Heat Transfer in Laminar Tube Flow of a Class of Non-Affine Viscoelastic Fluids

The fully developed thermal field in constant pressure gradient driven laminar flow of viscoelastic fluids in straight pipes of arbitrary contour ∂D is investigated. The nonlinear fluids considered are constitutively represented by a class of single mode, non-affine constitutive equations. The driving forces can be large and inertial effects are accounted for. Asymptotic series in terms of the Weissenberg number Wi are employed to represent the field variables. Heat transfer enhancement due to shear-thinning is identified together with the enhancement due to the inherent elasticity of the fluid. The latter is the result of secondary flows in the cross-section. Increasingly large enhancements are computed with increasing elasticity of the fluid as compared to its Newtonian counterpart. Large enhancements are possible even with dilute fluids. Isotherms for the temperature field are presented and discussed for several non-circular contours such as the ellipse and the equilateral triangle together with heat transfer behavior in terms of the Nusselt number Nu.

HERSCHEL–BULKLEY FLUIDS: EXISTENCE AND REGULARITY OF STEADY FLOWS

The equations for steady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved for the Dirichlet boundary-value problem. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient. Such a fluid stiffens if its local stresses do not exceed τ*, and it behaves like a non-Newtonian fluid otherwise. We address here a class of nonlinear fluids which includes shear-thinning p-law fluids with 9/5 < p ≤ 2. The flow equations are formulated in the stress-velocity setting (cf. Ref. 25). Our approach is different from that of Duvaut–Lions (cf. Ref. 10) developed for classical Bingham visco-plastic materials. We do not apply the variational inequality but make use of an approximation of the Herschel–Bulkley fluid with a generalized Newtonian fluid with a continuous constitutive law.