1. MAE 5130: VISCOUS FLOWS Examples Utilizing The Navier-Stokes Equations
September 16, 2010
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. THE NAVIER-STOKES EQUATIONS

3. N/S EQUATION FOR INCOMPRESSIBLE, CONSTANT m FLOW
Start with Newton’s 2nd Law for a fixed mass
Divide by volume
Introduce acceleration in Eulerian terms
Ignore external forces
Only body force considered is gravity
Express all surface forces that can act on an element
3 on each surface (1 normal, 2 perpendicular)
Results in a tensor with 9 components
Due to moment equilibrium (no angular rotation of element) 6 components are independent)
Employ a Stokes’ postulates to develop a general deformation law between stress and strain rate
White Equation 2-29a and 2-29b
Assume incompressible flow and constant viscosity

4. STREAMLINE AND STREAM FUNCTION EXAMPLE
y=0
y=1
y=2
f=0
f=1
f=2

5. STREAMLINE PATTERN: MATLAB FUNCTION
Physical interpretation of flow field
Flow caused by three intersecting streams
Flow against a 120º corner
Flow around a 60º corner
Patterns (2) and (3) would not be realistic for viscous flow, because the ‘walls’ are not no-slip lines of zero velocity

6. STREAMLINE PATTERN: MATLAB FUNCTION

7. STREAMLINE PATTERN: MATLAB FUNCTION

8. STREAMLINE PATTERN: MATLAB FUNCTION

9. u AND v VELOCITY COMPONENTS

10. VELOCITY MAGNITUDE

11. STATIC PRESSURE FIELD

12. TOTAL PRESSURE FIELD: P + ½rV2 = P + ½r(u2 +v2)½

13. ASIDE: MATLAB CAPABILITY FOR STREAMLINE PLOTTING
Altitude 1
Altitude 2
Altitude 3

14. ASIDE: MATLAB CAPABILITY FOR STREAMLINE PLOTTING

15. COEFFICIENT OF VISCOSITY, m
More fundamental approach to viscosity shows it is property of fluid which relates applied stress (t) to resulting strain rate (e)
Consider fluid sheared between two flat plates
Bottom plate is fixed
Top plate moving at constant velocity, V, in positive x-direction only
u = u(y) only
Geometry dictates that shear stress, txy, must be constant throughout fluid
Perform experiment → for all common fluids, applied shear is a unique function of strain rate
For given V, txy is constant, it follows that du/dy and exy are constant throughout fluid, so that resulting velocity profile is linear across plates
Newtonian fluids (air, water, oil): linear relationship between applied stress and strain
Coefficient of viscosity of a Newtonian fluid: m
Dimension: Ns/m2 or kg/ms
Thermodynamic property (related to molecular interactions) that varies with T&P