Chapter 5 – Fluid In Motion – The Bernoulli Equation

Chapter 5 – Fluid in Motion – The Bernoulli EquationBERNOULLI EQUATIONThe motion of a fluid is usually extremely complex.The study of a fluid at rest, or in relative equilibrium,was simplified by the absence of shear stress, butwhen a fluid flows over a solid surface or otherboundary, whether stationary or moving, the velocityof the fluid in contact with the boundary must be thesame that the boundary, and a velocity gradient iscreated at right angle to the boundary.The resulting change of velocity from layer to layerof fluid flowing parallel to the boundary gives rise toshear stresses in the fluid.Individual particles of fluid move as a result of theaction of forces set up by differences of pressure ofelevation.Their motion is controlled by their inertia and theeffect of the shear stresses exerted by the surroundingfluid.The resulting motion is not easily analysedmathematically, and it is often necessary tosupplement theory by experiment.1

Chapter 5 – Fluid in Motion – The Bernoulli EquationMotion of Fluid Particles and Streams1. Streamline is an imaginary curve in the fluid acrosswhich, at a given instant, there is no flow.Figure 12. Steady flow is one in which the velocity, pressureand cross-section of the stream may vary frompoint to point but do not change with time.If, at a given point, conditions do change with time,the flow is described as unsteady flow.3. Uniform flow occur if the velocity at a giveninstant is the same in magnitude and direction atevery point in the fluid.If, at the given instant, the velocity changes frompoint to point, the flow is described as non-uniformflow.2

Chapter 5 – Fluid in Motion – The Bernoulli EquationFigure 23

Chapter 5 – Fluid in Motion – The Bernoulli Equation4. Real fluid is a fluid which when it flows past aboundary, the fluid immediately in contact with theboundary will have the same velocity as theboundary.Ideal fluid is a fluid which is assumed to have noviscosity and in which there are no shear stresses.Figure 34

Chapter 5 – Fluid in Motion – The Bernoulli Equation5. Compressible fluid is a fluid which its density willchange with pressure.6. Laminar flow, sometimes known as streamlineflow, occurs when a fluid flows in parallel layers,with no disruption between the layers.Turbulent flow is a flow regime characterized bychaotic, stochastic property changes.From the observation done by Osborne Reynolds in1883, in straight pipes of constant diameter, flow canbe assumed to be turbulent if the Reynolds number,Re, exceeds 4000.Re ρvDµFigure 45

Chapter 5 – Fluid in Motion – The Bernoulli EquationBernoulli EquationFigure 5Mass per unit time flowing; ρAvRate of increase of momentum from AB to CD; ρAv[(v δv) v] ρAvδvForce due to pressure at surface AB; pA6

Chapter 5 – Fluid in Motion – The Bernoulli EquationForce due to pressure at surface CD; ( p δp)( A δA)Force due to pressure at side surface; psideδA (can be neglected)Force due to weight of the component; mg cosθ ρgVδz ρg ( A 12 A)δs δsNeglecting products of small quantities.Resultant force in the direction of motion Aδp ρgAδzApplying the Newton’s second law;ρAvδv Aδp ρgAδzDividing by ρAδs1 δpδvδz0 v gρ δsδsδs7

Chapter 5 – Fluid in Motion – The Bernoulli EquationIn the limit as δs 00 1 dpdvdz v gρ dsdsds(eq.1)This is known as Euler’s equation, giving, indifferential form, the relationship between pressure,velocity, density and elevation along a streamline forsteady flow.It cannot be integrated until the relationship betweendensity and pressure is known.For an incompressible fluid, for which density isconstant, integration of Euler’s equation (eq.1) alongthe streamline, with respect to s, gives;v2constant gzρ 2pIt can be written as;p v2constant zρg 2 g(eq.2)It is called Bernoulli equation.8

Chapter 5 – Fluid in Motion – The Bernoulli EquationTo use it correctly, we must constantly remember thebasic assumptions used in its derivation:1. Viscous effect are assumed negligible2. The flow is assumed to be steady3. The flow is assumed to be incompressible4. The equation is applicable along a streamlineIf Bernoulli equation (eq.2) is integrated along thestreamline between any two points indicated bysuffixes 1 and 2;21p1 vp2 v22 z1 z2ρg 2 gρg 2 g(eq.3)9

Chapter 5 – Fluid in Motion – The Bernoulli EquationPhysical InterpretationIntegration of the equation of motion to give eq.2actually corresponds to the work-energy principleoften used in the study of dynamics.The work doneon a particle by all forces acting on the particle isequal to the change of the kinetic energy of theparticle.Each of the terms in this equation has the units ofenergy per weight (LF/F L) or length (feet, meters)and represents a certain type of head.The elevation term, z, is related to the potentialenergy of the particle and is called the elevation head.The pressure term, p/ρg, is called the pressure headand represents the height of a column of the fluid thatis needed to produce the pressure p.The velocity term, V2/2g, is the velocity head andrepresents the vertical distance needed for the fluid tofall freely (neglecting friction) if it is to reachvelocity V from rest.The Bernoulli equation states that the sum of thepressure head, the velocity head, and the elevationhead is constant along a streamline.10

Chapter 5 – Fluid in Motion – The Bernoulli EquationStatic, Stagnation, Dynamic and Total PressureFigure 6The second term in the Bernoulli equation, V2/2g, istermed the dynamic pressure.Its interpretation can be seen in Figure 6 byconsidering the pressure at the end of a small tubeinserted into the flow and pointing upstream.After the initial transient motion has died out, theliquid will fill the tube to a height of H as shown. Thefluid in the tube, including that at its tip, (2), will bestationary. That is, V2 0, or point (2) is a stagnationpoint.11

Chapter 5 – Fluid in Motion – The Bernoulli EquationIf we apply the Bernoulli equation between points (1)and (2), using V2 0 and assuming that z1 z2, wefind thatp2 p1 12 ρv12Hence, the pressure at the stagnation point is greater2than the static pressure, p1, by an amount 12 ρv1 , thedynamic pressure.12

Chapter 5 – Fluid in Motion – The Bernoulli EquationFigure 7It can be shown that there is a stagnation point on anystationary body that is placed into a flowing fluid.Some of the fluid flows “over” and some “under” theobject. The dividing line (or surface fortwo-dimensional flows) is termed the stagnationstreamline and terminates at the stagnation point onthe body.For symmetrical objects (such as a sphere) thestagnation point is clearly at the tip or front of theobject as shown in Figure 7(a).For nonsymmetrical objects such as the airplaneshown in Figure 7(b), the location of the stagnationpoint is not always obvious.13

Chapter 5 – Fluid in Motion – The Bernoulli EquationKnowledge of the values of the static and stagnationpressures in a fluid implies that the fluid speed can becalculated.This is the principle on which the Pitot-static tube isbased H. de Pitot (1695–1771), as shown in Figure 8.Figure 814

The Bernoulli Equation for anIncompressible, Steady Fluid FlowIn 1738 Daniel Bernoulli (1700-1782) formulated the famous equation for fluid flow thatbears his name. The Bernoulli Equation is a statement derived from conservation ofenergy and work-energy ideas that come from Newton's Laws of Motion.An important and highly useful special case is where friction is ignored and the fluid isincompressible. This is not as unduly restrictive as it might first seem. The absence offriction means that the fluid flow is steady. That is, the fluid does not stick to the pipesides and has no turbulence. Most common liquids such as water are nearlyincompressible, which meets the second condition.Consider the case of water flowing though a smooth pipe. Such a situation is depicted inthe figure below. We will use this as our working model and obtain Bernoulli's equationemploying the work-energy theorem and energy conservation.

We examine a fluid section of mass m traveling to the right as shown in the schematicabove. The net work done in moving the fluid isWtotal W1 W2 F1 x1 F2 x 2Eq.(1)Where F denotes a force and an x a displacement. The second term picked up its negativesign because the force and displacement are in opposite directions.Pressure is the force exerted over the cross-sectional area, or P F/A. Rewriting this asF PA and substituting into Eq.(1) we find that W P1 A1 x1 P2 A2 x 2Eq.(2)The displaced fluid volume V is the cross-sectional area A times the thickness x. Thisvolume remains constant for an incompressible fluid, soV A1 x1 A2 x 2Eq.(3)Using Eq.(3) in Eq.(2) we have W ( P1 P2 )VEq.(4)

Since work has been done, there has been a change in the mechanical energy of the fluidsegment. This energy change is found with the help of the next diagram.The energy change between the initial and final positions is given by Potential energy Potential energy E andand Kinetic energy outlet Kinetic energy inlet11 E (mgh2 mv 22 ) (mgh1 mv12 )22Eq.(5)Here, the the kinetic energy K mv²/2 where m is the fluid mass and v is the speed of thefluid. The potential energy U mgh where g is the acceleration of gravity, and h isaverage fluid height.The work-energy theorem says that the net work done is equal to the change in thesystem energy. This can be written as W EEq.(6)Substitution of Eq.(4) and Eq.(5) into Eq.(6) yields( P1 P2 )V (mgh2 1 21mv2 ) (mgh1 mv12 )22Eq.(7)

Dividing Eq.(7) by the fluid volume, V, and replace m/V ρ gives usP1 P2 ρgh2 1 21ρv 2 ρgh1 ρv1222Eq.(8)Rearrange Eq.(8), gives us Eq.(9)P1 1 21ρv1 ρgh1 P2 ρv 22 ρgh222P1 v12Pv2 h1 2 2 h2ρg 2 gρg 2 gEq.(9)Finally, note that Eq.(9) is true for any two positions. Therefore, Equation (10) iscommonly referred to as Bernoulli's equation. Keep in mind that this expression wasrestricted to incompressible fluids and smooth fluid flows.P1 v12 h1 Constantρg 2 gEq.(10)

Chapter 5 – Fluid in Motion – The Continuity EquationCONTINUITY EQUATIONFigure 1Consider a fluid flowing through a fixed volume thathas one inlet and one outlet as shown in Figure 1.If the flow is steady so that there is no additionalaccumulation of fluid within the volume, the rate atwhich the fluid flows into the volume must equal therate at which it flows out of the volume.Otherwise, mass would not be conserved.The mass flowrate from an outlet is given as below;m& ρQ ρAVm& : Mass flowrateQ : Volume flowrateA : Outlet areaV : Average velocity1

Chapter 5 – Fluid in Motion – The Continuity EquationTo conserve mass, the inflow rate must equal theoutflow rate. If the inlet is designated as (1) and theoutlet as (2), it follows that;m& 1 m& 2Thus, conservation of mass requires;ρ1 A1V1 ρ 2 A2V2If the density remains constant, then ρ1 ρ 2 ,And the above equation becomes the continuityequation for incompressible flow, and shown as;A1V1 A2V2orQ1 Q22

Chapter 5 – Fluid in Motion – Examples of use of the Bernoulli equation.EXAMPLE OF USE OF THEBERNOULLI EQUATIONFree jetsFigure 1From the fact, we found that;z1 hz2 0andp1 p2 0v1 0andThus, the fluid leaves as a “free jets” with;v2 2 ghThis is introduced in 1643 by Torricelli (1608-1647)1

Chapter 5 – Fluid in Motion – Examples of use of the Bernoulli equation.Nozzle(a)(b)Figure 2We can safely use the centerline velocity at point (2)as a reasonable “average velocity”, as shown inFigure 2(a).If the exit is not a smooth, well-contoured nozzle, butrather a flat plate as shown in Figure 2(b), thediameter of the jet, dj will be less that the diameter ofthe hole, dh.This phenomenon is called a vena contracta effect, isa result of the inability of the fluid to turn the sharp90-degree corner indicated by the dotted line in thefigure.2