CHAPTER 4 EULERS EQUATION Engineering Fluid Mechanics 8/E by
Crowe, Elger, and Roberson Copyright 2005 by John Wiley & Sons,
Inc. All rights reserved. Dr. Ercan Kahya Fluid Mechanics
Slide 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
Slide 3
EULERS EQUATION To predict pressure variation in moving fluid
Eulers 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
Slide 4
EULERS 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.) Taking the limit of the two terms at left
side at a given time as l 0 When a = 0 Euler equation reduces to
hydrostatic equation! ACCELERATION IS IN THE DIRECTION OF
DECREASING PIEZOMETRIC PRESSURE!!!
Slide 5
EULERS EQUATION Open tank is accelerated to the right at a rate
a x 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 (ABCD) 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 An example of Euler
Equation is to the uniform acceleration of in a tank:
Slide 6
EULERS EQUATION The component of acceleration in the l
direction: a x cos Apply the above equation along AB Apply the
above equation along DC
Slide 7
Example 4.3: Eulers equation The truck carrying gasoline ( =
6.60 kN/m3) and is slowing down at a rate of 3.05 m/s 2. 1) What is
the pressure at point A? 2) Where is the greatest pressure & at
what value in that point?
Slide 8
Solution: Apply Eulers equation along the top of the tank; so z
is constant Assume that deceleration is constant Pressure does not
change with time Pressure variation is hydrostatic in the vertical
direction Along the top the tank Eulers equation in vertical
direction: (Note that a z =0)
Slide 9
Centripetal (Radial) Acceleration a r = centripetal (radial)
acceleration, m/s 2 V t = tangential velocity, m/s r = radius of
rotation, m = angular velocity, rad/s For a liquid rotating as a
rigid body: V = r
Slide 10
Pressure Distribution in Rotating Flow When flow is rotating,
fluid level will rise away from the direction of net acceleration
Pressure variation 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 Note that this is
not the Bernoulli equation
Slide 11
Example 4.4: Find the elevation difference between point 1 and
2 p 1 = p 2 = 0 and r 1 = 0, r 2 = 0.25m then z 2 z 1 = 0.051m
& Note that the surface profile is parabolic
Slide 12
Pressure Distribution in Rotating Flow p = pressure, Pa =
specific weight, N/m3 z = elevation, m = rotational rate,
radians/second r = distance from the axis of rotation Another
independent equation; The sum of water heights in left and right
arms should remain unchanged
Slide 13
Bernoulli Equation Integrating Eulers equation along a
streamline in a steady flow of an incompressible, inviscid fluid
yields the Bernoulli equation: z: Position p/ : Pressure head V 2
/2g: Velocity head C: Integral constant
Slide 14
Application of Bernoulli Equation Bernoulli Equation:
Piezometric pressure : p + z Kinetic pressure : V 2 /2 For the
steady flow of incompressible fluid inviscid fluid the sum of these
is constant along a streamline
Slide 15
Application of Bernoulli Equation: Stagnation Tube
Slide 16
Stagnation Tube V 2 =0 & z 1 = z 2
Slide 17
Application of Bernoulli Equation: Pitot Tube Bernoulli
equation btw static pressure pt 1 and stagnation pt 2; V 2 = 0 then
Pitot tube equation;
Slide 18
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
