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Unformatted text preview: 9 Fluid dynamics and RayleighB enard convection In these lectures we derive (mostly) the equations of viscous uid dynamics. We then show how they may be generalized to the problem of Rayleigh B enard convectionthe problem of a uid heated from below. Later we show how the RB problem itself may be reduced to the famous Lorenz equations. The highlights of these lectures are as follows: NavierStokes equations of uid dynamics (mass and momentum conser vation). Reynolds number Phenomenology of RB convection Rayleigh number Equations of RB convection Thus far we have dealt almost exclusively with the temporal behavior of a few variables. In these lectures we digress, and discuss the evolution of a continuum . 9.1 The concept of a continuum Real uids are made of atoms or molecules. The mean free path mfp is the characteristic length scale between molecular collisions. Let L hydro be the characteristic length scale of macroscopic motions. Fluids may be regarded as continuous fields if L hydro mpf . 82 When this condition holds, the evolution of the macroscopic field may be described by continuum mechanics , i.e., partial differential equations. To make this idea clearer, consider a thought experiment in which we measure the density of a uid over a length scale using some particularly sensitive device. We then move the device in the xdirection over a distance of roughly 10 . Suppose L 1 mpf . Then we expect the density to vary greatly in space as in Figure (a) below: density x/L 1 x/L 2 x/L hydro (a) (b) (c) We expect that the uctuations in (a) should decrease as increases. ( Statistics tells us that these uctuations should decrease like 1 /N 1 / 2 , where N 3 is the average number of molecules in a box of size . ) On the other hand, if L hydro (see (c)), variations in density should reect density changes due to macroscopic motions (e.g., a rising hot plume), not merely statistical uctuations. Our assumption of a continuum implies that there is an intermediate scale, L 2 , over which uctuations are small. Thus the continuum hypothesis implies a separation of scales between the molecular scale, L 1 mfp , and the hydrodynamic scale, L hydro . Thus, rather than dealing with the motion 10 23 molecules and therefore 6 10 23 ordinary differential equations of motion (3 equations each for position and momentum), we model the uid as a continuum. 83 The motion of the continuum is expressed by partial differential equations for evolution of conserved quantities. We begin with the conservation of mass. 9.2 Mass conservation Let = density of a macroscopic uid particle u = velocity Consider a volume V of uid, fixed in space: V dS u d s is an element of the surface, d s is its area, and it points in the outward   normal direction....
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 Fall '06
 DanielRothman
 Fluid Dynamics

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