This week we are still looking at viscous flow and more on the Navier-Stokes equations.
We also will touch on Newtonian viscous fluids. These satisfy Newton’s Law of Viscosity. For an incompressible fluid, this law takes the form of:
Note that the term is called the dynamic viscosity and is constant, at a fixed temperature and pressure, for a given Newtonian fluid. Also if this is equal to 0 then the inviscid form of sigma(ij) is immediately recovered. This has dimensions: mass/(length*time).
If we write the momentum conservation equation using suffix notation, we obtain:
But since V is fixed, we can write:
Also since sigma(i)=sigma(ij)n(j) then the momentum conservation equation can be written as:
But if we simplify this further and set the surface integrals equal to volume integrals and combining the terms, we get:
This equation expresses the momentum conservation principle for a general fluid.
If we insert the expression we have for sigma(ij) (at the start of this blog) gives the Navier-Stokes equations for an incompressible fluid, which can be written as:
These are the Navier-Stokes equations for an incompressible, constant viscosity fluid. This can be written in vector notation:
To find solutions of the Navier-Stokes and continuity equations, we require boundary conditions (and initial conditions if the problem is unsteady).
We also touched on simple shear flow which can be solved using the Navier-Stokes equations with boundary conditions, since we are not continuing on this next week, I will come back and explain this in more detail later in the year.