As we noted earlier there are two main physical principles involved i

As we noted earlier there are two main physical

This preview shows page 53 - 55 out of 164 pages.

As we noted earlier, there are two main physical principles involved: i ) conservation of mass and ii ) Newton’s second law of motion, the latter of which leads to a system of equations expressing the balance of momentum. In addition, we will utilize Newton’s law of viscosity in the guise of what will be termed a “constitutive relation,” and, of course, all of this will be done within the confines of the continuum hypothesis. We should also note that Newton’s second law of motion formally applies to point masses, i.e. , discrete particles, making its application to fluid flow seem difficult at best. But we will see that because we can define fluid particles (via the continuum hypothesis), the difficulties are not actually so great. We will begin with a brief discussion of the two types of reference frames widely used in the study of fluid motion, and provide a mathematical operator that relates these. We then review some additional mathematical constructs that will be needed in subsequent derivations. Once this groundwork has been laid we will derive the “continuity” equation which expresses the law of conservation of mass for a moving fluid, and we will consider some of its practical consequences. We then provide a similar analysis leading to the momentum equations, thus arriving at the complete set of equations known as the Navier–Stokes (N.–S.) equations. These equations are believed to represent all fluid motion within the confines of the continuum hypothesis. We will provide qualitative discussions of the physical importance of each of their various terms, and we will close the chapter with a treatment of dimensional analysis and similitude, first based on the equations of motion, and then via the more standard engineering approach—use of the Buckingham Π Theorem . 3.1 Lagrangian & Eulerian Systems; the Substantial Derivative In the study of fluid motion there are two main approaches to describing what is happening. The first, known as the Lagrangian viewpoint, involves watching the trajectory of each individual fluid parcel as it moves from some initial location, often described as “placing a coordinate system on each fluid parcel” and “riding on that parcel as it travels through the fluid.” At each instant in time the fluid particle(s) being studied will have a different set of coordinates within some global coordinate system, but each particle will be associated with a specific initial set of coordinates. The alternative is the Eulerian description. This corresponds to a coordinate system fixed in space, and in which fluid properties are studied as functions of time as the flow passes fixed spatial locations. 47
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48 CHAPTER 3. THE EQUATIONS OF FLUID MOTION 3.1.1 The Lagrangian viewpoint Use of Lagrangian-coordinate formulations for the equations of fluid motion is very natural in light of the fact that Newton’s second law applies to point masses, and it is reasonable to view a fluid parcel as such. The equations of motion that arise from this approach are relatively simple because they result from direct application of Newton’s second law. But their solutions consist merely of
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