A new approach for modeling viscoelastic thin film lubrication

Ahmed, Humayun; Biancofiore, Luca

doi:10.1016/j.jnnfm.2021.104524

Cited by 30 publications

(22 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Black triangles represent the results of the numerical simulation obtained from calculating the pressure drop along the centreline (). We note that while our analysis assumes , where Deborah number is the product of the relaxation time and the characteristic extensional rate of the flow, the Weissenberg number , which is the product of the relaxation time and the characteristic shear rate of the flow, is (see also Ahmed & Biancofiore 2021). To further highlight this point, we present our results both as a function of and .…”

Section: Results and Comparison With Finite-element Simulationsmentioning

confidence: 99%

“…2002; Saprykin et al. 2007; Ahmed & Biancofiore 2021). We note that in most studies all the components of the polymer stress tensor were scaled with the same scaling .…”

Section: Results and Comparison With Finite-element Simulationsmentioning

confidence: 99%

“…2002; Saprykin et al. 2007; Ahmed & Biancofiore 2021), where we have introduced the aspect ratio of the configuration, which is assumed to be small, the viscosity ratios, and the Deborah and Weissenberg numbers, The non-dimensional shape of the channel is denoted by and is an important parameter in our analysis.…”

Section: Problem Formulation and Governing Equationsmentioning

confidence: 99%

“…2007). The Weissenberg number is the product of the relaxation time scale of the fluid and the characteristic shear rate of the flow, and is related to the Deborah number through (Ahmed & Biancofiore 2021). We note that since we assume , can be small while keeping .…”

Section: Problem Formulation and Governing Equationsmentioning

confidence: 99%

“…In subsequent work, we will analyse more complex constitutive models, incorporating additional microscopic features of polymer solutions. Our approach for obtaining analytical solutions for velocity and pressure is motivated by studies on thin films and lubrication problems (Tichy 1996; Zhang, Matar & Craster 2002; Saprykin, Koopmans & Kalliadasis 2007; Ahmed & Biancofiore 2021). Such an approach relies on exploiting the narrowness of the geometry through the application of the lubrication approximation and a perturbation expansion in powers of the Deborah number , which is assumed to be small, , and solving order by order, often resulting in cumbersome calculations at high orders.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Pressure-driven flow of the viscoelastic Oldroyd-B fluid in narrow non-uniform geometries: analytical results and comparison with simulations

Boyko

Stone

2022

J. Fluid Mech.

View full text Add to dashboard Cite

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $\Delta p$ . We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ( $De$ ), we provide analytical expressions for the velocity, stress and the $q$ – $\Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$ . Furthermore, we exploit the reciprocal theorem derived by Boyko $\&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, L081301) to obtain the $q$ – $\Delta p$ relation at the next order, $O(De^3)$ , using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$ . We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.

show abstract

Section: Results and Comparison With Finite-element Simulationsmentioning

confidence: 99%

“…2002; Saprykin et al. 2007; Ahmed & Biancofiore 2021). We note that in most studies all the components of the polymer stress tensor were scaled with the same scaling .…”

Section: Results and Comparison With Finite-element Simulationsmentioning

confidence: 99%