Modelling enhanced diffusion

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Write $c$ for the concentration of the chyme. The partial derivative $\frac{\partial c}{\partial t}$ measures the rate of change in the concentration with respect to time. As before, write $U=(u,v,w)$ for the velocity of the chyme (with its eddies). The equation modelling enhanced diffusion is

  \[ \frac{\partial c}{\partial t}+U.\nabla c = \nu \nabla ^2 c. \]    
  \[ U.\nabla c = u \frac{\partial c}{\partial x}+ v \frac{\partial c}{\partial y} + w \frac{\partial c}{\partial z}. \]    
It represents the motion of the drug due to the convection in the chyme. The term $\nu \nabla ^2 c$ is defined as
  \[ \nu \nabla ^2 c = \nu \left(\frac{\partial ^2 c}{\partial x^2} + \frac{\partial ^2 c}{\partial y^2}+\frac{\partial ^2 c}{\partial z^2}\right). \]    
It represents the motion of the drug due to diffusion through the chyme. In this expression $\nu $ is a constant called the coefficient of molecular diffusion.

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