1. EULER’S EQUATION Fluid Mechanics CHAPTER 4 Dr . Ercan Kahya
Engineering Fluid Mechanics 8/E by Crowe, Elger, and Roberson Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

2. Review of Definitions
Steady flow: velocity is constant with respect to time
Unsteady flow: velocity changes with respect to time
Uniform flow: velocity is constant with respect to position
Non-uniform flow: velocity changes with respect to position
Local acceleration:
– change of flow velocity with respect to time
– occurs when flow is unsteady
• Convective acceleration:
– change of flow velocity with respect to position
– occurs when flow is non‐uniform

3. Assume that the viscous forces are zero
EULER’S EQUATION
To predict pressure variation in moving fluid
Euler’s Equation is an extension of the hydrostatic equation for accelerations other than gravitational
RESULTED FROM APPLYING NEWTON SECOND LAW TO A FLUID ELEMENT IN THE FLOW OF INCOMPRESSIBLE, INVISCID FLUID
Assume that the viscous forces are zero

4. EULER’S EQUATION ACCELERATION IS IN THE DIRECTION OF
Taking the limit of the two terms at left side at a given time as Δl → 0
ACCELERATION IS IN THE DIRECTION OF
DECREASING PIEZOMETRIC PRESSURE!!!
When “a = 0” → Euler equation reduces to hydrostatic equation!
In the x direction,
for example:
“2” and “1” refer to the location with respect to the direction l (When l = x direction, then “2” is the right-most point. When l = z direction, “2” is the highest point.)

5. EULER’S EQUATION Open tank is accelerated to the right at a rate ax
An example of Euler Equation is to the uniform acceleration of in a tank:
Open tank is accelerated to the right at a rate ax
For this to occur; a net force must act on the liquid in the x-direction
To accomplish this; the liquid redistributes itself in the tank (A’B’CD)
– The rise in fluid causes a greater hydrostatic force on the left than the right side
→ this is consistent with the requirement of “F = ma”
– Along the bottom of tank, pressure variation is hydrostatic in the vertical direction

6. EULER’S EQUATION
The component of acceleration in the l direction: ax cosα
Apply the above equation along A’B’
Apply the above equation along DC

7. Example 4.3: Euler’s equation
The truck carrying gasoline (γ = 6.60 kN/m3) and is slowing down at a rate of 3.05 m/s2.
1) What is the pressure at point A?
2) Where is the greatest pressure & at what value in that point?

8. Euler’s equation in vertical direction: (Note that az =0)
Solution:
Apply Euler’s equation along the top of the tank; so z is constant
Assume that deceleration is constant
Pressure does not change with time
Along the top the tank
Euler’s equation in vertical direction:
(Note that az =0)
Pressure variation is hydrostatic in the vertical direction

9. Centripetal (Radial) Acceleration
For a liquid rotating
as a rigid body:
V = ω r
ar = centripetal (radial) acceleration, m/s2
Vt = tangential velocity, m/s
r = radius of rotation, m
ω = angular velocity, rad/s

10. Pressure Distribution in Rotating Flow
A common type of rotating flow is the flow in which the fluid rotates as a rigid body.
Applying Euler Equation in the direction normal to streamlines and outward from the center of rotation (OR INTEGRATING EULER EQUATION IN THE RADIAL DIRECTION FOR A ROTATING FLOW) results in
Pressure variation in rotating flow
Note that this is not the Bernoulli equation
When flow is rotating, fluid level will rise away from the direction of net acceleration

11. Example 4.4: Find the elevation difference between point 1 and 2
p1 = p2 = 0 and r1 = 0 , r2= 0.25m then →
z2 – z1= 0.051m & Note that the surface profile is parabolic

12. Pressure Distribution in Rotating Flow
Another independent equation;
The sum of water heights in left and right arms should remain unchanged
p = pressure, Pa
γ = specific weight, N/m3
z = elevation, m
ω = rotational rate, radians/second
r = distance from the axis of rotation

13. Bernoulli Equation z: Position p/γ: Pressure head V2/2g: Velocity head
Integrating Euler’s equation along a streamline in a steady flow of an incompressible, inviscid fluid yields the Bernoulli equation:
z: Position
p/γ: Pressure head
V2/2g: Velocity head
C: Integral constant

14. Application of Bernoulli Equation
– Piezometric pressure : p + γz
– Kinetic pressure : ρV2/2
For the steady flow of incompressible fluid inviscid fluid the sum of these is constant along a streamline

15. Application of Bernoulli Equation: Stagnation Tube

16. Stagnation Tube
V2=0 & z1 = z2

17. Application of Bernoulli Equation: Pitot Tube
Bernoulli equation btw static pressure pt 1 and stagnation pt 2;
V2 = 0 then Pitot tube equation;

18. Solve for pressure differential
VENTURI METER
The Venturi meter device measures the flow rate or velocity of a fluid through a pipe. The equation is based on the Bernoulli equation, conservation of energy, and the continuity equation.
Solve for flow rate
Solve for pressure differential

19. Class Exercises: (Problem 4.42)

20. Class Exercises: (Problem 4.59)