Discussion Forum

Posted: 1 decade ago 12 lug 2010, 19:45 GMT-4

I am also interested in this problem of finding the shear stress from the velocity data in a microfluidic channel. I have also found that the equation for stress is part of the NS equations: tau = viscosity * (grad(U) + grad(U)[transpose]) Short of adding another multiphysics model, I see no way of adding an expression that generates the stress tensor. I would be happy with the maximum/minimum shear stresses as well (rather then the full tensor). Anyone have any ideas? Thanks!

Posted: 1 decade ago 13 lug 2010, 10:40 GMT-4

Hi Tyler The equation you are talking about is good for two dimensional calculations along the velocity profile. Do you have a 2D or a 3D model? If you have a 3D model you can extract one velocity slice and do further calculations in matlab. I calculated the shear stress along the X and Y axis, one result is attached. Of course, this result is partly wrong because only the effect of 2 of the 4 walls is noted. But I think that the calculations are good directly at the middle of the involved walls, because there is less effect of the other walls. As further as you go to the edges or in the middle of the channel, the effect of the not noted walls increases and the results get huge error. But this could be a way to get your maximum shear stress values with a small error because the maximum values should be located in the middle of each wall. The minimum value I am not quite sure about, but assuming laminar flow (micro fluidic) there should be no stress in the middle of the channel, but I will double check that. I still look for a way to visualize the overall shear stress profile in comsol, or a way how I can combine the results I created in matlab. I am thankful for any help. T. Gumbel

Attachments:

• A.png

Posted: 1 decade ago 14 lug 2010, 13:59 GMT-4

I've had some more time to look at this, and what you are saying makes sense. My system is only 2D, though I believe the equation I mention above should apply in 3D. I've found that in 2d, the equation generated a symmetrical tensor where the diagonal elements are the normal stresses and the other two are equal and represent the shear stress. In 3D there would be 3 non-diagonal elements, creating a problem for graphing the shear stress -- however in 2D, with only one value to plot, it's possible to show the shear stress through the 2D plane. This method could be applied, as you were saying, in 3D with 2D slices across the channel with some loss in accuracy due to ignoring some walls. Other than that, it may be possible to somehow find the magnitude of the 3 shear elements of the 3D stress tensor -- I would guess some type of modulus, but I'm no expert. As it is, I think I'll stick with pulling out the single tensor element I neeed -- namely uy+vx (comsol notation for dVx/dy+dVy/dx) Thanks for the help!

Posted: 1 decade ago 15 lug 2010, 14:24 GMT-4

Hi Tyler Regarding your answer I have a question. Is it allowed to just sum up the stress???? You mention that for 2D cases you use eta*(uy+vx). So if that is right, for 3D cases the formula eat*abs(vx) + eat*abs(uy) + eat*abs(wz) would be right as well???? I just implemented it in comsol with the result I added (first two pics). The velocity profile looks fine. The shear stress not really. The stress values at the bottom and top wall are acceptable but the stress on the sidewalls is wrong I think. So if I sum up all components eta0*abs(vy) +eta0*abs(uy) +eta0*abs(wy) … + eta0*abs(vx) +eta0*abs(ux) +eta0*abs(wx) … + eta0*abs(vz) +eta0*abs(uz) +eta0*abs(wz) the third profile is calculated, but I am not sure whether summing up all individual results is allowed? Do you have any advice? Thanks for any further help. T. Gumbel

Attachments:

• I_shear_middle_allvec.tif
• I_shear_middle.tif
• I_velocity_middle.tif

Posted: 1 decade ago 15 lug 2010, 14:40 GMT-4

I'm glad you found my answer somewhat helpful. Let me clarify, first. I'm not just adding the stress terms; rather the value ( uy+vx) is actually a single component in the tensor matrix I calculate. I suppose it depends on your definition of the stress tensor. Because I'm using a symmetric tensor definition (assumes no spin and is coordinate reference independent -- at least this is what I've been told), my two non-diagonal elements are the same. With a nonsymmetric definition of the stress tensor, tau=viscosity*grad(U), the non-diagonal elements will be different. Honestly, I'm unsure which is the most appropriate, but the best I can offer is this regarding symmetric/non-symmetric definitions: "This velocity gradient tensor is not exactly similar to the stress tensor in that the velocity gradient tensor is not symmetric, if you transpose rows and columns you will have something totally different. The physical interpretation of this is that the velocity gradient measures not only the deformation of the fluid body (by compression and/or shearing), but also the relative orientation of the fluid body (rotation). When I’ll have some more time, I will demonstrate this. Therefore, we cannot write a direct mathematical relationship between stress tensor (tau) and velocity gradient (grad(U)) since the stress tensor doesn’t care at all of the rotation of the body, which has nothing to do with pure deformation. However, the velocity gradient tensor like any similar tensor can be broken into two parts, one symmetric (D) and the other one, antisymmetric (S)." http://www.granular-volcano-group.org/viscous_stress.html The symmetric part of the velocity gradient tensor is used for the stress tensor (tau=2*visc*D). At any rate, back to your question: I don't think it's appropriate to simply add the non-diagonal components of the 3D tensor. My guess is that you want a magnitude rather than a simple sum. Something like the square root of the sum of the squares would make more sense, but again, I'm not for sure. I've heard some people refer to this as a simple "shear modulus", which seems to be another word for magnitude in this case. The only other idea I have is that you may only be concerned with the maximum of any of the shear elements -- maybe their combination doesn't matter, and you only care about the max? Hope this is helpful, I'm still learning about this myself. Please let me know if you find any clear answers! Tyler House

Posted: 1 decade ago 13 lug 2014, 02:48 GMT-4

hey, what further calculations you did in order to find the wall shear stress from the velocity profile slice. please help.