Finding the force acting on a fluid: compressible Navier-Stokes equation

The compressible Navier-Stokes equation is given by

$$
rho[frac{partialtextbf{v}}{partial t}+(textbf{v}cdotnabla)textbf{v}] = -nabla p +etaDeltatextbf{v}+[zeta+frac{eta}{3}]nabla(nablacdottextbf{v})
$$

where $eta$ is dynamic viscosity and $zeta$ is the second viscosity. (This equation is taken from Landau-Lifshitz Fluid Mechanics)

There are a couple of things I need help to figure out:

1. Why is there a $Deltatextbf{v}$ term? Since we are talking about a 3-dimensional vector, shouldn’t we take $nablatextbf{v}$ to determine how it changes over space?
2. I understand that analytical solutions to NS are difficult. But is there a closed form for the force acting on the fluid from the compressible Navier-Stokes equation? If so, could you please suggest a source?

Thank you very much.

PS: I am aware of this post but it does not really answer my question because it seems like viscosity was ignored.