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Unformatted text preview: 530.327  Introduction to Fluid Mechanics  Su Statics: Derivation of the basic equation. Reading: Text, Ā§ 3.1 Having defined a āļ¬uidā and covered the fundamental concepts of stress, viscosity, etc., weāre prepared to look at the first major category of problems in ļ¬uid mechanics, namely ļ¬uid statics , which is concerned with ļ¬uids that are stationary. (The other category of ļ¬uid mechanics problems is ļ¬uid dynamics, which naturally involves ļ¬uids in motion.) As is familiar in physical problems, our goal is to write the differential equation that is relevant to ļ¬uid statics. The approach we will take is to start with an infinitesimal ļ¬uid particle, integrate the forces on that particle, and take advantage of the infinitesimalness of the particle to end up with a differential equation. This approach is ubiquitous in ļ¬uid mechanics ā weāll use it again later in the semester ā and is also significant in numerous other areas of science and engineering, such as structural mechanics. The ļ¬uid particle weāre interested in is shown in Fig. 1. The particle is centered at the origin of Figure 1: A ļ¬uid particle with volume dV and density Ļ , centered at the origin of x ā y ā z space. Cartesian coordinate space, and has side lengths dx , dy and dz , volume dV = dx dy dz , and density Ļ . The positive zaxis points vertically upward. Assuming the particle is stationary, what forces act on the particle? Reviewing the possibilities as we know them: ā¢ Gravity (a body force): Yes, gravity acts on the particle. Gravity doesnāt care whether the ļ¬uid particle is moving or not. ā¢ Pressure (a surface force): Yes, pressure forces act on each of the particleās faces. Pressure is present in ļ¬uids also regardless of whether or not theyāre moving. ā¢ Shear stress (a surface force): There are no shear stresses. To understand this, recall our definition that ļ¬uids move under the action of any shear stress. Since the ļ¬uid particle is stationary, there can be no shear stresses....
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 Spring '08
 Su
 Statics, Fluid Dynamics, Fluid Mechanics, Force, Stress, fluid particle

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