Convective-derivative control of viscous heating in Gordon–Schowalter plane Poiseuille flow: thermal equivalence of the Maxwell limits and constitutive sensitivity
Year: 2026
Type: article
Abstract: We analyse viscous heating in steady plane Poiseuille flow of a Gordon–Schowalter fluid between isothermal walls and determine how the convective-derivative parameter $$a\in [-1,1]$$ controls the thermal response. For $$\beta =O(1)$$ , a regular expansion to first order in the Brinkman number Br and second order in the Weissenberg number Ws gives $$\Theta (\eta )=\Pi f_{0}(\eta )+(1-a^{2})\Psi f_{1}(\eta )+O(Br^{2},Br\,Ws^{4})$$ , where $$\Pi =Br\,G^{2}$$ , $$\Psi =Br\,Ws^{2}G^{4}$$ , and G is the dimensionless pressure gradient. Although the temperature-dependent viscosity changes the complete velocity field, $$\Theta =O(Br)$$ implies that this feedback first changes the temperature at $$O(Br^{2})$$ and therefore lies beyond the retained closed form. Two structural results do not depend on this truncation. The upper- and lower-convected Maxwell limits ( $$a=\pm 1$$ ) 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 $$1-a^{2}$$ . For the constant-viscosity reference, the Newtonian-connected homogeneous branch requires $$(1-a^{2})Ws^{2}G^{2}\le 1$$ . 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. Graphical abstract
Source: Korea-Australia Rheology Journal
Language: English
Subfield: Fluid Flow and Transfer Processes
Field: Chemical Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,790
OpenAlex ID: W7215953975