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