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.