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