Brief Notes on the Decay of Vortices in a Second Grade Fluid

G. I. Taylor [ 1 ] showed that the flow representing a double array of vortices which has the same periodicity in both the x and y directions is a solution to the equations of motion in two dimensions of a linearly viscous fluid. It was shown in [2] that such a result is also true for 'second order' fluids if time scale which characterizes the memory of the fluid and the size of the vortices satisfy certain apriori restrictions. In this note we show that the results established by Taylor [1] for the linearly viscous fluid are uncondi t ional ly true, irrespective of the time scale which characterizes the fluid or the size of the vortices in the case of incompressible second grade fluids provided they are thermodynamically compatible. Also, in this analysis we investigate the relationship between the rate of decay of the vortices, and the periodicity of the vortices. It is found that if the periodicity is increased in the x or y directions, the vortices decay faster. It is also found, as is to be expected, that the vortices decay faster as the coefficient of viscosity # increases, while the decay is slower if the normal stress moduli oq is larger. The Cauchy stress T in an incompressible second grade fluid is assumed to be related to the fluid motion in the following manner [3 ]