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7708d_c02_040-099

# 7708d_c02_040-099 - An image of hurricane Allen viewed via...

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An image of hurricane Allen viewed via satellite: Although there is considerable motion and structure to a hurricane, the pressure variation in the vertical direction is approximated by the pressure-depth relationship for a static fluid. 1 Visible and infrared image pair from a NOAA satellite using a technique developed at NASA/GSPC. 2 1 Photograph courtesy of A. F. Hasler [Ref. 7]. 2

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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 parti- cles. 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 con- cern 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. 41 2 F luid Statics 2.1 Pressure at a Point 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 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 rel- ative motion between adjacent elements. There are no shear- ing stresses present in a fluid at rest.
The equations of motion 1 Newton’s second law, 2 in the y and z directions are, respectively, where and are the average pressures on the faces, and are the fluid specific multiplied by an appropriate area to obtain the force generated by the pressure. It follows from the geometry that so that the equations of motion can be rewritten as Since we are really interested in what is happening at a point, we take the limit as and approach zero 1 while maintaining the angle 2 , and it follows that or The angle 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 present . This important result is known as

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