Studies on the Motion of Viscous Flows — III 



n = Unit vector directed inwards along the normal to an element ds of 

 the surface S bounding the fluid 



\, /J. = coefficients of viscosity. ' ^■-■''- ->'■"''— - 



After transforming the surface integrals into volume integrals, the above ex- 

 pression may be rewritten as 



h 



dr , 



(1.24b) 



where 



$ = Dissipation function 



2 



^-^ \dx. 2 ^ I Bx. dx. 



(1.25) 



From (1.25) we see that <t> and hence E are nonnegative. It is also clear 

 that $ and hence E can vanish only under the following two circumstances. 



(i) k = 0, ft = 0, i.e., when the fluid is regarded as ideal. 



(ii) 3ui/Bxi= 0, 3ui/3xj + Buj/Bxi = 0, i.e., when there is no deformation 

 and the fluid moves as a rigid body. 



For a real fluid for which \ > 0, /i > 0, E can vanish only under the second con- 

 dition and this is of little interest. We note that E = is the absolute minimum 

 which this function can attain. Under circumstances other than the one noted 

 there will always be some dissipation of energy. If we postulate that the flow 

 evolves to minimize dissipation [2,3,4], this is tantamount to a principle. 



PRINCIPLE OF MINIMUM DISSIPATION - . - ■ 



For all real flows, the rate of energy dissipation assumes the lowest attain- 

 able value consistent with the conservation principles and the boundary condi- 

 tions. 



To see the implications of this principle, let us write Eq. (1.24a) in the fol- 

 lowing two groups: 



E = (Ij + I^) + (I3 + I4) • •'- - (1.24c) 



= (Ij) + (I2 + I3 + I4) , ' -^ (1.24d) 



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