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Unformatted text preview: Differential Form of the Fundamental Laws of Hydraulics: NavierStokes Equations & their Applications By M.S. Ghidaoui, Spring 2002 Introduction: The general form of the fundamental physical laws of fluid mechanics/hydraulics for a control volume was provided in the previous chapter. The purpose of this chapter is to establish the general form of theses physical laws at a ``point in the fluid. The formulation of the fundamental physical laws for a ``point (x,y,z,t) in the fluid media can be obtained by (i) applying the fundamental laws for a control volume to a fluid element (i.e., control volume) of length x, width y and height z during a time interval t, (ii) taking the limit as ( x, y, z, t) (0,0,0) and (iii) assuming that density, velocity, pressure etc. are continuous quantities. The continuum assumption allows the use of the classical definition of limits (i.e., the relationship between limits and differential quantities). For example, the continuum assumption allows us to write expressions such as: x v ) , , , ( ) , , , ( lim x = + x t z y x v t z y x x v x v ) , , , ]( [ ) , , , ]( [ lim x = + x t z y x v t z y x x v y y t z y x v t z y y x v y = + v ) , , , ]( [ ) , , , ]( [ lim Note that the v/ x is simply a measure of the difference between the velocity at (x+ x,y,z,t) and the velocity at (x,y,z,t) when x is infinitesimal. In addition, the p/ x measures the difference between the pressure at (x+ x,y,z,t) and the pressure at (x+ x,y,z,t) when x is infinitesimal. Can you think of how you would estimate v/ x in a river? The Continuum Assumption: The assumption that the field variables such as velocity, pressure, temperature and density are continuous is central to the derivation of the differential form of the physical laws. What justifies the continuum assumption? To this end, consider a water droplet of radius 1 cm. To the naked eye, the water droplet looks smooth and continuous. However, if one were to magnify the water droplet 640,000,000 times so that its radius becomes 6400 Km (i.e. roughly the size of the earth), the droplet no longer looks smooth and continuous! Instead, the magnified droplet becomes a porous volume with a gigantic number of molecules jiggling and bouncing around with different paths and magnitudes. Le l define the average distance between molecules. Therefore, the average size of the pores in is of order than l 3 . Since the pores do not contain molecules, field variables such as density, pressure and velocity dont even exist inside the pores let alone there derivatives! That is, at the molecular scale, matter is discontinuous and the continuum assumption fails. However, we are fortunate enough that hydraulic engineering scales are much larger than the molecular scale! For example, the macroscopic length scale, L, for pipe flow is of the order of the pipe diameter (i.e., L= D). The pipe diameter is L, for pipe flow is of the order of the pipe diameter (i....
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