1. Eulerian Description
• A finite volume called a flow domain or control volume is
defined through which fluid flows in and out.
• There is no need to keep track of the position and
velocity of a fluid particle of fixed identity.
• Field variables such as pressure, velocity and
acceleration fields are defined.
• Pressure field:
Velocity field:
Acceleration field:

2. Difference Between Lagrangian and Eulerian Description
• Imagine a person standing beside a river measuring its
properties. In the Lagrangian approach, he throws in a
probe that moves downstream with the water. In the
Eulerian approach, he anchors the probe at a fixed
location in the water.
• Experimental measurements are more suitable to the
Eulerian Description.
• Equations of motion of fluid flow in Lagrangian
description are well defined (Newtons’ second law), but
needs to be carefully derived for the Eulerian description

3. Example 1 A Steady 2-dimensional Velocity Field
A steady, incompressible, two dimensional velocity field is
given by
where the x- and y-coordinates are in meters and the
magnitude of velocity is in m/s. A stagnation point is
defined as a point in the flow field where the velocity is zero. (a) Determine if there are any stagnation points in this flow field and, if so, where? (b) Sketch velocity vectors at several locations in the domain between x = -2 m to 2 m and y = 0 m to 5 m; qualitatively
•describe the flow field.

5. Acceleration Field
Transforming from the Lagrangian to the Eulerian frams of reference ,

6. Acceleration Field
For steady flow, local acceleration is zero, but advective acceleration is nonzero even for steady flows.

7. Example 2 • Nadeen is washing her car, using a nozzle. The nozzle is
3.90 in (0.325 ft) long, with an inlet diameter of in
( ft) and an outlet diameter of in. The volume
flow rate through the garden hose (and through the nozzle)
is V =0.841 gal/min ( ft3/s), and the flow is steady.
Estimate the magnitude of the acceleration of a fluid
particle moving down the centerline of the nozzle

8. Material Derivative Material derivative: Material acceleration:
Material derivative of pressure:
Material derivative is also called total, particle, Lagrangian, Eulerian and substantial derivative

9. Material Acceleration of a Steady Velocity Field
Consider the steady incompressible two-dimensional
velocity field. (a) Calculate the material acceleration at
the point (x=2 m, y= 3m).

10. Flow visualization Spinning baseball at speed
77 ft/s, rotated at 630 ppm.
Streamlines: A streamline is a curve that is everywhere
tangent to the instantaneous local velocity vector. They are useful indicators of the instantaneous direction of fluid motion.
Stremline in the xy - plane

11. Example 3 Consider the steady incompressible two-dimensional
velocity field. Plot several streamlines in the right half of
the flow (x>0)
Note: Velocity vectors are superimposed on the streamlines. The speed can not be determined directly from the streamlines alone.

12. Streamtube A streamtube consists of a bundle of streamlines. Since
streamlines are everywhere parallel to the local velocity,
fluid cannot cross a streamline.
• Fluid within a streamtube must remain there and cannot
cross the boundary of the streamtube.
• Both streamlines and streamtubes are instantaneous
quantities, defined at a particular instant in time according
to the velocity field at that instant.

13. • In an unsteady flow, the streamline pattern may change
significantly with time, but the mass flow rate passing
through any cross-sectional slice of a given streamtube
must remain the same.
• For example, in a converging portion of an incompressible
flow field, the diameter of the streamtube must decrease
as the velocity increases. Likewise, the streamtube
diameter increases in diverging portions of the
incompressible flow

14. Pathlines
• A pathline is the actual path traveled by an individual fluid
particle over some time period.
• A pathline is a Lagrangian concept in that we simply
follow the path of an individual particle.
• For steady flow pathlines are identical to streamlines

15. Streaklines
• A streakline is the locus of fluid particles that have passed
sequentially through a prescribed point in the flow.
• Streaklines are the most common flow pattern generated
in a physical experiment. If a small tube is inserted into a
flow and introduce a continuous stream of tracer fluid
(dye in a water flow or smoke in an airflow), the observed
pattern is a streakline.

16. • Streaklines are often confused with streamlines or pathlines.
• While the three flow patterns are identical in steady flow,
they can be quite different in unsteady flow.
• The main difference is that a streamline represents an
instantaneous flow pattern at a given instant in time,
while a streakline and a pathline are flow patterns that
have some age and thus a time history associated with
them.
• A streakline is an instantaneous snapshot of a time
integrated flow pattern.
• A pathline, on the other hand, is the time-exposed flow
path of an individual particle over some time period.