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Keywords:
implicit algebraic constitutive relations; viscoplastic fluid; seemingly viscoplastic fluid; flow between adjacent surfaces
implicit algebraic constitutive relations; viscoplastic fluid; seemingly viscoplastic fluid; flow between adjacent surfaces
Summary:
A simplified model is derived for pressure-driven flow between adjacent surfaces of materials modeled as seemingly viscoplastic or truly viscoplastic. The material response to external forces is traditionally described by constitutive relations in which the extra stress tensor ${\bf S}$ is expressed as a function of the symmetric part of the velocity gradient ${\bf D}$. However, for viscoplastic materials, ${\bf S}$ cannot, in general, be written as a function of ${\bf D}$, whereas ${\bf D}$ can be expressed in terms of ${\bf S}$. Motivated by this observation, a model based on constitutive relations of the form ${\bf D} = f({\bf S})$ is proposed, leading to a system of first-order partial differential equations. A local Poiseuille law is also formulated, and a reduced-dimensional equation for the pressure is derived. Explicit velocity profiles are obtained for selected cases.
References:
[1] Almqvist, A., Burtseva, E., Rajagopal, K. R., Wall, P.: On lower-dimensional models in lubrication. Part A: Common misinterpretations and incorrect usage of the Reynolds equation. Proc. Inst. Mech. Eng., Part J, J. Eng. Tribology 235 (2021), 1692-1702. DOI 10.1177/13506501209737
[2] Almqvist, A., Burtseva, E., Rajagopal, K. R., Wall, P.: On lower-dimensional models in lubrication. Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids. Proc. Inst. Mech. Eng., Part J, J. Eng. Tribology 235 (2021), 1703-1718. DOI 10.1177/1350650120973800
[3] Almqvist, A., Burtseva, E., Rajagopal, K. R., Wall, P.: On flow of power-law fluids between adjacent surfaces: Why is it possible to derive a Reynolds-type equation for pressure-driven flow, but not for shear-driven flow?. Appl. Eng. Sci. 15 (2023), Article ID 100145, 9 pages. DOI 10.1016/j.apples.2023.100145
[4] Almqvist, A., Burtseva, E., Rajagopal, K. R., Wall, P.: On lower-dimensional models of thin film flow. Part C: Derivation of a Reynolds type of equation for fluids with temperature and pressure dependent viscosity. Proc. Inst. Mech. Eng., Part J, J. Eng. Tribology 237 (2023), 514-526. DOI 10.1177/13506501221135269
[5] Almqvist, A., Burtseva, E., Rajagopal, K. R., Wall, P.: On modeling flow between adjacent surfaces where the fluid is governed by implicit algebraic constitutive relations. Appl. Math., Praha 69 (2024), 725-746. DOI 10.21136/AM.2024.0131-24 | MR 4841724 | Zbl 07980742
[6] Barnes, H. A., Walters, K.: The yield stress myth?. Rheol. Acta 24 (1985), 323-326. DOI 10.1007/BF01333960
[7] Bingham, E. C.: An investigation of the laws of plastic flow. Bull. Bur. Stand. 13 (1916), 309-353. DOI 10.6028/bulletin.304
[8] Bingham, E. C.: Fluidity and Plasticity. McGraw-Hill, New York (1922).
[9] Bulíček, M., Gwiazda, P., Málek, J., Rajagopal, K. R., Świerczewska-Gwiazda, A.: On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph. Mathematical Aspects of Fluid Mechanics London Mathematical Society Lecture Note Series 402. Cambridge University Press, Cambridge (2012), 23-51. DOI 10.1017/CBO9781139235792.003 | MR 3050290 | Zbl 1296.35137
[10] Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On steady flows of incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var. 2 (2009), 109-136. DOI 10.1515/ACV.2009.006 | MR 2523124 | Zbl 1233.35164
[11] Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 (2012), 2756-2801. DOI 10.1137/110830289 | MR 3023393 | Zbl 1256.35074
[12] Bulíček, M., Málek, J., Maringová, E.: On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary. J. Math. Fluid Mech. 25 (2023), Article ID 72, 29 pages. DOI 10.1007/s00021-023-00803-w | MR 4624542 | Zbl 1525.35198
[13] Bulíček, M., Maringová, E., Málek, J.: On nonlinear problems of parabolic type with implicit constitutive equations involving flux. Math. Models Methods Appl. Sci. 31 (2021), 2039-2090. DOI 10.1142/S0218202521500457 | MR 4344259 | Zbl 1492.35152
[14] Casson, N.: A flow equation for pigment-oil suspensions of the printing ink type. Rheology of Disperse Systems Pergamon Press, Oxford (1959), 84-104.
[15] Coussot, P.: Bingham's heritage. Rheologica Acta 56 (2017), 163-176. DOI 10.1007/s00397-016-0983-y
[16] Dean, E. J., Glowinski, R., Guidoboni, G.: On the numerical simulation of Bingham visco-plastic flow: Old and new results. J. Non-Newton. Fluid Mech. 142 (2007), 36-62. DOI 10.1016/j.jnnfm.2006.09.002 | Zbl 1107.76061
[17] Diening, L., Kreuzer, C., Süli, E.: Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal. 51 (2013), 984-1015. DOI 10.1137/120873133 | MR 3035482 | Zbl 1268.76030
[18] Dowson, D.: A generalized Reynolds equation for fluid-film lubrication. Int. J. Mech. Sci. 4 (1962), 159-170. DOI 10.1016/S0020-7403(62)80038-1
[19] Edgeworth, R., Dalton, B. J., Parnell, T.: The pitch drop experiment. Eur. J. Phys. 5 (1984), 198-200. DOI 10.1088/0143-0807/5/4/003
[20] Fabricius, J., Manjate, S., Wall, P.: Error estimates for pressure-driven Hele-Shaw flow. Q. Appl. Math. 80 (2022), 575-595. DOI 10.1090/qam/1619 | MR 4453782 | Zbl 1490.76020
[21] Fabricius, J., Manjate, S., Wall, P.: On pressure-driven Hele-Shaw flow of power-law fluids. Appl. Anal. 101 (2022), 5107-5137. DOI 10.1080/00036811.2021.1880570 | MR 4475758 | Zbl 1500.76017
[22] Fabricius, J., Miroshnikova, E., Tsandzana, A., Wall, P.: Pressure-driven flow in thin domains. Asymptotic Anal. 116 (2020), 1-26. DOI 10.3233/ASY-191535 | MR 4044383 | Zbl 1442.35335
[23] Farrell, P. E., Gazca-Orozco, P. A.: An augmented Lagrangian preconditioner for implicitly constituted non-Newtonian incompressible flow. SIAM J. Sci. Comput. 42 (2020), B1329--B1349. DOI 10.1137/20M1336618 | MR 4169754 | Zbl 1458.65147
[24] Farrell, P. E., Gazca-Orozco, P. A., Süli, E.: Numerical analysis of unsteady implicitly constituted incompressible fluids: 3-field formulation. SIAM J. Numer. Anal. 58 (2020), 757-787. DOI 10.1137/19M125738X | MR 4066569 | Zbl 1434.76065
[25] Fusi, L., Rajagopal, K. R.: Flow past a porous plate of a new class of fluids with limiting stress: Analytical results and linear stability analysis. Eur. J. Mech., B, Fluids 112 (2025), 58-64. DOI 10.1016/j.euromechflu.2025.02.007 | Zbl 08069168
[26] Garimella, S. M., Anand, M., Rajagopal, K. R.: A new model to describe the response of a class of seemingly viscoplastic materials. Appl. Math., Praha 67 (2022), 153-165. DOI 10.21136/AM.2021.0163-20 | MR 4396682 | Zbl 1549.74145
[27] Garimella, S. M., Anand, M., Rajagopal, K. R.: Start-up shear flow of a shear-thinning fluid that approximates the response of viscoplastic fluids. Appl. Math. Comput. 412 (2022), Article ID 126571, 8 pages. DOI 10.1016/j.amc.2021.126571 | MR 4300330 | Zbl 1510.76020
[28] Gazca-Orozco, P. A.: A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow. ESAIM, Math. Model. Numer. Anal. 55 (2021), 2679-2703. DOI 10.1051/m2an/2021068 | MR 4337453 | Zbl 1483.65182
[29] Herschel, W. H., Bulkley, R.: Konsistenzmessungen von Gummi-Benzollösungen. Kolloid Zeit. 39 (1926), 291-300 German. DOI 10.1007/BF01432034
[30] Housiadas, K. D., Georgiou, G. C.: The analytical solution for the flow of a Papanastasiou fluid in ducts with variable geometry. J. Non-Newton. Fluid Mech. 319 (2023), Article ID 105074, 13 pages. DOI 10.1016/j.jnnfm.2023.105074 | MR 4612283
[31] Janečka, A., Málek, J., Průša, V., Tierra, G.: Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor. Acta Mech. 230 (2019), 729-747. DOI 10.1007/s00707-019-2372-y | MR 3918562 | Zbl 1428.76017
[32] Kreuzer, C., Süli, E.: Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. ESAIM, Math. Model. Numer. Anal. 50 (2016), 1333-1369. DOI 10.1051/m2an/2015085 | MR 3554545 | Zbl 1457.65201
[33] Málek, J.: Mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations. ETNA, Electron. Trans. Numer. Anal. 31 (2008), 110-125. MR 2569596 | Zbl 1182.35182
[34] Mitsoulis, E., Tsamopoulos, J.: Numerical simulations of complex yield-stress fluid flows. Rheologica Acta 56 (2017), 231-258. DOI 10.1007/s00397-016-0981-0
[35] Papanastasiou, T. C.: Flows of materials with yield. J. Rheol. 31 (1987), 385-404. DOI 10.1122/1.549926 | Zbl 0666.76022
[36] Rajagopal, K. R.: On implicit constitutive theories. Appl. Math., Praha 48 (2003), 279-319. DOI 10.1023/A:1026062615145 | MR 1994378 | Zbl 1099.74009
[37] Rajagopal, K. R.: On implicit constitutive theories for fluids. J. Fluid Mech. 550 (2006), 243-249. DOI 10.1017/S0022112005008025 | MR 2263984 | Zbl 1097.76009
[38] Süli, E., Tscherpel, T.: Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids. IMA J. Numer. Anal. 40 (2020), 801-849. DOI 10.1093/imanum/dry097 | MR 4092271 | Zbl 1464.65131
[39] Widdicombe, A. T., Ravindrarajah, P., Sapelkin, A., Phillips, A. E., Dunstan, D., Dove, M. T., Brazhkin, V. V., Trachenko, K.: Measurement of bitumen viscosity in a room-temperature drop experiment: Student education, public outreach and modern science in one. Phys. Educ. 49 (2014), 406-411. DOI 10.1088/0031-9120/49/4/406
[40] You, Z., Huilgol, R. R., Mitsoulis, E.: Application of the Lambert $W$ function to steady shearing flows of the Papanastasiou model. Int. J. Eng. Sci. 46 (2008), 799-808. DOI 10.1016/j.ijengsci.2008.02.002 | MR 2427932 | Zbl 1213.74022
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