1. Navier-Stokes

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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.

11. 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

12. 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 Cauchy’s equation.

13. Euler’s 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 Euler’s equation of motion for a fluid.

14. 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

15. 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

16. 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.

17. Bernoulli’s Equation Gravity is an external force.
Gradient of potential
No time dependence
The result is Bernoulli’s equation.
Steady flow no time change
Integrate to a constant

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

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

20. Navier-Stokes Equation
The stress and strain relations can be combined with the equation of motion.
Reduces to Euler for no viscosity.

21. Bernoulli Rederived
Make assumptions about flow to approximate fluid motion.
Incompressible
Inviscid
Irrotational
Force from gravity
Apply to Navier-Stokes
The result is Bernoulli’s equation.