## THERMAL SCIENCE

International Scientific Journal

### A NUMERICAL SCHEME FOR DARCY-FORCHHEIMER FLOW OF NON-NEWTONIAN NANOFLUID UNDER THE EFFECTS OF CONVECTIVE AND ZERO MASS FLUX BOUNDARY CONDITIONS

**ABSTRACT**

This research aims to propose a numerical scheme for solving boundary value problems. It is a two-stage, third-order accurate scheme known as a predictorcorrector scheme. The two main results are finding the region of the scheme where it is stable and determining the stability criterion for a set of linearized first-order differential equations. In addition, a mathematical model for heat and mass transfer of Darcy-Forchheimer flow of non-Newtonian nanofluid over the sheet is presented. The similarity transformations reduce PDE into a system of ODE for easier manipulation. The results are compared with the past research and those obtained by MATLAB SOLVER BVP4C. The results show that the velocity profile slightly decays by enhancing the Weisenberg number.

**KEYWORDS**

PAPER SUBMITTED: 2022-12-30

PAPER REVISED: 2023-03-02

PAPER ACCEPTED: 2023-03-18

PUBLISHED ONLINE: 2023-04-22

**THERMAL SCIENCE** YEAR

**2023**, VOLUME

**27**, ISSUE

**Issue 6**, PAGES [4581 - 4595]

- Peralta, J. M., et al., Analytical Solutions for the Free-Draining Flow of a Carreau-Yasuda Fluid on a Vertical Plate, Chem. Eng. Sci., 16831 (2017), Aug., pp. 391-402
- Khan, Z., et al., Effect of Thermal Radiation and Chemical Reaction on Non-Newtonian Fluid through a Vertically Stretching Porous Plate with Uniform Suction, Res. Phys., 9 (2018), June, pp. 1086-1095
- Khani, F., et al., Analytical Investigation for Cooling Turbine Disks with a Non-Newtonian Viscoelastic Fluid, Comput. Mathe. Appl., 61 (2011), 7, pp. 1728-1738
- Mahmood, R., et al., A Comprehensive Finite Element Examination of Carreau Yasuda Fluid Model in a Lid Driven Cavity and Channel with Obstacle by Way of Kinetic Energy and Drag and Lift Coefficient Measurements, J. Mater. Res. Technol., 9 (2020), 2, pp. 1785-1800
- Khan, M. I., et al., Theoretical and Numerical Investigation of Carreau-Yasuda Fluid-flow Subject to Soret and Dufour Effects, Comput. Methods Programs Biomed., 186 (2020), 105145
- Kefayati, G. R., Tang, H., Three-Dimensional Lattice Boltzmann Simulation on Thermosolutal Convection and Entropy Generation of Carreau-Yasuda Fluids, Int. J. Heat Mass Transf., 131 (2019), Mar., pp. 346-364
- Khan, M. I., et al., Activation Energy for the Carreau-Yasuda Nanomaterial Flow: Analysis of the Entropy Generation Over a Porous Medium, J. Mol. Liq., 297 (2020), 111905
- Tallmadge, J. A., Gutfinger, C., Entrainment of liquid films. Drainage, withdrawal, and Removal, Ind. Eng. Chem., 59 (1967), 11, pp. 18-34
- Sherwood, J. D., Optimal Shapes for Best Draining, Phys. Fluids, 21 (2009), 11, 113102
- Ungarish, M., Sherwood, J. D., Draining of a thin Film on the Wall of a Conical Container Set into Rapid Rotation about its Vertical Axis, Phys. Fluids, 24 (2012), 2, 023602
- Ali, A., et al., Self-Drainage of Viscous Liquids in Vertical and Inclined Pipes, Food Bioprod. Process. 99 (2016), July, pp. 38-50
- Schunk, P. R., et al., Free-Meniscus Coating Processes, in: Liquid Film Coating. Scientific Principles and their Technological Implications, (Kistler, S. F., Schweizer, P. M. Eds.), Springer Netherlands, Dordrecht, 1997, pp. 673-708
- Weinstein, S. J., Palmer, H. J., Capillary Hydrodynamics and Interfacial Phenomena, in: Liquid Film Coating Scientific Principles and Their Technological Implications, (Kistler, S. F., Schweizer, P. M. Eds.), Springer Netherlands, Dordrecht, 1997, pp. 19-62
- Sochi, T., Analytical Solutions for the Flow of Carreau and Cross Fluids in Circular Pipes and Thin Slits, Rheol. Acta, 54 (2015), 8, pp. 745-756
- Eley, R. R., Schwartz, L. W., Interaction of Rheology, Geometry, and Process in Coating Flow, J. Coat. Technol., 74 (2002), 9, pp. 43-53
- Carreau, P. J., Rheological Equations from Molecular Network Theories, J. Rheol. (Melville, NY, U. S.) 16 (1972), 1, 99
- Yasuda, K., Investigation of the Analogies Between Isometric and Linear Viscoelastic Properties of Polystyrene Fluids. Ph. D., Massachusetts Institute of Technology. Dept. of Chemical Engineering., Boston, Mass., USA, 1979
- Surana, K. S., Advanced Mechanics of Continua, Applied and Computational Mechanics Series, CRC Press, Boca Raton, Fla., USA, 2014
- Genovese, D. B., Shear Rheology of Hard-Sphere, Dispersed, and Aggregated Suspensions, and Filler- Matrix Composites, Adv. Colloid Interface Sci., 171-172 (2012), Mar.-Apr., pp. 1-16
- Spikes, H., Jie, Z., History, Origins and Prediction of Elastohydrodynamic Friction, Tribol. Lett., 56 (2014), 1, pp. 1-25
- Brummer, R., Rheology Essentials of Cosmetic and Food Emulsions, Springer, Berlin; London, 2006
- Hayat, T., et al., Hall and Ohmic Heating Effects On the Peristaltic Transport of Carreau-Yasuda Fluid in an Asymmetric Channel, Z. Naturforschung, 69 (2014), 1-2, pp. 43-51
- Abbasi, F. M., et al., Numerical Analysis for MHD Peristaltic Transportof Carreau-Yasuda Fluid in a Curved Channel with Hall Effects, J. Magn. Magn. Mater., 382 (2015), May, pp. 104-110
- Peralta, J. M., et al., Analytical Solutions for the Free-Draining Flow of a Carreau-Yasuda Fluid on a Vertical Plate, Chemical Engineering Science, 168 (2017), Aug., pp. 391-402
- Rehman, M. I. U., et al., Theoretical Investigation of Darcy-Forchheimer Flow of Bioconvection Casson Fluid in the Presence of Chemical Reaction Effect, On-line first, Biomass Conv. Bioref., doi.org/10.1007/s13399-022-03060-5, 2022
- Jonnalagadda, J. M., et al., Existence and Stability of Solutions for Nonlinear Impulsive Nabla Fractional Boundary Value Problems of order Less Than One, Discontinuity, Non-Linearity, and Complexity, 12 (2023), 2, pp. 231-244
- Alzabut, J., et al., Second Order Iterative Dynamic Boundary Value Problems with Mixed Derivative Operators with Applications, Qual. Theory Dyn. Syst. 22 (2023), Jan., 32
- Ijaz Khan, M., et al., Estimation of Entropy Optimization in Darcy Forchheimer flow of Carreau-Yasuda fluid (Non-Newtonian) with First Order Velocity Slip, Alexandria Engineering Journal, 59 (2020), 5, pp. 3953-3962
- Hayat, T., et al., Heat and Mass Transfer for Soret and Dufour's Effect On Mixed Convection Boundary Layer Flow Over a Stretching Vertical Surface in a Porous Medium Filled with a Viscoelastic Fluid, Commun Non-Linear Sci Numer Simulat, 15 (2010), 5, pp. 1183-1196
- Yih, K. A., Free Convection Effect on MHD Coupled Heat and Mass Transfer of a Moving Permeable Vertical Surface, Int. Commun Heat Mass Transfer, 26 (1999), 1, pp. 95-104
- Pasha, S. A., et al., The Modified Homotopy Perturbation Method with an Auxiliary Term for the Non- Linear Oscillator with Discontinuity, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 3-4, pp. 1363-1373
- Arif, M. S., et al., A Stochastic Numerical Analysis for Computer Virus Model with Vertical Transmission Over the Internet, Computers, Materials & Continua, 61 (2019), 3, pp. 1025-1043
- Shatanawi, W., et al., Design of Non-Standard Computational Method for Stochastic Susceptible-Infected- Treated-Recovered Dynamics of Coronavirus Model, Advances in Difference Equations, 2020 (2020), 505, pp. 1-15
- Bibi, M., et al., A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid-flow Over a Linearly Stretching Surface, Computers, Materials & Continua, 62 (2020), 2, pp. 657-677
- Nawaz, Y., Arif, M., A New Class of A-Stable Numerical Techniques for Odes: Application to Boundary Layer Flow, Thermal Science, 25 (2020), 3A, pp. 1665-1675