Numerical Modeling of Non-Affine Viscoelastic Fluid Flow Including Viscous Dissipation Through a Square Cross-Section Duct: Heat Transfer Enhancement due to the Inertia and the Elastic Effects

It is known that non-isothermal, incompressible, laminar flow of non-affine viscoelastic fluid in tubes of arbitrary contour gives rise to secondary recirculations which are due both to the anisotropy of the second difference in normal stresses and to the non-symmetry of cross-section.

Together, the effect of elasticity and the rheofluidicitation of the fluid result in an increase in the Nusselt number compared to the flow of a Newtonian fluid. For low Peclet numbers (Re P r < 50) , the contribution of elasticity to the increase in the Nusselt number is negligible, the improvement in heat transfer rate is only related to the rheofluidification.

The flow through the tube is governed by the conservation equations of energy, mass, momentum associated with to one non–affine rheological model mentioned above. The stress tensor of the solvent obeys a linear Newtonian behaviour. Our numerical approach is based on the finite difference method.

The mixed type of the governing system of equations (elliptic–parabolic–hyperbolic) requires coupling between discretisation methods designed for elliptic–type equations and techniques adapted to transport equations.

The system of governing is discretised in time by the BDF2 scheme where the convection with the non–linear terms receive explicit treatment and the Laplacian, gradient and divergence operators undergo an implicit approach. To allow appropriate spatial iscretisation of the convection terms, the system is rewritten in a quasi-linear first-order and  homogeneous form without the continuity and energy equations. For each spatial direction only subject to the positivity of the conformation tensor the equation system is said to be hyperbolic.

With the rheological models of the UCM, Oldroyd-B and Giesekus type, the conformation tensor is by definition symmetrical and positive-definite, with the PTT model the hyperbolicity condition is subject to restrictions related to the rheological parameters. Based on this hyperbolicity condition, the contribution of the hyperbolic part is approximated by applying the characteristic method to extract pure advection terms which are then discretized by high ordre schemes WENO and HOUC.

The incompressibility constraint is imposed using a semi-implicit incremental projection method. The Laplacian, gradient and divergence operators are spatially discretised by conventional second order centered differentiations. The algorithm thus developed makes it possible, on the one hand, to avoid the problems of instabilities related to the high Weissenberg number without the use of any stabilization method, and on the other hand, it ensures the maintenance of the positivity of the conformation tensor. Finally, a Nusselt number analysis is given as a function of inertia, elasticity, viscous dissipation, solvent viscosity ratio and various material and rheological parameters.