327lec-statics1 - 530.327 Introduction to Fluid Mechanics...

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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 “fluid” and covered the fundamental concepts of stress, viscosity, etc., we’re prepared to look at the first major category of problems in fluid mechanics, namely fluid statics , which is concerned with fluids that are stationary. (The other category of fluid mechanics problems is fluid dynamics, which naturally involves fluids in motion.) As is familiar in physical problems, our goal is to write the differential equation that is relevant to fluid statics. The approach we will take is to start with an infinitesimal fluid 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 fluid mechanics – we’ll use it again later in the semester – and is also significant in numerous other areas of science and engineering, such as structural mechanics. The fluid particle we’re interested in is shown in Fig. 1. The particle is centered at the origin of Figure 1: A fluid particle with volume dV and density ρ , 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 ρ . 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 doesn’t care whether the fluid particle is moving or not. • Pressure (a surface force): Yes, pressure forces act on each of the particle’s faces. Pressure is present in fluids also regardless of whether or not they’re moving. • Shear stress (a surface force): There are no shear stresses. To understand this, recall our definition that fluids move under the action of any shear stress. Since the fluid particle is stationary, there can be no shear stresses....
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327lec-statics1 - 530.327 Introduction to Fluid Mechanics...

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