Perturbative modelling of Arrhenius reactive transport in non-isothermal Gordon– Schowalter Couette–Poiseuille flow

<title>Abstract</title> <p>We analyse coupled heat and mass transfer with first-order homogeneous Arrhenius kinetics in plane Couette–Poiseuille flow of a Gordon–Schowalter viscoelastic fluid between isothermal, perfectly reactive walls, in the small-Brinkman, weakly viscoelastic regime. Three results are established. (i) The leading viscoelastic correction to the thermal–reactive response is proportional to the mixed group \(\:Br\hspace{0.17em}W{s}^{2}\left(1-{a}^{2}\right)\), where \(\:a\) is the convected-derivative parameter: it vanishes identically in the upper- and lower-convected Maxwell limits (\(\:a=\pm\:1\)) and is maximal for the corotational case (\(\:a=0\)), so the convected derivative acquires a measurable signature in temperature rise, axial decay rate and conversion length, and not only in the stress field. (ii) The viscoelastic perturbation reaches the concentration field through two additive first-order routes of opposite sign: an advective route, in which the rate-thinning response at fixed pressure gradient increases the through-flow and lengthens the conversion zone, and a thermal–reactive route, in which enhanced viscous heating accelerates the Arrhenius kinetics and shortens it. (iii) Their competition is governed by a single crossover number \(\:K\); the net effect of viscoelasticity on reactor sizing reverses sign at \(\:K=1\). Closed-form expressions are obtained for the velocity and temperature corrections, and the eigenvalue shift of the underlying reactive Graetz problem is validated against direct numerical solution, with first-order accuracy better than 1.9% and route additivity within 0.13%. The crossover criterion provides a compact design rule for reactive coating, polymer-processing and channel-reactor applications.</p>