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.

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

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

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.)

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

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

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?

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

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

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

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

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

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

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

Application of Bernoulli Equation: Stagnation Tube

Stagnation Tube V2=0 & z1 = z2

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

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

Class Exercises: (Problem 4.42)

Class Exercises: (Problem 4.59)