Lumley 



particularly interesting because it does not depend on our choice of viscosity. It 

 exhibits a distinct and consistent variation of this parameter with concentration. 



Hence it is clear that the non-Newtonian character of this fluid has a direct 

 effect on the stability of its motion. Possibly this effect is a result of shear- 

 induced normal stresses or anisotropy in the relation between stress and rate 

 of strain, which is implied by the fact that the fluid is birefringent under shear. 



DISCUSSION 



A BASIC THEORY THAT COULD EXPLAIN DRAG REDUCTION 

 IN A FLOW CARRYING ADDITIVES 



A. Cemal Eringen 



Purdue University 



Lafayette, Indiana \ 



Lumley [l], Hoyt and Fabula [2], and Vogel and Patterson [3] gave excellent 

 experimental demonstrations of the phenomena of drag reduction by minute 

 amount of additives to fluid surrounding a moving object. We do not possess as 

 yet a theory explaining this phenomena. Classical Stokesian fluids do not contain 

 a mechanism which could provide the desired mathematical treatment. In fact, 

 I do not believe that even the modern theories of visco-elastic fluids [4] can 

 throw light into this phenomena. Quite by accident, a new theory, "Simple Micro- 

 fluids," introduced by Eringen [5], in a different context, seems to have just the 

 proper mechanism for this purpose. 



The theory of simple micro-fluids requires that we determine nineteen iin- 

 knowns p, i^^ = ^ mk > ^kz ^^^ ^k ^V solving nineteen partial differential eq;ia- 

 tions given in [5] subject to appropriate boundary and initial conditions. Here 

 p, i^^, v^i and v^ are respectively the mass density, the micro-inertia, the 

 gyration tensor and the velocity vector. The micro-inertia i^m provides a 

 mechanism for the inertial anisotropy. Roughly speaking, it is similar to the 

 inertia tensor of rigid dynamics. The gyration tensor provides a mechanism for 

 the local micromotions and small vortices. 



The present theory is shown [5], [6] to contain the celebrated Navier-Stokes 

 Theory of fluid dynamics and the theory of anisotropic fluids. A theory of tur- 

 bulence based on this theory is as yet lacking. 



Some sample calculations made are indicative of the above mentioned drag 

 reductions. However, presently this work is too naive for publication and the 

 possible application of the theory of simple micro-fluids to the problem of drag 

 reduction by additives is brov^ht to your attention as a conjectirre. 



944 



