Taylor dispersion or Taylor diffusion or Shear-induced dispersion is an apparent or effective diffusion of some scalar field arising on the large scale due to the presence of a strong, confined, zero-mean shear flow on the small scale. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it spreads in that direction. The effect is named after the British fluid dynamicist G. I. Taylor, who described the shear-induced dispersion for large Peclet numbers. The analysis was later generalized by Rutherford Aris for arbitrary values of the Peclet number, and hence the process is sometimes also referred to as Taylor-Aris dispersion. The canonical example is that of a simple diffusing species in uniform Poiseuille flow through a uniform circular pipe with no-flux boundary conditions, but is relevant in many other contexts, including the spread of pollutants in rivers and of drugs in blood flow and rivulet flow.
Description We use z as an axial coordinate and r as the radial coordinate, and assume axisymmetry. The pipe has radius a, and the fluid velocity is:
u = w z ^ = w 0 ( 1 − r 2 / a 2 ) z ^ {\displaystyle {\boldsymbol {u}}=w{\hat {\boldsymbol {z}}}=w_{0}(1-r^{2}/a^{2}){\hat {\boldsymbol {z}}}}
The concentration of the diffusing species is denoted c and its diffusivity is D. The concentration is assumed to be governed by the linear advection–diffusion equation:
∂ c ∂ t + w ⋅ ∇ c = D ∇ 2 c {\displaystyle {\frac {\partial c}{\partial t}}+{\boldsymbol {w}}\cdot {\boldsymbol {\nabla }}c=D\nabla ^{2}c}
The concentration and velocity are written as the sum of a cross-sectional average (indicated by an overbar) and a deviation (indicated by a prime), thus:
w ( r ) = w ¯ + w ′ ( r ) {\displaystyle w(r)={\bar {w}}+w'(r)}
c ( r , z ) = c ¯ ( z ) + c ′ ( r , z ) {\displaystyle c(r,z)={\bar {c}}(z)+c'(r,z)}
Under some assumptions (see below), it is possible to derive an equation just involving the average quantities:
∂ c ¯ ∂ t + w ¯ ∂ c ¯ ∂ z = D ( 1 + a 2 w ¯ 2 48 D 2 ) ∂ 2 c ¯ ∂ z 2 {\displaystyle {\frac {\partial {\bar {c}}}{\partial t}}+{\bar {w}}{\frac {\partial {\bar {c}}}{\partial z}}=D\left(1+{\frac {a^{2}{\bar {w}}^{2}}{48D^{2}}}\right){\frac {\partial ^{2}{\bar {c}}}{\partial z^{2}}}}
Observe how the effective diffusivity multiplying the derivative on the right hand side is greater than the original value of diffusion coefficient, D. The effective diffusivity is often written as:
D e f f = D ( 1 + P e 2 48 ) , {\displaystyle D_{\mathrm {eff} }=D\left(1+{\frac {{\mathit {Pe}}^{2}}{48}}\right)\,,}
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