Navier-Stokes

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Navier-Stokes
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Eulerian View

• In the Lagrangian view each body is described at
  each point in space.
• Difficult for a fluid with many particles
• In the Eulerian view the points in space are
  described.
• Bulk properties of density and velocity

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Streamlines

• A streamline follows the tangents to fluid
  velocity.
• Lagrangian view
• Dashed lines at left
• Stream tube follows an area
• A streakline (blue) shows the current position of
  a particle starting at a fixed point.
• A pathline (red) tracks an individual particle.

Wikimedia image
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Fluid Change

• A change in a property like pressure depends on
  the view.
• In the Lagrangian view the total time derivative
  depends on position and time.
• The Eulerian view uses just the partial
  derivative with time.
• Points in space are fixed

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Jacobian Tensor

• A general coordinate transformation can be
  expressed as a tensor.
• Partial derivatives between two systems
• Jacobian N?N real matrix
• Inverse for nonsingular Jacobians
• Cartesian coordinate transformations have an
  additional symmetry.
• Not generally true for other transformations

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Volume Element

• An infinitessimal volume element is defined by
  coordinates.
• dV dx1dx2dx3
• Transform a volume element from other
  coordinates.
• components from the transformation
• The Jacobian determinant is the ratio of the
  volume elements.

x3
x2
x1
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Compressibility

• A change in pressure on a fluid can cause
  deformation.
• Compressibility measures the relationship between
  volume change and pressure.
• Usually expressed as a bulk modulus B
• Ideal liquids are incompressible.

V
p
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Volume Change

• Consider a fixed amount of fluid in a volume dV.
• Cubic, Cartesian geometry
• Dimensions dx, dy, dz
• The change in dV is related to the divergence.
• Incompressible fluids - no velocity divergence

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Balance Equations

• The equation of motion for an arbitrary density
  in a volume is a balance equation.
• Current J through the sides of the volume
• Source s inside the volume
• Additional balance equations describe
  conservation of mass, momentum and energy.
• No sources for conserved quantities

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Mass Conservation

• A mass element must remain constant in time.
• Conservation of mass
• Combine with divergence relationship.
• Write in terms of a point in space.

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Pressure Force

• Each volume element in a fluid is subject to
  force due to pressure.
• Assume a rectangular box
• Pressure force density is the gradient of pressure

dV
dz
dy
p
dx
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Equation of Motion

• A fluid element may be subject to an external
  force.
• Write as a force density
• Assume uniform over small element.
• The equation of motion uses pressure and external
  force.
• Write form as force density
• Use stress tensor instead of pressure force
• This is Cauchys equation.

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Eulers Equation

• Divide by the density.
• Motion in units of force density per unit mass.
• The time derivative can be expanded to give a
  partial differential equation.
• Pressure or stress tensor
• This is Eulers equation of motion for a fluid.

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Momentum Conservation

• The momentum is found for a small volume.
• Euler equation with force density
• Mass is constant
• Momentum is not generally constant.
• Effect of pressure
• The total momentum change is found by
  integration.
• Gauss law

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Energy Conservation

• The kinetic energy is related to the momentum.
• Right side is energy density
• Some change in energy is related to pressure and
  volume.
• Total time derivative
• Volume change related to velocity divergence

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Work Supplied

• The work supplied by expansion depends on
  pressure.
• Potential energy associated with change in volume
• This potential energy change goes into the energy
  conservation equation.

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Bernoullis Equation

• Gravity is an external force.
• Gradient of potential
• No time dependence
• The result is Bernoullis equation.
• Steady flow no time change
• Integrate to a constant

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Strain Rate Tensor

• Rate of strain measures the amount of deformation
  in response to a stress.
• Forms symmetric tensor
• Based on the velocity gradient

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Stress and Strain

• There is a general relation between stress and
  strain
• Constants a, b include viscosity
• An incompressible fluid has no velocity
  divergence.

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Navier-Stokes Equation

• The stress and strain relations can be combined
  with the equation of motion.
• Reduces to Euler for no viscosity.

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Bernoulli Rederived

• Make assumptions about flow to approximate fluid
  motion.
• Incompressible
• Inviscid
• Irrotational
• Force from gravity
• Apply to Navier-Stokes
• The result is Bernoullis equation.