# Analysis of non-Newtonian fluid with phase flow model

Document Type : Review Article

Authors

1 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

2 - Mathematics and Its Applications in Life Sciences Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam. - Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

3 Al-Farabi Kazakh National University, Faculty of Mechanics and Mathematics, av. al-Farabi 71, Almaty, Kazakhstan

4 Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia

5 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

Abstract

We considered a stagnation point of Non-Newtonian Fluid with Phase Flow Model over a stretching surface with slip conditions. Two types of the nanoparticle used, namely Cu and 〖Al〗_2 O_3 with base fluid H_2 O. Acceptable to theoretical study, the mathematical model has been constructed through flow assumptions. Partial differential equations are made by applying the boundary layer approximations on the momentum and energy equations. The suitable similarity transformations are applied to the partial equations which are converted into ordinary differential equations. These equations are solved by numerical scheme, namely BVP4C method. The involving physical parameters effect is shown by graphs and tables. Our work shows a good agreement with the decay literature. The expressions F''(0) and -θ'(0) achieve fewer values by hybrid nanofluid than that of nanofluid. Moreover F''(0) and -θ'(0) increase for large values of the dimensionless parameter (N) where as F'(ξ) and θ(ξ).increase for large values of Φ_2.

Keywords

#### References

1. References

1. Rivlin, R. S., & Ericksen, J. L. Stress-deformation relations for isotropic materials. In Collected Papers of RS Rivlin (pp. 911-1013). Springer, New York, NY (1997).
2. Labropulu, F. A few more exact solutions of a second grade fluid via inverse method. Mechanics Research Communications, 27(6), 713-720 (2000).
3. Yürüsoy, M., Pakdemirli, M., & Noyan, Ö. F. Lie group analysis of creeping flow of a second grade fluid. International Journal of Non-Linear Mechanics, 36(6), 955-960 (2001).
4. Shkoller, S. Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations. Applied Mathematics Letters, 14(5), 539-543 (2001).
5. Labropulu, F. D'Alembert motions for non-Newtonian second grade fluid. International journal of non-linear mechanics, 38(7), 1027-1036 (2003).
6. Nadeem, S., Hussain, A., Malik, M. Y., & Hayat, T. Series solutions for the stagnation flow of a second-grade fluid over a shrinking sheet. Applied Mathematics and Mechanics, 30(10), 1255 (2009).
7. Mehmood, R., Nadeem, S., & Akbar, N. S. Non-orthogonal stagnation point flow of a micropolar second grade fluid towards a stretching surface with heat transfer. Journal of the Taiwan Institute of Chemical Engineers, 44(4), 586-595 (2013).
8. Akinbobola, T. E., & Okoya, S. S. The flow of second grade fluid over a stretching sheet with variable thermal conductivity and viscosity in the presence of heat source/sink. Journal of the Nigerian Mathematical Society, 34(3), 331-342 (2015).
9. Majeed, A., Javed, T., & Ghaffari, A. Numerical investigation on flow of second grade fluid due to stretching cylinder with Soret and Dufour effects. Journal of Molecular Liquids, 221, 878-884 (2016).
10. Khan, M., ur Rahman, M., & Manzur, M. Axisymmetric flow and heat transfer to modified second grade fluid over a radially stretching sheet. Results in physics, 7, 878-889 (2017).
11. Ghadikolaei, S. S., Hosseinzadeh, K., Yassari, M., Sadeghi, H., & Ganji, D. D. Analytical and numerical solution of non-Newtonian second-grade fluid flow on a stretching sheet. Thermal Science and Engineering Progress, 5, 309-316 (2018).
12. Alamri, S. Z., Khan, A. A., Azeez, M., & Ellahi, R.. Effects of mass transfer on MHD second grade fluid towards stretching cylinder: A novel perspective of Cattaneo–Christov heat flux model. Physics Letters A, 383(2-3), 276-281 (2019).
13. Bilal, S., Mustafa, Z., Rehman, K. U., & Malik, M. Y. MHD Second Grade NanoFluid Flow Induced by a Rotatory Cone. Journal of Nanofluids, 8(4), 876-884 (2019).
14. Elkoumy, S. R., Barakat, E. I., & Abdelsalam, S. I. Hall and transverse magnetic field effects on peristaltic flow of a Maxwell fluid through a porous medium. Global J. Pure Appl. Math9(2), 187-203 (2013).
15. Mekheimer, K. S., Hasona, W. M., El-Shekhipy, A. A., & Zaher, A. Z. Electrokinetics of Dielectric Non-Newtonian Bio Fluids with Heat Transfer Through a Flexible Channel: Numerical Study. Computational Methods in Science and Technology23(4), 331-341 (2017).
16. Mekheimer, K. S., Hasona, W. M., Abo-Elkhair, R. E., & Zaher, A. Z. Peristaltic blood flow with gold nanoparticles as a third grade nanofluid in catheter: Application of cancer therapy. Physics Letters A382(2-3), 85-93 (2018).
17. Abdelsalam, S. I., Bhatti, M. M., Zeeshan, A., Riaz, A., & Beg, O. A. Metachronal propulsion of electrically-conducting viscoelastic particle-fluid suspension in a ciliated channel under transverse magnetic field: mathematical modelling. Physica Scripta94, 115301-115314 (2019).
18. Eldesoky, I. M., Abdelsalam, S. I., El-Askary, W. A., & Ahmed, M. M. The Integrated Thermal Effect in Conjunction with Slip Conditions on Peristaltically Induced Particle-Fluid Transport in a Catheterized Pipe. Journal of Porous Media23(7) (2020).
19. Crane, L. J. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21(4), 645-647 (1970).
20. Mekheimer, K. S., Salem, A. M., & Zaher, A. Z. Peristaltically induced flow due to a surface acoustic wavy moving wall. Chinese Journal of Physics51(5), 968-982 (2013).
21. Mekheimer, K. S., Salem, A. M., & Zaher, A. Z. Peristatcally induced MHD slip flow in a porous medium due to a surface acoustic wavy wall. Journal of the Egyptian Mathematical Society22(1), 143-151 (2014).
22. Malvandi, A., Hedayati, F., & Ganji, D. D. Nanofluid flow on the stagnation point of a permeable non-linearly stretching/shrinking sheet. Alexandria Engineering Journal (2017).
23. Abdelsalam, S. I., & Mekheimer, K. S. Couple stress fluid flow in a rotating channel with peristalsis. Journal of Hydrodynamics30(2), 307-316 (2018).
24. Abdelsalam, S. I., & Bhatti, M. M. The study of non-Newtonian nanofluid with hall and ion slip effects on peristaltically induced motion in a non-uniform channel. RSC advances8(15), 7904-7915 (2018).
25. Mekheimer, K. S., Zaher, A. Z., & Hasona, W. M. Entropy of AC electro-kinetics for blood mediated gold or copper nanoparticles as a drug agent for thermotherapy of oncology. Chinese Journal of Physics (2020).
26. Bhatti, M. M., Marin, M., Zeeshan, A., Ellahi, R., & Abdelsalam, S. I. Swimming of Motile Gyrotactic Microorganisms and Nanoparticles in Blood Flow Through Anisotropically Tapered Arteries. Frontiers in Physics8, 95 (2020).
27. Khan, M., Salahuddin, T., Tanveer, A., Malik, M. Y., & Hussain, A. Change in internal energy of thermal diffusion stagnation point Maxwell nanofluid flow along with solar radiation and thermal conductivity. Chinese Journal of Chemical Engineering. (2019).
28. Nadeem, S., Ahmed, Z., & Saleem, S. Carbon nanotubes effects in magneto nanofluid flow over a curved stretching surface with variable viscosity. Microsystem Technologies, 1-8 (2020).
29. Choi, U. S. Enhancing thermal conductivity of fluids with nanoparticles, Development and Applications of Non-Newtonian flows edited by Siginer, DA and Wang, HP, EFD-Vol. 231/MD-Vol. 66. ASME (1995).
30. Mahian, O., Kianifar, A., Kalogirou, S. A., Pop, I., & Wongwises, S. A review of the applications of nanofluids in solar energy. International Journal of Heat and Mass Transfer, 57(2), 582-594 (2013).
31. Turkyilmazoglu, M. Nanofluid flow and heat transfer due to a rotating disk. Computers & Fluids, 94, 139-146 (2014).
32. Abbas, N., Saleem, S., Nadeem, S., Alderremy, A. A., & Khan, A. U. On stagnation point flow of a micro polar nanofluid past a circular cylinder with velocity and thermal slip. Results in Physics, 9, 1224-1232 (2018).
33. Abdelsalam, S. I., & Bhatti, M. M. The impact of impinging TiO2 nanoparticles in Prandtl nanofluid along with endoscopic and variable magnetic field effects on peristaltic blood flow. Multidiscipline Modeling in Materials and Structures (2018).
34. Abdelsalam, S. I., & Bhatti, M. M. The impact of impinging TiO2 nanoparticles in Prandtl nanofluid along with endoscopic and variable magnetic field effects on peristaltic blood flow. Multidiscipline Modeling in Materials and Structures (2018).
35. Eldesoky, I., Abdelsalam, S., El-Askary, W., El-Refaey, A., & Ahmed, M. Joint effect of thermal energy and magnetic field on particulate fluid suspension in a catheterized tube. Bionanoscience9(3), 723-739 (2019).
36. Abdelsalam, S. I., & Bhatti, M. M. New insight into AuNP applications in tumour treatment and cosmetics through wavy annuli at the nanoscale. Scientific reports9(1), 1-14 (2019).
37. Bhatti, M. M., Zeeshan, A., Ellahi, R., Bég, O. A., & Kadir, A. Effects of coagulation on the two-phase peristaltic pumping of magnetized Prandtl biofluid through an endoscopic annular geometry containing a porous medium. Chinese Journal of Physics58, 222-234 (2019).
38. Abd Elmaboud, Y., & Abdelsalam, S. I. (2019). DC/AC magnetohydrodynamic-micropump of a generalized Burger's fluid in an annulus. Physica Scripta94(11), 115209.
39. Abd Elmaboud, Y., Abdelsalam, S. I., Mekheimer, K. S., & Vafai, K. Electromagnetic flow for two-layer immiscible fluids. Engineering Science and Technology, an International Journal22(1), 237-248 (2019).
40. Sohail, M., Naz, R., & Abdelsalam, S. I. Application of non-Fourier double diffusions theories to the boundary-layer flow of a yield stress exhibiting fluid model. Physica A: Statistical Mechanics and its Applications537, 122753 (2020).
41. Abumandour, R. M., Eldesoky, I. M., Kamel, M. H., Ahmed, M. M., & Abdelsalam, S. I. Peristaltic thrusting of a thermal-viscosity nanofluid through a resilient vertical pipe. Zeitschrift für Naturforschung A75(8), 727-738 (2020).
42. Momin, G. G. Experimental investigation of mixed convection with water-Al2O3 & hybrid nanofluid in inclined tube for laminar flow. Int. J. Sci. Technol. Res, 2, 195-202 (2013).
43. Suresh, S., Venkitaraj, K. P., Selvakumar, P., & Chandrasekar, M. Synthesis of Al2O3–Cu/water hybrid nanofluids using two step method and its thermo physical properties. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 388(1-3), 41-48 (2011).
44. Nadeem, S., Abbas, N., & Khan, A. U. Characteristics of three dimensional stagnation point flow of Hybrid nanofluid past a circular cylinder. Results in physics, 8, 829-835 (2018).
45. Nadeem, S., & Abbas, N. On both MHD and slip effect in Micropolar Hybrid nanofluid past a circular cylinder under stagnation point region. Canadian Journal of Physics, (ja) (2018).

45   Eldesoky, I. M., Abdelsalam, S. I., El-Askary, W. A., & Ahmed, M. M. Concurrent development of thermal energy with magnetic field on a particle-fluid suspension through a porous conduit. BioNanoScience9(1), 186-202 (2019).

46   Abdelsalam, S. I., & Bhatti, M. M. Anomalous reactivity of thermo-bioconvective nanofluid towards oxytactic microorganisms. Applied Mathematics and Mechanics41(5), 711-724 (2020).

47   Abdelsalam, S. I., & Sohail, M. Numerical approach of variable thermophysical features of dissipated viscous nanofluid comprising gyrotactic micro-organisms. Pramana: Journal of Physics94(1) (2020).

48   Abdelsalam, S. I., Mekheimer, K. S., & Zaher, A. Z. Alterations in blood stream by electroosmotic forces of hybrid nanofluid through diseased artery: Aneurysmal/stenosed segment. Chinese Journal of Physics (2020).

49   Sadaf, H., & Abdelsalam, S. I. Adverse effects of a hybrid nanofluid in a wavy non-uniform annulus with convective boundary conditions. RSC Advances10(26), 15035-15043 (2020).

50   Sohail, M., Naz, R., & Abdelsalam, S. I. On the onset of entropy generation for a nanofluid with thermal radiation and gyrotactic microorganisms through 3D flows. Physica Scripta95(4), 045206 (2020).

51   Abdelsalam, S. I., Mekheimer, K. S., & Zaher, A. Z. Alterations in blood stream by electroosmotic forces of hybrid nanofluid through diseased artery: Aneurysmal/stenosed segment. Chinese Journal of Physics (2020).

52   Ariel, P. D. On extra boundary condition in the stagnation point flow of a second grade fluid. International journal of engineering science, 40(2), 145-162 (2002).

53   Wang, C. Y. Stagnation flow towards a shrinking sheet. International Journal of Non-Linear Mechanics, 43(5), 377-382 (2008).

54   Bachok, N., Ishak, A., & Pop, I. Stagnation-point flow over a stretching/shrinking sheet in a nanofluid. Nanoscale Research Letters, 6(1), 623 (2011).

55   Malvandi, A., Hedayati, F., & Ganji, D. D. Nanofluid flow on the stagnation point of a permeable non-linearly stretching/shrinking sheet. Alexandria Engineering Journal (2017).

56   Yacob, N. A., Ishak, A., & Pop, I. Falkner–Skan problem for a static or moving wedge in nanofluids. International Journal of Thermal Sciences, 50(2), 133-139 (2011).