2
$\begingroup$

Consider an arbitrary 3D fluid flow:

$$\vec{\nu}=\vec{V}\left( \vec{x} ,t \right) \tag{1}$$

where velocity at each point $\vec{\nu}$ is a function $\vec{V}$ of position $\vec{x}$ and time $t$ (non-steady). Due to the viscosity $\mu$ there is heat being generated at each point in space. I know this heat should be a function of velocity, viscosity and div/curl of velocity:

$$ \dot{q}=F\left(\vec{\nabla},\vec{\nu},\mu \right) \tag{2}$$

But I can't just find it out.

For a 2D problem with unidirectional horizontal velocity of:

$$ \nu_x=V_x\left(x,y,t\right) \tag{3}$$

I think I can write

$$ \dot{q}=\mu \frac{\partial \nu_x}{\partial y}\nu_x \tag{4}$$

Where $\tau_x=\mu \frac{\partial \nu_x}{\partial y} $ is the shear force due to viscosity. But I'm not quite sure if that's correct.

I would appreciate if you could help me know what is the correct form for equation 2.

P.S. I'm not quite sure but I think it should be something like:

$$ \dot{q}=\mu \left( \vec{\nu} \times \vec{\nabla}\right).\vec{\nu} \tag{5}$$

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

It's not what you have written. The rate of viscous heat generation per unit volume is given by $$2\mu\mathbf{E}:\mathbf{E}$$ where $\mathbf{E}$ is the rate of deformation tensor. For the shear flow you mentioned, this reduces to $\mu (dv/dy)^2$. See Transport Phenomena by Bird, et al for the full development.

Improve this answer
$\endgroup$
20
  • 1
    $\begingroup$ I don't understand this notation. how should I write $E$? and what type of operator $:$ is? $\endgroup$
    – Foad
    Commented Feb 19, 2018 at 13:50
  • 1
    $\begingroup$ $\mathbf{E}=(\nabla \mathbf{v}+(\nabla \mathbf{v})^T)/2$ (the velocity gradient tensor plus its transpose divided by 2), and $\mathbf{E}:\mathbf{E}$ is the double contraction of the rate of deformation tensor with itself. See the reference I provided, if you are not familiar working with tensors. $\endgroup$
    – Chet Miller
    Commented Feb 19, 2018 at 14:30
  • 1
    $\begingroup$ Appendix A covers vector and tensor analysis. Consider particularly section A.3 which covers dyadic tensor notation and manipulation, and how to get dot products and double-dot products of tensors. $\endgroup$
    – Chet Miller
    Commented Feb 19, 2018 at 16:09
  • 1
    $\begingroup$ Yes. That looks correct. $\endgroup$
    – Chet Miller
    Commented Feb 19, 2018 at 19:29
  • 1
    $\begingroup$ By the way, Nice Job. $\endgroup$
    – Chet Miller
    Commented Feb 20, 2018 at 13:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.