thermodynamics - heat generation due to viscosity in a 3D fluid flow

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:

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