To allow for heat effects in a fluid, you need to couple the Navier-Stokes equations, (momentum conservation) which BTW contain the continuity equation for mass conservation too, to the energy (or temperature) equation (energy conservation).

Momentum dissipation in the momentum equation

$$ \frac{\partial v}{\partial t} + (\vec{v}\cdot\nabla)\vec{v} = -\frac{\nabla p}{\rho} + \frac{1}{\rho}\nabla S $$

is more correctly described by the divergence of the symmetric stress tensor $S$

$$ S = \rho \,\nu (\nabla \circ\vec{v} + (\nabla \circ\vec{v})^{T}) + \rho \,\eta\, I (\nabla \cdot \vec{v}) $$

The coefficients $\mu$ and $\nu$ denote the dynamic and kinematic viscosity respectively, $\circ$ is the tensor (outer) product, and $I$ is the unity tensor.

In the energy equation, the momentum dissipation leads to a corresponding positive definite dissipation (frictional heating) $\epsilon$

$$ \epsilon = \frac{1}{\rho}(S \, \nabla) \cdot \vec{v} $$

The energy equation can, dependent on the system considered, have different sources of diabatic heating apart from the radiative heating (of which Newtonian cooling es a spacial case), such as latent heating due to phase transitions for example.