High-Order Algorithm for Simulating Heat Transfer in Complex Yield-Stress Fluid Flows

Document Type : Research Paper

Authors
Bashaer K. Al-Bahrani
Alaa H. Al-Muslimawi
Department of Mathematics, University of Basrah, Basrah, Iraq
10.22055/jacm.2025.48167.5026
Abstract
This paper examines the numerical simulation of non-isothermal flows of Herschel-Bulkley fluids under stick-slip boundary conditions and a constant wall temperature. To solve the governing equations, we propose a decoupled transient solution algorithm utilizing high-order time-stepping methods based on the Taylor-Galerkin approach combined with pressure correction in a finite element framework. This numerical approach exhibits high accuracy in achieving convergence. The results obtained for the steady-state solution emphasize the significant impact of temperature changes on the flow pattern. Their effect on the topology of yielded and unyielded regions, along with the efficiency of heat transfer and the friction factor of Fanning as functions of yield stress, power-law index, Reynolds number, and Péclet number, underscores the significant interaction among these parameters. Notably, the effects of stick-slip conditions are particularly pronounced at the transition to the slipping zone within the singularity. Here, a peak in the shear rate is observed, leading to significant changes in the stress distribution within the channel. Moreover, the numerical method was validated in good agreement with previous numerical findings. 
Keywords
Finite element method
Non-isothermal flow
Stick-slip
Viscoplastic fluid
Yield stress
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Bingham, E.C., Fluidity and plasticity, McGraw-Hill, 1922.
[2] Herschel, W.H., Bulkley, R., Konsistenzmessungen von gummi-benzollösungen, Kolloid-Zeitschrift, 39, 1926, 291-300.
[3] Bird, R.B., Dai, G.C., Yarusso, B.J., The rheology and flow of viscoplastic materials, Reviews in Chemical Engineering, 1, 1983, 1-70.
[4] Vradis, G.C., O tugen, M.V., The axisymmetric sudden expansion flow of a non-Newtonian viscoplastic fluid, Journal of Fluids Engineering, 1997, 119, 193-200.
[5] Burgos, G.R., Alexandrou, A.N., Flow development of Herschel-Bulkley fluids in a sudden three-dimensional square expansion, Journal of Rheology, 43, 1999, 485-498.
[6] Vinay, G., Wachs, A., Agassant, J.-F., Numerical simulation of non-isothermal viscoplastic waxy crude oil flows, Journal of Non-Newtonian Fluid Mechanics, 128, 2005, 144-162.
[7] Bognár, G., Similarity solution of boundary layer flows for non-Newtonian fluids, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 2009, 1555-1566.
[8] Bognár, G., Kovács, J., Non-isothermal steady flow of power-law fluids between parallel plates, International Journal of Mathematical Models and Methods in Applied Sciences, 6, 2012, 122-129.
[9] Santhosh, N., Radhakrishnamacharya, G., Chamkha, A.J., Effect of slip on Herschel-Bulkley fluid flow through narrow tubes, Alexandria Engineering Journal, 54, 2015, 889-896.
[10] Jing, Z., Wang, S., Zhai, Z., Effects of slip and rheological parameters on the flow and heat transfer of a Herschel-Bulkley fluid, International Journal of Numerical Methods for Heat & Fluid Flow, 27, 2017, 981-999.
[11] Yasir, R.Y., Al-Muslimawi, A.H., Jassim, B.K., Numerical simulation of non-Newtonian inelastic flows in channel based on artificial compressibility method, Journal of Applied and Computational Mechanics, 6, 2020, 271-283.
[12] Singh, J., Vishalakshi, A.B., Mahabaleshwar, U.S., Bognar, G., MHD Casson fluid flow with Navier's and second order slip due to a perforated stretching or shrinking sheet, Plos One, 17, 2022, e0276870.
[13] Mitsoulis, E., Abdali, S.S., Markatos, N.C., Flow simulation of Herschel-Bulkley fluids through extrusion dies, The Canadian Journal of Chemical Engineering, 71, 1993, 147-160.
[14] Fusi, L., Non-isothermal flow of a Bingham fluid with pressure and temperature dependent viscosity, Meccanica, 52, 2017, 3577-3592.
[15] Baranovskii, E.S., On Flows of Bingham-Type Fluids with Threshold Slippage, Advances in Mathematical Physics, 2017, 2017, 7548328.
[16] Vaidya, H., Gudekote, M., Choudhari, R., Prasad, K.V., Role of slip and heat transfer on peristaltic transport of Herschel-Bulkley fluid through an elastic tube, Multidiscipline Modeling in Materials and Structures, 14, 2018, 940-959.
[17] Pakhomov, M.A., Zhapbasbayev, U.K., RANS modeling of the transition of a non-isothermal flow of a Newtonian fluid to a viscoplastic state in a pipe, Institute of Metallurgy and Ore Beneficiation Satbayev University, 2023.
[18] Zhapbasbayev, U., Bekibayev, T., Pakhomov, M., Ramazanova, G., Numerical Modeling of Non-Isothermal Laminar Flow and Heat Transfer of Paraffinic Oil with Yield Stress in a Pipe, Energies, 17, 2024, 2080.
[19] Hawken, D.M., Tamaddon-Jahromi, H.R., Townsend, P., Webster, M.F., A Taylor-Galerkin-based algorithm for viscous incompressible flow, International Journal for Numerical Methods in Fluids, 10, 1990, 327-351.
[20] Ngamaramvaranggul, V., Webster, M.F., Viscoelastic simulations of stick-slip and die-swell flows, International Journal for Numerical Methods in Fluids, 36, 2001, 539-595.
[21] Belblidia, F., Haroon, T., Webster, M.F., The Dynamics of Compressible Herschel-Bulkley Fluids in Die-Swell Flows, Damage and Fracture Mechanics: Failure Analysis of Engineering Materials and Structures, 2009, 425-434.
[22] Echendu, S.O.S., Belblidia, F., Tamaddon-Jahromi, H.R., Webster, M.F., Modelling with viscous and viscoplastic materials under combining and separating flow configurations, Mechanics of Time-Dependent Materials, 15, 2011, 407-428.
[23] Ngamaramvaranggul, V., Webster, M.F., Simulation of coating flows with slip effects, International Journal for Numerical Methods in Fluids, 33, 2000, 961-992.
[24] Al-Haboobi, A., Fadhel, I.A., Sharhan, A.A., Al-Muslimawi, A.H., Computational simulation of compressible Newtonian flow past circular and square bluff bodies, Ocean Systems Engineering, 14, 2024, 315.
[25] Sharhan, A., Al-Muslimawi, A., Inelastic Solution for Power Law Fluid with Taylor Galerkin-Pressure Correction Finite Element Method: Axisymmetric Contraction Flows, Journal of Applied Fluid Mechanics, 16, 2023, 2411-2423.
[26] Shareef, A., Al-Muslimawi, A., Numerical Investigation on the Conical Flow of Viscoelastic Fluids, CFD Letters, 16, 2024, 85-97.
[27] Ding, D., Townsend, P., Webster, M.F., A computer code for simulating transient non-isothermal flows of non-Newtonian fluids, Third European Rheology Conference, Springer, Netherlands, 1990, 132-134.
[28] Ding, D., Townsend, P., Webster, M.F., The development of flow simulation software for polymer processing applications, Revista Portuguesa De Hemorreologia, 4, 1990, 31-38.
[29] Ding, D., Townsend, P., Webster, M.F., A study of some model problems for validating polymer processing computer codes, University of Wales (Swansea), Mathematics and Computer Science Division, 1990.
[30] Tamaddon ‐Jahromi, H.R., Ding, D., Webster, M.F., Townsend, P., A Taylor— Galerkin finite element method for non-Newtonian flows, International Journal for Numerical Methods in Engineering, 34, 1992, 741–757.
[31] Ding, D., Townsend, P., Webster, M.F., Computer modelling of transient thermal flows of non-Newtonian fluids, Journal of Non-Newtonian Fluid Mechanics, 47, 1993, 239–265.
[32] Ding, D., Townsend, P., Webster, M.F., On computation of two and three-dimensional unsteady thermal non-Newtonian flows, International Journal of Numerical Methods for Heat & Fluid Flow, 5, 1995, 495–510.
[33] Ding, D., Townsend, P., Webster, M.F., Finite element simulation of an injection moulding process, International Journal of Numerical Methods for Heat & Fluid Flow, 7, 1997, 751–766.
[34] Al-Bahrani, B.K., Al-Muslimawi, A.H.A., Coupled effects of stick-slip and thermal boundary conditions on viscous dissipation for inelastic power-law fluids, Engineering Computations, 42, 2025, 929-950.
[35] Benmoussa, A., Páscoa, J.C., A Chamfered Anchor Impeller Design for Enhanced Efficiency in Agitating Viscoplastic Fluids, Fluids, 9, 2024, 288.
[36] Mokhefi, A., Hydrodynamic and thermal performance analysis of an inclined anchor impeller designed for yield stress food mixing applications, Food and Bioproducts Processing, 143, 2024, 255-270.
[37] Oliveira, P.J., A shear and elongational decomposition approach of the rate-of-deformation tensor for non-Newtonian flows with mixed kinematics, Journal of Fluid Mechanics, 1001, 2024, A17.
[38] Latif, M.J., Heat Convection, Springer, 2006.
[39] Graebel, W., Advanced Fluid Mechanics, Academic Press, Elsevier, 2007.
[40] Papanastasiou, T.C., Flows of materials with yield, Journal of Rheology, 31, 1987, 385-404.
[41] Alexandrou, A.N., McGilvreay, T.M., Burgos, G., Steady Herschel-Bulkley fluid flow in three-dimensional expansions, Journal of Non-Newtonian Fluid Mechanics, 100, 2001, 77-96.
[42] Huilgol, R.R., Variational inequalities in the flows of yield stress fluids including inertia: theory and applications, Physics of Fluids, 14, 2002, 1269-1283.
[43] Al-Haboobi, A., Al-Muslimawi, A.H., Novel algorithm for compressible Newtonian axisymmetric thermal flow, International Journal of Modern Physics C, 2450025, 2024, 17.
[44] Ferrás, L.L., Nóbrega, J.M., Pinho, F.T., Analytical solutions for Newtonian and inelastic non-Newtonian flows with wall slip, Journal of Non-Newtonian Fluid Mechanics, 175, 2012, 76-88.
[45] Nouar, C., Lebouché, M., Devienne, R., Riou, C., Numerical analysis of the thermal convection for Herschel-Bulkley fluids, International Journal of Heat and Fluid Flow, 16, 1995, 223-232.
[46] Al-Bahrani, B.K., Al-Muslimawi, A.H., High-order artificial compressibility algorithm for simulating non-Newtonian fluid flows, International Journal of Modern Physics C, 2025, doi.10.1142/S0129183125500548.
Journal of Applied and Computational Mechanics
Volume 12, Issue 1
Winter 2026
Pages 212-225