FC-CIV MECAFLUI: Fluid Mechanics Session 24:
Navier-Stokes Equation
Eusebio Ingol Blanco, Ph.D.
Civil Engineering Program, San Ignacio de Loyola University
Objective
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Understand the meaning of Navier-Stokes (N-S) equation and its derivation.
Apply the N-S equation to fluid mechanics problems
Navier-Stokes Equation
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
For fluids at rest, the only stress on a fluid element is
the hydrostatic pressure, which always acts inward and
normal to any surface.
ij, called the viscous stress tensor or the
deviatoric stress tensor
Mechanical pressure is the mean normal
stress acting inwardly on a fluid element.
Newtonian versus Non-Newtonian Fluids
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Rheology: The study of the deformation of
flowing fluids.
Newtonian fluids: Fluids for which the shear
stress is linearly proportional to the shear
strain rate.
Non-Newtonian fluids: Fluids for which the
shear stress is not linearly related to the shear
strain rate.
Viscoelastic: A fluid that returns (either fully
or partially) to its original shape after the
applied stress is released.
Some non-Newtonian fluids are called shear
thinning fluids or pseudoplastic fluids,
because the more the fluid is sheared, the less
viscous it becomes.
Plastic fluids are those in which the shear
thinning effect is extreme.
Rheological behavior of fluids
shear stress as a function of shear
strain rate.
Derivation of the NavierStokes Equation for Incompressible, Isothermal Flow
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
The incompressible flow approximation implies constant density,
and the isothermal approximation implies constant
viscosity.
ij is the strain rate tensor
Derivation of the NavierStokes Equation for Incompressible, Isothermal Flow
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
In Cartesian coordinates the stress tensor becomes:
We substitute the previous equation into the three Cartesian components of Cauchys
equation. For the x-component
This is the continuity equation = 0
The Laplacian of
Velocity component u
Derivation of the NavierStokes Equation for Incompressible, Isothermal Flow
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
The Laplacian operator, shown here in both Cartesian and
cylindrical coordinates, appears in the viscous term of the
incompressible NavierStokes equation.
Derivation of the NavierStokes Equation for Incompressible, Isothermal Flow
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
We combine the last three components into one vector equation, to get
the Navier-Stokes equation for incompressible flow with constant
viscosity
Continuity and NavierStokes Equations in Cartesian Coordinates
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Continuity and NavierStokes Equations in Cylindrical Coordinates
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Continuity and NavierStokes Equations in Cylindrical Coordinates
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
The six independent components of the viscous stress tensor in cylindrical coordinates:
Differential Analysis of Fluid Flow Problems
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
There are two types of problems for which the differential equations (continuity and
NavierStokes) are useful:
Calculating the pressure field for a known velocity field
Calculating both the velocity and pressure fields for a flow of known geometry and known boundary conditions
Example 9-13
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-13
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-13
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-14
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-14
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-14
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-14
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Example 9-14
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
The two-dimensional line
vortex is a simple
approximation of a tornado; the
lowest pressure is at the center
of the vortex.
Exact Solutions of the Continuity and NavierStokes Equations
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Boundary Conditions
A piston moving at speed VP in a cylinder. A thin film of oil is sheared between the piston and the cylinder; a
magnified view of the oil film is shown. The no-slip boundary condition requires that the velocity of fluid
adjacent to a wall equal that of the wall.
Exact Solutions of the Continuity and NavierStokes Equations
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
At an interface between two fluids,
the velocity of the two fluids must
be equal. In addition, the shear
stress parallel to the interface must
be the same in both fluids.
Along a horizontal free surface of
water and air, the water and air
velocities must be equal and the shear
stresses must match. However, since
air
Exact Solutions of the Continuity and NavierStokes Equations
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Boundary conditions along a plane of symmetry are defined so as to ensure that
the flow field on one side of the symmetry plane is a mirror image of that on the
other side, as shown here for a horizontal symmetry plane.
Problems 9-90 and 9-91
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Review solved examples, 9-16, 9-17, 9-18
Problem 9-90
Problem 9-91
Problem 9-106
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Summary
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Navier-Stokes Equation and Derivation
Navier-Stokes equation in Cartesian and cylindrical coordinates
Differential analysis of fluid flow problems
Applications
Homework
Universidad San Ignacio de Loyola Eusebio Ingol Blanco, Ph.D.
Study sections: 9-5 and 9-6
Hw8 is going to be posted soon.