Model development and verification for mass transport to Escherichia coli cells in a turbulent flow

Theoretical studies imply that fluid motion does not significantly increase the molecular diffusive mass flux toward and away from microscopic organisms. This study presents experimental and theoretical evidence that small-scale turbulence modulates enhanced mass transport to Escherichia coli cells in a turbulent flow. Using the technique of inner region and outer region expansions, a model for dissolved oxygen and glucose uptake by E. coli was developed. The mass transport to the E. coli was modeled by the Sherwood (Sh)-Péclet (Pe) number relationship with redefined characteristic length and velocity scales. The model Sh = (1 + Pe^1/2 + Pe) agreed with the laboratory measurements well. The Péclet number that quantifies the role and function of small-scale turbulence on E. coli metabolism is defined by Pe = (E_zzη_KL_c/D) where E_zz is the root mean square of fluid extension in the direction of local vorticity, η_K is the Kolmogorov length scale, L_c is the length scale of E. coli, and D is the molecular diffusion coefficient. An alternative formulation for the redefined Pe is given by Pe = (ū_KL _c/D) where ūK = 0.5(εν)^1/4 is the Kolmogorov velocity averaged over the Kolmogorov length scale, ε is dissipation of turbulent kinetic energy, and ν is the kinematic viscosity of fluid. The dissipation of turbulent kinetic energy was estimated directly from measured velocity gradients and was within the reported range in engineered and natural aquatic ecosytems. The specific growth of E. coli was up to 5 times larger in a turbulent flow in comparison to the still water controls. Dissolved oxygen and glucose uptake were enhanced with increased ε in the turbulent flow.