Diffusion with a slow second$-$order reaction Consider a second$-$order, irreversible reaction occurring at steady state nini a liquid film of thickness L$.$ The concentration of thereactant $($species A$)$ at $x=0$ is $C_0,$ and there is no transport across the surface at $x=L.$ The volumetric reaction rate is given by

$R_{VA}=-k{C}_A^2.$

Assuming that the reaction is slow relative to diffusion, use a regular perturbation scheme to determine $C_A(x)$ and the flux of $A$ at $x=0.$ The small parameter is the Damk$$hler number.

Obtain three terms in the expansion for $=\frac{C_A}{C_0}.$ That is$,$ include terms which are $O(D{a}^2).$