Piacsek 



The relevant equations of momentum and heat transport are formulated in 

 accordance with the following assumptions: 



1. The z axis of the cylindrical coordinate system coincides with the axis 

 of the cylindrical walls. 



2. The rotation vector is assumed to point in the positive z direction, 

 and the gravitational acceleration g in the negative z direction. 



3. The boundaries of the annulus are defined by the cylindrical surfaces 



r = a and r = b, and by the horizontal surfaces z = and z = d, respectively. 



4. The motion is described in a rotating system, so that all velocities 

 represent motion with respect to the cylinders. 



5. Only small rotation rates are considered, so that centrifugal body forces 

 may be neglected. 



6. Only small temperature contrasts are considered, so that the variation 

 of the coefficients of viscosity and heat conductivity with temperature may be 

 neglected, and the usual Boussinesq approximation concerning the density of an 

 incompressible fluid in natural convection may be applied, i.e., density varia- 

 tions are neglected everywhere except in the gravitational body force term, 

 giving rise to buoyancy effects. 



Taking the cylindrical coordinates (r, (p, z) with the corresponding unit 

 vectors r, $, z, and a velocity vector u = (u, v, w ), we may write the equa- 

 tions of state, continuity and momentum, and heat transport as: 



P - p^{\ - aCT-Tp)] = Po(l-«Ti) (1) 



V • u = (2) 



Bu Bu 3u Bu /„, u \ 1 ^P /v ^^\ ,„ ^ 



Bv Bv V Bv Bv /„, V \ 1 '^P /v ^^\ 



Bt Br "■ B(p Bz \ x^ j ^o' Bq, \^ j (3b) 



Bw Bw V Bw Bw „„ ^ 1 ^P _ 



Bt Br ^ Bcp Bz ^o Bz ^ (3c) 



BT, BT, BT, BT, 



1 1 V 1 1 V79T I .K 



Bt Br •■ Bq) Bz ^ ^ ' 



where T^ is defined by Eq. (1), T^ is the "mean" temperature (T^ + T3)/2 and 

 pQ the corresponding density, and p is the dynamic pressure (total minus 

 hydrostatic). 



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