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Diffusion of chemically reactive species in a porous medium over a stretching sheet. (English) Zbl 1104.35038

The authors study the flow and mass transfer of a chemically reactive species of a viscoelastic fluid over a stretching sheet, using the proper sign for the material constant arising in the Cauchy stress in an incompressible homogeneous fluid of second grade. By knowing the mathematical equivalence of the mass concentration boundary layer problem with the thermal boundary analogue, the results obtained for mass transfer characteristics can be carried directly to the heat transfer characteristics by replacing the Schmidt number with the Prandtl number. For the flow of a second-order fluid past a flat sheet coinciding with the plane $y=0$, the governing equations can be simplified and as a result, the problem is reduced to solving the following boundary value problem

$\begin{array}{cc}& {u}_{tt}+{t}^{\frac{1}{\theta }-1}{u}_{t}=\frac{1}{\theta {t}^{2}}\beta {u}^{n},\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right)\hfill \\ & u\left(0\right)=0,\phantom{\rule{1.em}{0ex}}u\left(1\right)=1,\hfill \end{array}$

where ${e}^{-y}={t}^{1/\theta }$. In general, the equation is degenerate for $t$ near zero. A refined analysis allows to overcome the singular nature of the resulting nonlinear boundary value problem. Using the Brouwer fixed point theorem, existence results are established. Moreover, the exact analytical solutions are obtained. The results obtained for the diffusion characteristics reveal many interesting behaviors that warrant further study of the effects of reaction rate on the transfer of chemically reactive species.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 92C45 Kinetics in biochemical problems 35C05 Solutions of PDE in closed form 34B16 Singular nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 76A10 Viscoelastic fluids