1. CP502 Advanced Fluid Mechanics
Flow of Viscous Fluids
Set 01

2. What do we mean by ‘Fluid’?
Physically: liquids or gases
Mathematically:
A vector field u (represents the fluid velocity)
A scalar field p (represents the fluid pressure)
fluid density (d) and fluid viscosity (v)
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3. Recalling vector operations
Del Operator:
Laplacian Operator:
Gradient:
Vector Gradient:
Divergence:
Directional Derivative:
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4. Continuity equation for incompressible (constant density) flow
- derived from conservation of mass
where u is the velocity vector
u, v, w are velocities in x, y, and z directions
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5. Continuity equation derivation
Mass flux out of differential volume
Rate of change of mass in differential volume
Mass flux into differential volume
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6. Continuity equation derivation
Mass flux into differential volume
= Mass flux out of differential volume
+ Rate of change of mass in differential volume = +
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7. known as one dimensional Continuity equation
Continuity equation in 1-dimension
known as one dimensional Continuity equation
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8. Continuity equation in 3-dimension
where u, v, w are velocities in x, y, and z directions
divergence
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9. Continuity equation for incompressible flow
Density is constant for incompressible flow: or
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10. kinematic viscosity (constant)
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
kinematic viscosity (constant)
density (constant)
pressure
external force
(such as gravity)
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11. Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ ρ υ
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12. Acceleration term: change of velocity with time
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
Acceleration term: change of velocity with time
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13. Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
Advection term:
force exerted on a particle of fluid by the other particles of fluid surrounding it
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14. Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
viscosity (constant) controlled
velocity diffusion term:
(this term describes how fluid motion is damped)
Highly viscous fluids stick together (honey)
Low-viscosity fluids flow freely (air)
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15. Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
Pressure term: Fluid flows in the direction of largest change in pressure
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16. Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
Body force term: external forces that act on the fluid (such as gravity, electromagnetic, etc.)
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17. Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum ρ υ
change in velocity
with time
body force =
advection +
diffusion +
pressure +
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18. Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluidρ υ
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19. Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid
in Cartesian coordinates
Continuity:
Navier-Stokes:
x - component:
y - component:
z - component:
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20. Steady, incompressible flow of Newtonian fluid in an infinite channel with stationery plates - fully developed plane Poiseuille flow
Fixed plate
Fluid flow direction h x y
Steady, incompressible flow of Newtonian fluid in an infinite channel with one plate moving at uniform velocity - fully developed plane Couette flow
Fixed plate
Moving plate h x y
Fluid flow direction
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21. Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid
in cylindrical coordinates
Continuity:
Navier-Stokes:
Radial component:
Tangential component:
Axial component:
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22. Steady, incompressible flow of Newtonian fluid in a pipe - fully developed pipe Poisuille flow
Fixed pipe 2a φ r z
Fluid flow direction 2a
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23. Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder - fully developed pipe Couette flow φ aΩ a b r
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