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Unformatted text preview: 9 Fluid dynamics and RayleighB´ enard convection In these lectures we derive (mostly) the equations of viscous ﬂuid dynamics. We then show how they may be generalized to the problem of Rayleigh B´ enard convection—the problem of a ﬂuid 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 ﬂuid 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 ﬂuids are made of atoms or molecules. The mean free path κ 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 κ 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 ﬂuid over a length scale κ using some particularly sensitive device. We then move the device in the xdirection over a distance of roughly 10 κ . Suppose κ L 1 κ 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 ﬂuctuations in (a) should decrease as κ increases. ( Statistics tells us that these ﬂuctuations should decrease like 1 /N 1 / 2 , where N ε 3 is the average number of molecules in a box of size ε . ) On the other hand, if κ L hydro (see (c)), variations in density should reﬂect density changes due to macroscopic motions (e.g., a rising hot plume), not merely statistical ﬂuctuations. Our assumption of a continuum implies that there is an intermediate scale, κ L 2 , over which ﬂuctuations are small. Thus the continuum hypothesis implies a separation of scales between the molecular scale, L 1 κ 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 ﬂuid 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 δ = density of a macroscopic ﬂuid particle ψu = velocity Consider a volume V of ﬂuid, fixed in space: V dS u d ψs is an element of the surface, d ψs is its area, and it points in the outward   normal direction....
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 Fall '06
 DanielRothman
 Fluid Dynamics

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