Viscosity and The Navier Stokes Equations

1
Viscosity and The Navier Stokes Equations
2
Viscosity

• Molten Metals 110-3 to 110-2 Pa-s
• Rate Activated Process
• Arrhenius Type Behaviour
• Ceramics
• Molten Glass, other ceramics
• Difficult to perform high T flow experiments
• Slurries

3
Viscosity

• Slurries Multicomponent mixtures where flow is
  affected by composition
• The ability to generally characterize fluid
  behaviour is important.
• Viscosity modification for proper flow, settling,
  etc.
• Polymers MW, Temp Dependence

4
Viscosity

• Natural Materials
• Cheese Whiz
• Honey
• Molasses
• Blood (Cardiology)
• Oil (Pipeline Processing)

5
How Measured?

• Cup and Bob Viscometer
• Cone and Plate Viscometer
• Other qualitative gauges like melt flow index

6
Navier Stokes Equations

• Simplified way of grouping terms together
  between material and process specific, and
  geometric constraints
• Equation of Continuity
• Equation of Motion (Energy Balance)

7
Equation of Continuity

• Mass Balance over a control volume
• dM/dt rate of mass in - rate of mass out
• 3 dimensions over which flow can occur
• d?/dt -d/dx (?vx) d/dy(?vy) d/dz(?vz)
• Compare for simple flows
• Compare for an incompressible fluid (liquids are
  generally much easider to consider

8
Equation of Motion

• Force Balance
• dMomentum/dt Momentum (Rate in - Rateout )?
  Forces acting on control V
• See p.107 (Gaskell)
• Break out individual components

9
Equation of Energy

• There is a corresponding equation of mechanical
  energy comparing the rate of energy change in a
  fluid to convection, work(pv), heating the fluid
  (friction), and gravity
• Show Equation (BSL)

10
Navier Stokes Equations

• So by performing balances of mass, momentum, and
  energy, under steady state conditions, there are
  no accumulation terms dM/dt, DMomentum/dt DE/dt
  0
• It also shows from a first principles perspective
  how fluid flow can be packaged into a
  comprehensive approach

11
Simplifications to NS

• Constant ???? can pull out of the differentials
• 1D Flow...vx and vy 0, vz not a fct of z
• Steady State Dynamic terms 0
• Geometry Coordinate transformations between
  rectilinear and cylindrical e.g.
• Look over pages 113, 115 for relation to already
  solved problems