CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics
Chapter 3: Newtonian Fluids
CM4650 Polymer Rheology Michigan Tech
Navier-Stokes Equation
v vv p 2 v g t
1 © Faith A. Morrison, Michigan Tech U.
Chapter 3: Newtonian Fluid Mechanics
TWO GOALS
•Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields
QUICK START V W
x First, before we get deep into 2 v (x ) H derivation, let’s do a Navier-Stokes 1 2 x1 problem to get you started in the x3 mechanics of this type of problem solving.
2 © Faith A. Morrison, Michigan Tech U.
1 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics
EXAMPLE: Drag flow between infinite parallel plates •Newtonian •steady state •incompressible fluid •very wide, long V •uniform pressure W
x2
v1(x2) H
x1 x3
3
EXAMPLE: Poiseuille flow between infinite parallel plates
•Newtonian •steady state •Incompressible fluid •infinitely wide, long W x2
2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL
4
2 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics
Engineering Quantities of In more complex flows, we can use Interest general expressions that work in all cases. (any flow)
volumetric ⋅ flow rate
∬ ⋅ | average 〈 〉 velocity ∬
Using the general formulas will Here, is the outwardly pointing unit normal help prevent errors. of ; it points in the direction “through”
5 © Faith A. Morrison, Michigan Tech U.
The stress tensor was Total stress tensor, Π: invented to make the calculation of fluid stress easier. Π ≡
b (any flow, small surface) dS nˆ Force on the S ⋅ Π surface V (using the stress convention of Understanding Rheology)
Here, is the outwardly pointing unit normal of ; it points in the direction “through”
6 © Faith A. Morrison, Michigan Tech U.
3 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics
To get the total force on the macroscopic surface , we integrate over the entire surface of interest.
‐Π
Fluid force on ⋅ the surface S