fluid_mechanics
2 Fluid Statics CHAPTER OPENING PHOTO: Floating iceberg: An iceberg is a large piece of fresh water ice that originated as snow in a glacier or ice shelf and then broke off to float in the ocean. Although the fresh water ice is lighter than the salt water in the ocean, the difference in densities is relatively small. Hence, only about one ninth of the volume of an iceberg protrudes above the ocean’s surface, so that what we see floating is literally “just the tip of the iceberg.” (Photograph courtesy of Corbis Digital Stock/Corbis Images) Learning Objectives After completing this chapter, you should be able to: ■ determine the pressure at various locations in a fluid at rest. ■ explain the concept of manometers and apply appropriate equations to determine pressures. ■ calculate the hydrostatic pressure force on a plane or curved submerged surface. ■ calculate the buoyant force and discuss the stability of floating or submerged objects. In this chapter we will consider an important class of problems in which the fluid is either at rest or moving in such a manner that there is no relative motion between adjacent particles. In both instances there will be no shearing stresses in the fluid, and the only forces that develop on the surfaces of the particles will be due to the pressure. Thus, our principal concern is to investigate pressure and its variation throughout a fluid and the effect of pressure on submerged surfaces. The absence of shearing stresses greatly simplifies the analysis and, as we will see, allows us to obtain relatively simple solutions to many important practical problems. 2.1 Pressure at a Point 38 As we briefly discussed in Chapter 1, the term pressure is used to indicate the normal force per unit area at a given point acting on a given plane within the fluid mass of interest. A question that immediately arises is how the pressure at a point varies with the orientation of the plane passing
2.1 Pressure at a Point 39 z p s δx δs θ p y δx δz δ s δ z δ y θ δ x y x δx ________ δyδz γ 2 p z δx δy F I G U R E 2.1 Forces on an arbitrary wedge-shaped element of fluid. through the point. To answer this question, consider the free-body diagram, illustrated in Fig. 2.1, that was obtained by removing a small triangular wedge of fluid from some arbitrary location within a fluid mass. Since we are considering the situation in which there are no shearing stresses, the only external forces acting on the wedge are due to the pressure and the weight. For simplicity the forces in the x direction are not shown, and the z axis is taken as the vertical axis so the weight acts in the negative z direction. Although we are primarily interested in fluids at rest, to make the analysis as general as possible, we will allow the fluid element to have accelerated motion. The assumption of zero shearing stresses will still be valid so long as the fluid element moves as a rigid body; that is, there is no relative motion between adjacent elements. The equations of motion 1Newton’s second law, F ma2 in the y and z directions are, respectively, a F dx dy dz y p y dx dz p s dx ds sin u r a 2 y a F dx dy dz dx dy dz z p z dx dy p s dx ds cos u g r a 2 2 z where p s , p y , and p z are the average pressures on the faces, g and r are the fluid specific weight and density, respectively, and a y , a z the accelerations. Note that a pressure must be multiplied by an appropriate area to obtain the force generated by the pressure. It follows from the geometry that dy ds cos u dz ds sin u so that the equations of motion can be rewritten as The pressure at a p y p s ra dy y point in a fluid at 2 rest is independent p z p s 1ra z g2 dz of direction. 2 Since we are really interested in what is happening at a point, we take the limit as dx, dy, and dz approach zero 1while maintaining the angle u2, and it follows that p y p s p z p s or p s p y p z . The angle u was arbitrarily chosen so we can conclude that the pressure at a point in a fluid at rest, or in motion, is independent of direction as long as there are no shearing stresses p y p z present. This important result is known as Pascal’s law, named in honor of Blaise Pascal 11623– 16622, a French mathematician who made important contributions in the field of hydrostatics. Thus, as shown by the photograph in the margin, at the junction of the side and bottom of the beaker, the p z pressure is the same on the side as it is on the bottom. In Chapter 6 it will be shown that for moving fluids in which there is relative motion between particles 1so that shearing stresses develop2, the p y normal stress at a point, which corresponds to pressure in fluids at rest, is not necessarily the same
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2 Fluid Statics<br />
CHAPTER OPENING PHOTO: Floating iceberg: An iceberg is a large piece of fresh water ice that originated as<br />
snow in a glacier or ice shelf and then broke off to float in the ocean. Although the fresh water ice is lighter<br />
than the salt water in the ocean, the difference in densities is relatively small. Hence, only about one ninth of<br />
the volume of an iceberg protrudes above the ocean’s surface, so that what we see floating is literally “just the<br />
tip of the iceberg.” (Photograph courtesy of Corbis Digital Stock/Corbis Images)<br />
Learning Objectives<br />
After completing this chapter, you should be able to:<br />
■ determine the pressure at various locations in a <strong>fluid</strong> at rest.<br />
■ explain the concept of manometers and apply appropriate equations to<br />
determine pressures.<br />
■ calculate the hydrostatic pressure force on a plane or curved submerged surface.<br />
■ calculate the buoyant force and discuss the stability of floating or submerged<br />
objects.<br />
In this chapter we will consider an important class of problems in which the <strong>fluid</strong> is either at rest<br />
or moving in such a manner that there is no relative motion between adjacent particles. In both<br />
instances there will be no shearing stresses in the <strong>fluid</strong>, and the only forces that develop on the surfaces<br />
of the particles will be due to the pressure. Thus, our principal concern is to investigate pressure<br />
and its variation throughout a <strong>fluid</strong> and the effect of pressure on submerged surfaces. The<br />
absence of shearing stresses greatly simplifies the analysis and, as we will see, allows us to obtain<br />
relatively simple solutions to many important practical problems.<br />
2.1 Pressure at a Point<br />
38<br />
As we briefly discussed in Chapter 1, the term pressure is used to indicate the normal force per<br />
unit area at a given point acting on a given plane within the <strong>fluid</strong> mass of interest. A question that<br />
immediately arises is how the pressure at a point varies with the orientation of the plane passing
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