The problem of the squeezing of a fluid film with a complex rheology is studied. The fluid is represented by the Phan-Tien and Tanner (PTT) constitutive equation. The goal of the paper is to develop an analytical approach which allows one to describe and quantify this process, which is often present in classical lubrication problems, but particularly in human joints. The present approach can potentially supplement time-consuming and costly purely numerical methods which are also susceptible to misleading artifacts. A parametric study reveals the effect of fluid elasticity compared to viscous effects and some explanation is suggested as to their possible links with joint diseases.

In this paper, a modified Reynolds equation in polar coordinates using the Phan-Thien and Tanner rheological model is combined with a modified Darcy law describing a poroelastic confining surface. The Phan-Thien and Tanner model is commonly used to describe the viscoelastic behaviour of synovial fluid, while the poroelastic model can portray articular cartilage. Together, they may provide a simulation basis for lubrication of the tribology of the knee contact under squeezing. The results obtained for the axisymmetric case are analysed and compared for certain non-Newtonian shear phenomena on the basis of works from the literature. NOMENCLATUREpermeability of lower disc (m 2 ) K volume average modulus of elasticity (Pa m À3 ) ℓ displacement vector of pores (m) m upper disc mass (kg) p pressure (Pa) p atm atmospheric pressure (Pa) p* Phan-Thien and Tanner (PTT) film fluid pressure, in porous media (Pa) p N Newtonian pressure (Pa) r radial coordinate (m) R disc radius (m) t time (s) v 0 initial radial velocity component (ms À1 ) u N Newtonian radial velocity component (ms À1 ) U 1 , U 2 radial velocity components of lower/upper disc (ms À1 ) U Ã 1 ; V Ã 1 PTT velocity components, in porous media/fluid film interface (radial, axial) (m s À1 ) V 1 axial (normal) velocity component of lower disc (ms À1 ) v* PTT fluid particles relative velocity vector, in porous medium (m s À1 ) v f PTT fluid particles absolute velocity vector, in porous medium (m s À1 ) W 1 , W 2 azimuthal velocity components of lower/upper disc (ms À1dynamic viscosity of PTT solvent (Pa s) η r effective viscosity in z-plane, r-direction (Pa s) η θ effective viscosity in z-plane, θ-direction (Pa s) θ angular coordinate (À) λ relaxation time (s) ξ PTT slip parameter π zz total axial (normal) stress component (Pa) τ rz extra stress component in z-plane, r-direction (Pa)

In this contribution, we reworked the Bell-Delaware technique for a HB-STHX to systematically study its thermohydraulic performance behavior in the shell side. The helical baffles are continuous quadrant sectors with five inclination angles 25°, 30°, 35°, 40° and 45° and are exanimated for six different values of Reynolds number  from  up to 40 . We have observed an increase in global heat transfer rate while a decrease in pressure loss, in continuous helical baffle case compared to segmental one. we have found that the baffle inclination 40° has the largest heat transfer coefficient, 25° gives the lowest pressure drop, whereas 45°, the largest inclination we have used, displays the greatest performance evaluation factor.