Physics for Scientists and Engineers, Vol. 1
Physics for Scientists and Engineers, Vol. 1
6th Edition
ISBN: 9781429201322
Author: Paul A. Tipler, Gene Mosca
Publisher: Macmillan Higher Education
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Chapter 13, Problem 63P
To determine

To Derive:The Bernoulli Equation in more general way

To Show: P1+ρgh1+12ρv12=P2+ρgh2+12ρv22 .

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Explanation of Solution

Introduction:

Bernoulli’s equation describes the quantitative relationship between the pressure and velocity in fluids. This is derived by Daniel Bernoulli, a Swiss Physicist and Mathematician in the year 1738.

  Physics for Scientists and Engineers, Vol. 1, Chapter 13, Problem 63P

FIGURE: 1

Consider the fluid flowing through a tube which varies in elevation as well as in cross-sectional area, as shown in the figure 1.The work-energy theorem can be applied to a parcel of fluid between the points 1 and 2. Let Δt be the time taken by this parcel of liquid to move along the tube to the region between the points 1' and 2' .Let ΔV be the volume of the fluid passing through the point 1' in time Δt . The same volume of fluid passes through the point 2' in the same time. Let ρ be the density of the fluid. Then,the mass of the fluid (Δm) with volume ΔV can be written as Δm=ρΔV .

The expression for work-energy theorem can be written as:

  Wtotal=ΔU+ΔK  (1)

Where, ΔU is change in the potential energy of the parcel which is given by,

  ΔU=(Δm)gh2(Δm)gh1

  =(Δm)g(h2h1)

Substituting for Δm ,

  ΔU=ρΔVg(h2h1)  (2)

Where, g is acceleration due to gravity, g=9.81 m/s2

  ΔK is change in kinetic energy of the parcel which is given by,

  ΔK=12(Δm)v2212(Δm)v12

  =12(Δm)(v22v12)

Substituting for Δm ,

  ΔK=12ρΔV(v22v12)  (3)

The fluid behind the parcel (that is fluid in parcel’s left in the diagram) pushes on the parcel with a force of magnitude F1=P1A1 and hence work done by the fluid behind the parcel can be expressed as,

  W1=F1Δx1W1=P1A1Δx1

Since area × length = volume, above expression becomes,

  W1=P1ΔV  (4)

Where, P1 is the pressure at the point 1 .

The fluid in the front of the parcel (that is fluid in parcel’s left in the diagram) pushes on the parcel with a force of magnitude F2=P2A2 and hence work done by the fluid in the front of the parcel can be expressed as,

  W2=F2Δx2W2=P2A2Δx2

Since area × length = volume, above expression becomes,

  W2=P2ΔV  (5)

Where, P1 is the pressure at the point 2 .

This work done by the fluid in the front of the parcel is negative because the applied force and displacement are in the direction opposite to the flow of fluid.

Thus, the total work done is,

  Wtotal=W1+W2Wtotal=P1ΔVP2ΔVWtotal=(P1P2)ΔV  (6)

Substituting for Wtotal , ΔU , and ΔK in equation (1) ,

  (P1P2)ΔV=ρΔVg(h2h1)+12ρΔV(v22v12)

  (P1P2)=ρg(h2h1)+12ρ(v22v12)

On rearranging,

  P1+ρgh1+12ρv12=P2+ρgh2+12ρv22

This is the Bernoulli’s equation for steady, non-viscous flow of an incompressible fluid.

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Physics for Scientists and Engineers, Vol. 1

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