Lumley 



viscoelasticity, since a particle of long characteristic time in a flow of short 

 time scale will tend to remain motionless as the flow sloshes past it. Thus an 

 unsteady fluid motion will be more dissipative than a steady one, as in a visco- 

 elastic fluid having an effective viscosity increasing with the frequency of a 

 temporally sinusoidal simple shear.* The particles will tend to store energy 

 associated with steady, organized motions (steady from a Lagrangian viewpoint) 

 and to oppose unsteady motion. This was probably first mentioned by Saffman 

 (1962). See also Hino (1963). 



The presence of particles, or colloidal suspensions, can of course, be even 

 more effective if the suspended material tends to combine with itself to form 

 elastic structures capable of resisting small shear. This is possible with 

 Bentonite, and may explain observations in flocculated thoria (Eissenberg and 

 Bogue (1963)) and in flows of fine aqueous suspensions of wax-laden oil droplets. 

 The suspended material then behaves somewhat as a Bingham- Plastic and need 

 not depend on a long time constant to make unsteady motions of the fluid more 

 dissipative than steady ones at low shear. 



Changing the Sublayer 



Finally, we may change the boundary layer by changing R. The effect of a 

 small change in R at constant \Jx/v is given by 



R d/i* j_L* (l 



= ( R 



/.* dR D \¥. 



obtained by differentiating the expression for drag, assuming self-preservation, 

 negligible laminar length, and indefinitely large Reynolds number. At the value 

 of R associated with the normal turbulent boundary layer, this is negative, and 

 of the order of one half. 



R may be changed in a number of ways. If a viscoelastic medium is used, 

 the effective viscosity of which in a temporally sinusoidal simple shear increases 

 with frequency, we may expect that a disturbance which is unsteady (from the 

 Lagrangian viewpoint) will be more dissipative than would be indicated by the 

 viscosity at the steady shear rate. Since R (based on the steady state viscosity) 

 is determined by that thickness below which all disturbances must import energy, 

 we may expect R to be increased.* In a similar way, particles may be intro- 

 duced in the sublayer. If their time scale is large they also will make unsteady 

 motions more dissipative and thus increase R. If they can form elastic struc- 

 tures, like flocculated thoria, (Eissenberg & Bogue (1963)), the effect is even 

 more pronounced. If the time and length scales are such that the energy con- 

 taining eddies in the turbulent flow outside the sublayer are unaffected, then the 

 familiar "law of the wall" will remain, K will be iinaffected, but the logarithmic 

 part of the profile will be displaced upward. This effect is illustrated by Fig. 5, 

 the mean velocity profile in a flow containing a low concentration of flocculated 

 thoria, reproduced from Eissenberg and Bogue (1963). 



"'But real viscoelastic media appear to display the opposite behavior. 



926 



