Fluid Mechanics Viscous Dissipation

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SUMMARY

The discussion focuses on the expression for viscous dissipation in the context of Burger's vortex velocity field, specifically in cylindrical coordinates. The correct equation for viscous dissipation is established as τ:S = 2μS:S, where τ represents the viscous stress tensor and S is the rate of deformation tensor defined as S = (∇V + ∇VT)/2. The confusion arises from varying interpretations of the equations, but all lead to the same conclusion regarding viscous dissipation in fluid mechanics.

PREREQUISITES
  • Understanding of Navier-Stokes equations
  • Familiarity with cylindrical coordinates in fluid dynamics
  • Knowledge of the rate of deformation tensor (S)
  • Basic concepts of viscous stress tensor (τ)
NEXT STEPS
  • Study the derivation of the Navier-Stokes equations in cylindrical coordinates
  • Learn about the properties and applications of the rate of deformation tensor (S)
  • Research the relationship between viscous stress tensor (τ) and velocity gradients
  • Explore advanced topics in fluid mechanics, such as turbulence and energy dissipation
USEFUL FOR

This discussion is beneficial for students and professionals in fluid mechanics, particularly those studying viscous flows, as well as researchers focusing on vortex dynamics and energy dissipation in fluids.

Jade Sola
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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!
 
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Um.. I think you need to be a bit more specific. What form are you working with?
 
a_potato said:
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
upload_2014-11-26_18-20-1.png

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.
 
Jade Sola said:
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+∇VT)/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
 

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