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Research Papers: Lubricants

# From the Phan–Thien–Tanner/Oldroyd-B Non-Newtonian Model to the Double Shear Thining Rabinowisch Thin Film Model

[+] Author and Article Information

Institut Camille Jordan (CNRS UMR 5208) and LaMCoS (CNRS UMR 5514),  Université de Lyon, INSA-Lyon, Bât. Léonard de Vinci, 21 Avenue Jean Capelle, F-69621 Villeurbanne cedex, Franceguy.bayada@insa-lyon.fr

Laurent Chupin

Laboratoire de Mathématiques, CNRS-UMR 6620,  Université Blaise Pascal, Clermont-Ferrand II, Campus des Cézeaux, F-63177 Aubière cedex, Francelaurent.chupin@math.univ-bpclermont.fr

Sébastien Martin

Laboratoire de Mathématiques, CNRS-UMR 8628,  Université Paris-Sud 11, Bâtiment 425 (Mathématique), F-91405 Orsay cedex, Francesebastien.martin@math.u-psud.fr

J. Tribol 133(3), 031802 (Jul 25, 2011) (13 pages) doi:10.1115/1.4003860 History: Received June 02, 2010; Revised March 17, 2011; Published July 25, 2011

## Abstract

In this paper, an asymptotic expansion is used to derive a description of Phan–Tien– Tanner (PTT)/Oldroyd-B flows in the thin film situation without the classical “upper convective maxwell”(UCM) assumption. We begin with a short presentation of the Phan–Thien–Tanner/Oldroyd-B models, which introduce viscoelastic effects in a solute–solvent mixture. The three-dimensional flow is described using five parameters, namely the Deborah number (De) (or the relaxation parameter $λ$), the viscosity ratio $r$, the bulk fluid viscosity $η$, the material slip parameter $a$ related to the “convected derivative” and an elongation number $κ$. Then we focus on the thin film assumption and the related asymptotic analysis that allows us to derive a reduced model. A perturbation procedure for “not too small” values of $κ$ allows us to obtain further results such as an asymptotic “effective viscosity/ shear rate” law, which appears to be a perturbation of the double Rabinowisch model, whose parameters are completely defined by those of the original three-dimensional model. And last a numerical procedure is proposed based on a penalized Uzawa method, to compute the corresponding solution. This algorithm can also be used for any generalized double Newtonian shear thinning Carreau law.

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## Figures

Figure 1

Physical domain

Figure 2

Influence of the elongation correction on the relative effective viscosity as a function of the shear rate ( a = 0.8, Dẽ = 0.5, and r = 0.8)

Figure 3

Pressure distribution (left) and effective viscosity including Oldroyd contribution: (x1,x2,z)→ηeffR(x1,x2,z) (right) in the linear converging profile with r=0.8, Dẽ=1.3, and a=0.8

Figure 4

Stress (x1,x2,z)→(σ11-σ33)(x1,x2,z)(left) and (x1,x2,z)→σ13(x1,x2,z)(right) in the linear converging profile with r=0.8, Dẽ=1.3, and a=0.8

Figure 5

Pressure distribution x1→p(x1,x20), following the streamline x20:=D/2, in the (linear) converging profile. Influence of the elongation number (other parameters are: a=0.80 and Dẽ=0.72

Figure 6

Stress component x1→(σ13)(x1,x20,z0), following the streamline x20:=D/2 at the shearing surface z0:=0, in the (linear) converging profile. Influence of the elongation number (other parameters are: r=0.88, a=0.80, and Dẽ=0.72)

Figure 7

Stress component x1→(σ11-σ33)(x1,x20,z0), following the streamline x20:=D/2 at the shearing surface z0:=0, in the (linear) converging profile. Influence of the elongation number (other parameters are: r=0.88, a=0.80, and Dẽ=0.72)

Figure 8

Convergence of the load difference LDẽ-L0 with respect to the Deborah number, with a retardation parameter r=0.85 and in the geometry HPrime=-0.8, HPrimePrime=-1, and with a=0.8. For small values of Dẽ, the slope of the graph is approximately equal to 2

Figure 9

Pressure distribution x1→p(x1,x20), following the streamline x20:=D/2, in the (linear) converging profile with Dẽ=1.3 and for different values of the retardation parameter r=0.1,0.2,...,0.8

Figure 10

Component of the constraint tensor x1→σ13(x1,x20,z0)(left), x1→(σ11-σ33)(x1,x20,z0) (on right) following the streamline x20:=D/2 at the shearing surface z0:=0, in the (linear) converging profile with Dẽ=1.3 and for different values of the retardation parameter r=0.1,0.2,...,0.8.

Figure 11

Pressure distribution x1→p(x1,x20), following the streamline x20:=D/2, in the (linear) converging profile with r=0.8 and for different values of the Deborah number Dẽ=0.1,0.2,...,5.0

Figure 12

Stress component x1→(σ11-σ33)(x1,x20,z0), following the streamline x20:=D/2 at the shearing surface z0:=0, in the (linear) converging profile with r=0.7 and for different values of the Deborah number Dẽ=0.00,0.17,0.33,0.50

Figure 13

Stress component x1→(σ11-σ33)(x1,x20,z0), following the streamline x20:=D/2 at the shearing surface z0:=0, in the (linear) converging profile with r=0.7 and for different values of the Deborah number Dẽ=0.5,0.65,0.80,0.95,1.15

Figure 14

Stress component x1→(σ11-σ33)(x1,x20,z0), in the (linear) converging profile with r=0.7 and for different values of the Deborah number Dẽ=1.15,1..33,1.50,1.60,3.33,5.00;+∞

Figure 15

(Left) Percentage distribution of the effective viscosity in the three-dimensional thin film flow, for different values of Hmin , in a non-Newtonian regime (Dẽ=0.72, r = 0.7 and a = 0.8). (Right) Percentage distribution of the average effective viscosity in the two-dimensional thin film flow, for different values of Hmin , in a non-Newtonian regime (Dẽ=0.72, r = 0.7 and a = 0.8). Effective viscosity varies from 1 − r to 1.

Figure 16

Pressure distribution in a converging–diverging profile (from top left to bottom right): r = 0.0, r = 0.2, r = 0.5 and r = 0.7, obtained with Dẽ=1.3 and a = 0.8

Figure 17

Spatial discretization and position of the unknowns pij , uij  = (uij , vij )

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