CP502 Advanced Fluid Mechanics

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CP502 Advanced Fluid Mechanics
Flow of Viscous Fluids and Boundary Layer Flow
10 Lectures 3 Tutorials
Computational Fluid dynamics (CFD)
project Midsemester (open book) examination
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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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Recalling vector operations

• Del Operator
• Laplacian Operator
• Gradient

• Vector Gradient
• Divergence
• Directional Derivative

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
5
Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum
kinematic viscosity (constant)
density (constant)
pressure
external force (such as gravity)
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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum
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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum
Acceleration term change of velocity with time
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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum
Advection term force exerted on a particle of
fluid by the other particles of fluid surrounding
it
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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum

• 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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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum

• Pressure term Fluid flows in the direction of
  largest change in pressure

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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum
Body force term external forces that act on the
fluid (such as gravity, electromagnetic, etc.)
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Navier-Stokes equation for incompressible flow of
Newtonian (constant viscosity) fluid
- derived from conservation of momentum
change in velocity with time
body force
advection
diffusion
pressure




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Continuity and Navier-Stokes equations for
incompressible flow of Newtonian fluid
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Continuity and Navier-Stokes equations for
incompressible flow of Newtonian fluid
in Cartesian coordinates
Continuity
Navier-Stokes
x - component
y - component
z - component
15
Steady, incompressible flow of Newtonian fluid in
an infinite channel with stationery plates-
fully developed plane Poiseuille flow
Steady, incompressible flow of Newtonian fluid in
an infinite channel with one plate moving at
uniform velocity - fully developed plane Couette
flow
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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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Steady, incompressible flow of Newtonian fluid in
a pipe- fully developed pipe Poisuille flow
Fixed pipe
r
z
Fluid flow direction
2a
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Steady, incompressible flow of Newtonian fluid
between a stationary outer cylinder and a
rotating inner cylinder- fully developed pipe
Couette flow