Stokes' and Einstein's Law for Nonuniform Viscosity

Abstract

The equations of hydrodynamics have been solved for the case of a viscosity dependence upon the distance r to a particle of radius a, suspended in a continuous fluid, as given by ${\eta }_{\infty }{\text{/η(r)=1−ε}}_m{\mathrm{(a}}_m{\mathrm{/r)}}^m$ . When $\text{|ε|<0.2,\hspace{0.17em}m⩾0\hspace{0.17em}}\mathrm{or}\text{\hspace{0.17em}ε⩾1,\hspace{0.17em}m>6}$ the omissions are less than 25% of the new terms. The force upon a spherical particle is given by ${\text{6πv}}_{\infty }{\eta }_{\infty }{a}_m{\text{[1+ε}}_m{\text{/(m+1)·(m}}^2\text{+4m+15)/(m+3)(m+5)]}$ . The torque upon a spherical particle is ${\text{8πω}}_{\infty }{\eta }_{\infty }{a}_m^3{\text{[1+ε}}_m\text{·3/(m+3)]}$ . The viscosity of a dilute suspension with 1 particle per sphere of radius R is ${\mathrm{\eta /\eta }}_{\infty }{\text{=1+ε}}_m{\mathrm{(a}}_m{\mathrm{/R)}}^m{\text{\hspace{0.17em}3/(3−m)+5/2(a}}_m{\mathrm{/R)}}^3{\text{·[1+ε}}_m{\text{3/(m−3)·(m}}^3{\text{+5m}}^2{\text{+47m−21)/(m+3)(m+5)(m+7)+4/5ε}}_0]$ . This enables a study of the diffusion constants of a suspended particle, its effect upon the viscosity of a fluid, and the transition of a liquid into a gel. Experimental data concerning proton magnetic relaxation times of diamagnetic solutions and self‐diffusion of protein and solvent molecules are discussed. The effect of slipping is examined.