Generation of dynamic biochemical signals with a tube mixer: effect of dispersion in an oscillatory flow

The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients $K_i(\tau)(i=1, 2,\ldots)The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients K_i(t)  (i=1, 2,?)K_i(\tau)(i=1, 2,\ldots) are determined as the functions of dimensionless time τ. With the assumptions of quasi-steady flow and steady flow, the dispersion coefficients K_i(t)  (i=1, 2)K_i(\tau)(i=1, 2) are simplified. The effects of the frequencies of the oscillatory flow and the oscillatory biochemical signal, and the length of the mixing tube on the average solute concentrations, θ _m , over the tube cross-section at the outlet of the mixing tube are analyzed by numerical simulations with and without the assumptions of the quasi-steady and steady flows. It is concluded that for the dispersion in an oscillatory flow, an excellent accuracy can be achieved by using quasi-steady flow assumption while the steady flow assumption would lead to inaccurate results. However, if the frequency of oscillatory flow is sufficiently high, the steady flow assumption can be used to further simplify the calculation while still maintaining sufficient accuracy. These results are of practical importance in producing dynamic biochemical signals as the stimuli of biological cells by a tube mixer.