If the molecular collisions are elastic will there be any dissipation in a fluid? Viscosity arises due to collisions of the molecules of one layer of a fluid with another in contact. But viscosity is a dissipative element leading to heating and dissipation. Where does it heat come from? Does it come from the molecular collisions being inelastic? If the collisions were elastic, would there be no viscosity or dissipation in a fluid?
 A: What is happening is that molecules from one layer collide with those of adjacent layers, transferring both momentum and kinetic energy.  If there is an organized motion of the molecules (e.g., mean velocity gradient), the kinetic energy of the organized motion is continually converted to random kinetic energy as a result of the collisions.  This happens even if the collisions between molecules are elastic. The increase in random kinetic energy is equivalent to an increase in internal energy of the fluid (i.e., temperature).  So, what the viscous behavior of the fluid does is convert more useable mechanical energy to internal energy.  An example of this is steady shear of a fluid between parallel plates, where mechanical energy (shear work) is continually applied at the boundary, but is continually being converted to random kinetic energy within the fluid.
A: The viscosity arises because of the elastic collisions, not in spite of them.
When temperature in the fluid rises, it is because the molecules are moving faster, due to elastic collisions. The energy is called "heat" because the motion is disorganized (random).
Its entropy has increased.
A: Viscosity arises due to collisions of the molecules of one layer of a fluid with another in contact.
Viscosity is due to intermolecular forces that resist relative motion within a fluid. The viscosity of a liquid can be defined as the force of friction between layers of the liquid that move relative to each other at different velocities and that resists that relative motion.
But viscosity is a dissipative element leading to heating and dissipation. Where does it heat come from?
Yes viscosity is a dissipative effect. It is the action of the friction force between the layers of liquid that increases the kinetic energy (temperature) of the liquid molecules at the interface. Heat transfer then occurs between the fluids at the interface to fluid at lower temperature away from the interface. A dry friction analogy is vigorously rubbing ones hands together to warm them. The friction force acting along the surface of the epidermis (friction work) elevates the temperature of the epidermis. 
Does it come from the molecular collisions being inelastic? If the collisions were elastic, would there be no viscosity or dissipation in a fluid?
I don’t believe collision elasticity is the principle factor involved, although there is an exchange of molecular momentum between the layers due to collisions. I believe the main reason for viscosity is the intermolecular forces between the fluid molecules. These forces are responsible for the motion of one layer of fluid attempting to drag along an adjacent layer. The greater these forces are (greater viscosity) the greater the resistance to relative motion between fluid layers. 
Hope this helps.
