APPENDIX 

 PRESSURE IN A MOVING FRAME OF REFERENCE 



In a moving frame of reference, Euler's equation of motion of an inviscid 



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 and incompressible fluid can be expressed as follows (see Batchelor ): 



DV , ( d r dfJ I 



= __LVp + F-/ — ^ + ^— X r + 2f] X V + n X (Q X r)) (12) 



Dt P ) dt^ dt ------ j 



where -;r— = material derivative defined by -=— = -r— ■ + V • V 



Dt L)t dt 



V = total velocity with respect to the moving reference frame 



p = fluid density 



p = pressure 



¥^ = body force per unit mass 



r = position vector of the origin of the moving frame 

 — o 



9, = angular velocity of the moving frame about the origin 



^ = position vector of a field point in the moving frame 



The last two terms, 2J} x V and g x (J^ x r ) , are called the Coriolis force and the 

 centrifugal force, respectively. 



If we take r = and 9, = constant. Equation (12) becomes: 

 — o — 



-T— + (V'V)V = ^ Vp + F - 2g X V - ^ X (g X £) (13) 



o t P 



Assuming that the gravity force is the only body force acting on the fluid, one 

 can express F^ by: 



F=V(-gy^) (14) 



where g is the gravitational acceleration and y is the vertical coordinate in the 



nonrotating coordinate system (x ,y ,z ) as shown in Figure 1. It is to be noted 

 '=' ^ o o o 



that this term is time-dependent in the rotating frame of reference. 

 Utilizing Equation (14) and the vector identities, 



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