Convective-derivative control of viscous heating in Gordon–Schowalter plane Poiseuille flow: thermal equivalence of the Maxwell limits and constitutive sensitivity

We analyse viscous heating in steady plane Poiseuille flow of a Gordon–Schowalter fluid between isothermal walls and determine how the convective-derivative parameter a∈ [-1,1] controls the thermal response. For β =O(1) , a regular expansion to first order in the Brinkman number Br and second order in the Weissenberg number Ws gives Θ (η )=Π f₀(η )+(1-a²)Ψ f₁(η )+O(Br²,Br\,Ws⁴) , where Π =Br\,G² , Ψ =Br\,Ws²G⁴ , and G is the dimensionless pressure gradient. Although the temperature-dependent viscosity changes the complete velocity field, Θ =O(Br) implies that this feedback first changes the temperature at O(Br²) and therefore lies beyond the retained closed form. Two structural results do not depend on this truncation. The upper- and lower-convected Maxwell limits ( a=± 1 ) have the same simple-shear law as a Newtonian fluid with the same temperature-dependent viscosity; consequently, their coupled momentum–energy problems and temperature fields coincide exactly whenever a steady solution exists. For $$|a|<1$$ , including the Johnson–Segalman and Jaumann cases, the leading viscoelastic correction is positive at fixed pressure gradient and is scaled by 1-a² . For the constant-viscosity reference, the Newtonian-connected homogeneous branch requires (1-a²)Ws²G²≤ 1 . The closed form is verified against a non-truncated prescribed-stress solution and independent second-order finite-difference and finite-volume schemes. The Maxwell–Newtonian equivalence also survives in genuine Couette–Poiseuille flow, but it is specific to steady simple-shear kinematics and need not persist when extensional rates enter the dissipation.