Numerical Simulation of Non-Newtonian Inelastic Flows in Channel based on Artificial Compressibility Method

Document Type: Research Paper


Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq


In this study, inelastic constitutive modelling is considered for the simulation of shear-thinning fluids through a circular channel. Numerical solutions are presented for power-law inelastic model, considering axisymmetric Poiseuille flow through a channel. The numerical simulation of such fluid is performed by using the Galerkin finite element approach based on artificial compression method (AC-method). Usually, the Naiver-Stoke partial differential equations are used to describe fluid activity. These models consist of two partial differential equations; a continuity equation (mass conservation) and time-dependent conservation of momentum, which are maintained in the cylindrical coordinate system (axisymmetric) flow in current study. The effects of many factors such as Reynolds number (Re) and artificial compressibility parameter (ßac) are discussed in this study. In particular, this study confirms the effect of these parameters on the convergence level. To meet the method analysis, Poiseuille flow along a circular channel under an isothermal state is used as a simple test problem. This test is conducted by taking a circular section of the pipe. The Findings reveal that, there is a significant effect from the inelastic parameters upon the the velocity temporal convergence-rates of velocity, while for pressue, the change in convergence is modest. In addition, the rate of convergence is increased as the values of artificial compressibility parameter (ßac) are decreased.


Main Subjects

[1] Madsen, P.A., Schaffer, H.A., A discussion of artificial compressibility, Coastal Engineering, 53(1), 2006, 93-98.

[2] Peyret, R., Taylor, T.D., Computational methods for Fluid Flow, New York:Springer Verlag, 1983.

[3] Kao, P.H., Yang, R.J., A segregated-implicit scheme for solving the incompressible navier-stokes equations, Computers & Fluids, 36(6), 2007, 1159-1161.

[4] Steger, J.L., Kutler, P., Implicit finite-difference procedures for the computation of vortex wakes, AIAA Journal, 15(4), 1977, 581-590.

[5] Chang, J.L., Kwak, D., On the method of pseudo-compressibility for numerically solving incompressible flows, AIAA Journal, 15, 1984, 84-0252.

[6] Choi, D., Merkle, C.L., Application of time-iterative schemes to incompressible flow, AIAA Journal, 23(10), 1985, 1518-1524.

[7] Rizzi, A., Eliksson, L.E., Computational of inviscid incompressible flow with rotation, Fluid Mechanics, 153, 1985, 275-312.

[8] Massarotti, N., Arpino, F., Nithiarasu, P., Fully explicit and semi-implicit cbs procedures for incompressible flows, International Journal for Numerical Methods in Engineering, 66(10), 2006, 1618-1640.

[9] Peyret, R., Unsteady evolution of a horizontal jet in a stratified fluid, Journal of Fluid Mechanics, 78(1), 1976, 49-63.

[10] Merkle, C.L., Athavale, M., Time-accurate unsteady incompressible flow algorithms based on artificial compressibility, AIAA Journal, 87, 1987, 1137-1147.

[11] Rogers, S.E., Kwak, D., Kaul, U., On the accuracy of the pseudocompressibility method in solving the incompressible navier-stokes equations, Applied Mathematical Modeling, 11(1), 1987, 35-44.

[12] Soh, W.Y., Goodrich, J.W., Unsteady solution of incompressible Navier Stokes equations, Journal of Computational Physics, 79(1), 1988, 113-134.

[13] Rosenfeld, M., Kwak, D., Vinokur, M., A Solution Method for Unsteady, Incompressible Navier-Stokes Equations in Generalized Curvilinear Coordinate Systems, Journal of Computational Physics, 94, 1991, 102-137.

[14] Mateescu, D., Paidoussis, M.P., Belanger, F., A time-integration method using artificial compressibility for unsteady viscous flows, Journal of Sound and Vibration, 177(2), 1994, 197-205.

[15] McHugh, P.R., Ramshaw, J.D., Damped artificial compressibility iteration scheme for implicit calculations of unsteady incompressible flow, International Journal for Numerical Methods in Fluids, 21(2), 1995, 141-153.

[16] Gatiganti, R.M., Badcock, K.J., Cantariti, F., Dubuc, L., Woodgate, M., Richards, B.E., Evaluation of an unfactored method for the solution of the incompressible flow equations using artificial compressibility, Applied Ocean Research, 20(3), 1998, 179-187.

[17] Rathish Kumar, B.V., Yamaguchi, T., Liu, H., Himeno, R., A parallel 3D unsteady incompressible flow solver on VPP 700, Parallel Computing, 27(13), 2001, 1687-1713.

[18] de Jouette, C., Laget, O., Le Gouez, J.M., Viviand, H., A dual time stepping method for fluid structure interaction problems, Computers & Fluids, 31(4-7), 2002, 509-537.

[19] Dejam, M., Derivation of dispersion coefficient in an electro-osmotic flow of a viscoelastic fluid through a porous-walled micro channel, Chemical Engineering Science, 204, 2019, 298-309.

[20] Dejam, M., Advective-diffusive-reactive solute transport due to non-Newtonian fluid flows in a fracture surrounded by a tight porous medium, International Journal of Heat and Mass Transfer, 128, 2019, 1307-1321.

[21] Dejam, M., Dispersion in non-Newtonian fluid flows in a conduit with porous Walls, Chemical Engineering Science, 189, 2018, 296-310.

[22] Kou, Z., Dejam, M., Dispersion due to combined pressure-driven and electroosmotic Flows in a channel surrounded by a permeable porous medium, Physics of Fluids, 31(5), 2019, 056603.

[23] Dejam, M., Hydrodynamic dispersion due to a variety of flow velocity profiles in a porous-walled microfluidic channel, International Journal of Heat and Mass Transfer, 136, 2019, 87-98.

[24] Dejam, M., Hassanzadeh, H., Chen, Z., Shear dispersion in a rough-walled Fracture, Society of Petroleum Engineers Journal, 23, 2018, 1669-1688.

[25] Dejam, M., Hassanzadeh, H., Chen, Z., A reduced-order model for chemical species transport in a tube with a constant wall concentration, The Canadian Journal of Chemical Engineering, 96(1), 2018, 307-316.

[26] Al-Muslimawi, A.H., Numerical analysis of partial differential equations for viscoelastic and free surface flows, Ph.D. Thesis, Department of Mathematics, Swansea University, 2013.

[27] Davies, A.J., The finite element method: An introduction with partial differential equations, OUP Oxford, 2011.

[28] Al-Muslimawi, A., Tamaddon-Jahromi, H.R., Webster, M.F., Numerical simulation of tube-tooling cable-coating with polymer melts, Korea-Australia Rheology Journal, 25(4), 2013, 197-216.

[29] López-Aguilar, J.E., Webster, M.F., Al-Muslimawi, A.H., Tamaddon-Jahromi, H.R., Williams, R., Hawkins, K., Askill, C., Ch’ng, C. L., Davies, G., Ebden, P., Lewis, K., A computational extensional rheology study of two biofluid systems, Rheologica Acta, 54(4), 2015, 287-305.

[30] Al-Muslimawi, A.H., Numerical study for differential constitutive equations with polymer melts by using a hybrid finite-element/volume method, Journal of Computational and Applied Mathematics, 308, 2016, 488–498.

[31] Al-Muslimawi, A.H., Taylor Galerkin Pressure Correction (TGPC) Finite Element Method for Incompressible Newtonian Cable-Coating Flows, Journal of Kufa for Mathematics and Computer, 5(2), 2018, 13-21.

[32] Anderson, D.A., Tannehill, J.C., Pletcher, R.H, Computational Fluid Dynamics and Heat Transfer, Washington DC: Taylor and Francis, 1798.

[33] Coelho, P.M., Pinho, F.T., Vortex shedding in cylinder flow of shear thinning fluids, I. Identification and demarcation of flow regime, Journal of Non-Newtonian Fluid Mechanics, 110(2-3), 2003, 143–176.

[34] Coelho, P.M., Pinho, F.T., Vortex shedding in cylinder flow of shear thinning fluids. III. Pressure measurements, Journal of Non-Newtonian Fluid Mechanics, 121(1), 2004, 55–68.

[35] Sivakumar, P., Bharti, R.P., Chhabra, R.P., Effect of power-law index on critical parameters for power-law flow across an unconfined circular cylinder, Chemical Engineering Science, 61(18), 2006, 6035-6046.