Fluid Mechanics Viscous Dissipation

[Jade Sola]

I am trying to find an expression for viscous dissipation for burger's vortex Velocity field which only has velocity component in the V theta direction. I'm confused as to which equation for viscosity dissipation is correct. I am seeing a lot of different things tau:S, tau:delV..which one is correct?

Thanks!

 

[a_potato]

Um.. I think you need to be a bit more specific. What form are you working with?

 

[Jade Sola]

Um.. I think you need to be a bit more specific. What form are you working with?

cylindrical coordintates...so this is what the velocity field is

what I am confused about is what exactly is the equation for viscous dissipation ... My professor said in class that it is 2muS:S (from Navierstokes eq.3) but I am seeing differen things online and I also heard from another classmate of mine that it has something to do with Sxtau components. I am just confused ...sorry if I am confusing the matter here.

 

[Chestermiller]

cylindrical coordintates...so this is what the velocity field is
View attachment 75868
what I am confused about is what exactly is the equation for viscous dissipation ... My professor said in class that it is 2muS:S (from Navierstokes eq.3) but I am seeing differen things online and I also heard from another classmate of mine that it has something to do with Sxtau components. I am just confused ...sorry if I am confusing the matter here.

If S is the rate of deformation tensor (which it certainly appears to be), then:
S = (∇V+∇V^T)/2
and
τ = 2μS
So, the rate of viscous dissipation is τ:∇V = τ:S = 2μS:S. All the relations you wrote are the same, and should give the same results.

Chet

 

[alphy]

Hello,

Thank you for your question. Viscous dissipation is an important aspect of fluid mechanics, and it is often expressed in different ways depending on the specific flow conditions and equations used. In the case of Burger's vortex velocity field, which only has a velocity component in the V theta direction, the correct equation for viscous dissipation would be the expression tau:delV, where tau is the shear stress and delV is the velocity gradient in the direction of the shear stress.

The expression tau:S, where S is the rate of strain tensor, is also a valid equation for viscous dissipation, but it is typically used for more complex flow fields that have multiple velocity components. In Burger's vortex, since there is only one velocity component, the tau:delV expression would be more appropriate.

It is important to note that both expressions are essentially equivalent and can be derived from the same underlying principles. The choice of which one to use may depend on the specific problem at hand and the available equations and data.

I hope this clarifies your confusion. If you have any further questions, please do not hesitate to ask. Best of luck in your research!