomboid.
> 
> So I still don’t get the deviation to the theoretical value…
> 
> Greetings Markus
> 
>  
> 
>  
> 
>  
> 
> *Von:address@hidden
> [mailto:address@hidden *Im Auftrag von *Ivan Cimrak
> *Gesendet:* Dienstag, 15. März 2016 13:22
> *An:* address@hidden
> *Betreff:* Re: [ESPResSo-users] No conservation of momentum/mass in LBM ??
> 
>  
> 
> Hi Markus,
> 
>  
> 
>     Hello Everybody,
> 
>      
> 
>     so far, in the LBM scheme only the body force is implemented and no 
> velocity/pressure boundary condition. So I was thinking on a way of
>     mimicking a “velocity boundary” condition without changing the source 
> code. I am aware that one should, as a proper approach, using Zou/He
>     boundary conditions and adjusting the distribution functions according to 
> the boundary conditions.
> 
>      
> 
>     For that I have constructed a channel with rectangular cross section and 
> put the fluid inside. Furthermore, two rhomboids where put inside,
>     one at the inlet of the channel, one at the outlet. The cross section of 
> the two rhomboids is equal to the cross section of the channel,
>     furthermore they have a constant velocity v0.
> 
>     My idea was, that, since the no-slip boundary condition is implemented, I 
> force the fluid nodes adjacent to the rhomboids to have a constant
>     velocity, thus achieving constant velocity inlet/outlet condition.
> 
>      
> 
>     As a result I achieve a poiseuille profile in the middle of the channel 
> (fully developed flow after inlet/outlet effects). The qualitative
>     pressure gradient looks proper, too.
> 
>     Nevertheless, the maximum velocity is not the same as I expected (factor 
> 3 to the expected one).
> 
> How did you compute the expected maximum velocity? As far as I know, the 
> poisseuille flow has an exact expression for the velocity in the case
> of channel with circular cross section, and you have a rectangular one.
> 
> 
> I have checked the mean velocity. I would expect, that the mean velocity of 
> the fluid should be the velocity v0 of the rhomboid (due to
> mass/momentum conservation), I get less (10 %).
> 
> This is strange. The amount of fluid at the inlet (integral of velocity over 
> the inlet surface, in this case is the velocity constant over the
> inlet surface) should be the same as integral over the middle cross section, 
> as well as integral over the outlet surface.... Supposing you
> computed the average velocity as sum of velocities over the LB nodes at 
> middle cross section divided by number of these nodes, you should have
> obtained the velocity at the inlet...
> 
> Now I read the comment of Florian - does that mean that amount of fluid is 
> decreasing when no-slip is prescribed?
> 
> Ivan
> 
>  
> 
> What is wrong with my idea stated here? Obviously, something is not correct, 
> but I have no idea, what the reason for that is. Where does the
> momentum vanish?
> 
>  
> 
> Does anybody have an idea?
> 
>  
> 
> Sincerely,
> 
>  
> 
> Markus
> 
>  
> 
>  
>

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