An upper bound for passive scalar diffusion in shear flows

This study is concerned with the diffusion of a passive scalar Theta(r,t) advected by general n-dimensional shear flows u=u(y,z,...,t)x having finite mean-square velocity gradients. The unidirectionality of the incompressible flows conserves the streamwise scalar gradient, partial derivative(x)Theta, allowing only the cross-stream components to be amplified by shearing effects. This amplification is relatively weak because an important contributing factor, partial derivative(x)Theta, is conserved, effectively rendering a slow diffusion process. It is found that the decay of the scalar variance <Theta(2)> satisfies d <Theta(2)>/dt >=-C kappa(1/3), where C>0 is a constant, depending on the fluid velocity gradients and initial distribution of Theta, and kappa is the molecular diffusivity. This result generalizes to axisymmetric flows on both the plane and sphere having finite mean-square angular velocity gradients. (c) 2007 American Institute of Physics.