×

Investigation of dual solutions in flow of a non-Newtonian fluid with homogeneous-heterogeneous reactions: critical points. (English) Zbl 1408.76014

Summary: In the current letter, we present a numerical study to review the impacts of homogeneous-heterogeneous reactions on the stagnation point flow of Carreau fluid. In addition, an investigation is considered for the flow impelled by a shrinking sheet along with uniform suction on the wall. We explored the prototype model of homogeneous-heterogeneous reactions in which the diffusion coefficients of reactant and catalyst are identical. With the aid of non-dimensional variables, we get a non-linear system of differential equations which is numerically integrated using the MATLAB builtin routine bvp4c. The flow and concentration are exceptionally impacted by the pertinent parameters, like, the Weissenberg number, shrinking parameter, mass transfer parameter, homogeneous/heterogeneous reactions parameter and Schmidt number. Likewise, we inspected that dual solutions for the velocity and concentration fields exist in the case of a shrinking sheet and for a fixed range of other parameters. Our review indicates that the momentum boundary layer thickness rises significantly with an increase in the shrinking parameter for the second solution. Besides, the strength of homogeneous reaction is extremely useful to reduce the concentration of reaction. Under some special assumptions, the consequences of the present study demonstrate a splendid relationship with prior works.

MSC:

76A05 Non-Newtonian fluids
80A20 Heat and mass transfer, heat flow (MSC2010)

Software:

Matlab; bvp4c
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wang, C. Y., Liquid film on an unsteady stretching sheet, Quart. Appl. Math., 48, 601-610, (1990) · Zbl 0714.76036
[2] Miklavcic, M.; Wang, C. Y., Viscous flow due to a shrinking sheet, Quart. Appl. Math., 64, 2, 283-290, (2006) · Zbl 1169.76018
[3] Wang, C. Y., Stagnation flow towards a shrinking sheet, Int. J. Non-Linear Mech., 43, 5, 377-382, (2008)
[4] Fang, T., Boundary layer flow over a shrinking sheet with power-law velocity, Int. J. Heat Mass Transfer, 51, 5838-5843, (2008) · Zbl 1157.76010
[5] Fang, T.; Zhang, J., Thermal boundary layer over a shrinking sheet: an analytical solution, Acta Mech., 209, 325-343, (2010) · Zbl 1381.76056
[6] Nadeem, S.; Haq, R. U.; Lee, C., MHD boundary layer flow over an unsteady shrinking sheet: analytical and numerical approach, J. Braz. Soc. Mech. Sci. Eng., 37, 1339-1346, (2015)
[7] Williams, W. W.; Zhao, J.; Schmidt, L. D., Ignition and extinction of surface and homogeneous oxidation of and, Am. Inst. Chem. Eng. (AIChE), 37, 641-649, (1991)
[8] Merkin, J. H., A model for isothermal homogeneous-heterogenous reactions in boundary layer flow, Math. Comput. Modelling, 24, 125-136, (1996) · Zbl 0884.76090
[9] Chaudhary, M. A.; Merkin, J. H., A simple isothermal model for homogeneous-heterogeneous reactions in boundary layer flow: I. equal diffusivities, Fluid Dyn. Res., 16, 311-333, (1995) · Zbl 1052.80505
[10] Bachok, N.; Ishak, A.; Pop, I., On the stagnation-point flow towards a stretching sheet with homogeneous-heterogeneous reactions effects, Commun. Nonlinear Sci. Numer. Simul., 16, (2011), 4396-4302 · Zbl 1301.76079
[11] Khan, W. A.; Pop, I., Effects of homogeneous-heterogeneous reactions on the viscoelastic fluid towards a stretching sheet, ASME J. Heat Transf., 134, 1-5, (2012)
[12] Khan, W. A.; Pop, I., Flow near the two-dimensional stagnation-point on an infinite permeable wall with a homogeneous-heterogeneous reaction, Commun. Nonlinear Sci. Numer. Simul., 15, 3435-3443, (2010)
[13] Hayat, T.; Farooq, M.; Alsaedi, A., Homogeneous-heterogeneous reactions in the stagnation point flow of carbon nanotubes with Newtonian heating, AIP Adv., 5, 027130, (2015)
[14] Hashim; Khan, M., On Cattaneo-Christov heat flux model for carreau fluid flow over a slendering sheet, Results Phys., 7, 310, (2017)
[15] Carreau, P. J., Rheological equations from molecular network theories, Trans. Soc. Rheol., 116, 99-127, (1972)
[16] Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager, O., Dynamics of Polymeric Liquids, (1987), Wiley New York
[17] Shadid, J. N.; Eckert, E. R.G., Viscous heating of a cylinder with finite length by a high viscosity fluid in steady longitudinal flow. II. non-Newtonian carreau model fluids, Int. J. Heat Mass Transfer, 35, 27, 39-49, (1992) · Zbl 0825.76040
[18] Khellaf, K.; Lauriat, G., Numerical study of heat transfer in a non-Newtonian carreau-fluid between rotating concentric vertical cylinders, J. Non-Newton. Fluid Mech., 89, 45-61, (2000) · Zbl 0963.76008
[19] Raju, C. S.K.; Sandeep, N., Falkner-Skan flow of a magnetic-carreau fluid past a wedge in the presence of cross diffusion effects, Eur. Phys. J. Plus, 131, 267, (2016)
[20] Khan, M.; Hashim, Boundary layer flow and heat transfer to carreau fluid over a nonlinear stretching sheet, AIP Adv., 5, 10723, (2015)
[21] Khan, M.; Hashim; Alshomrani, A. S., MHD stagnation-point flow of a carreau fluid and heat transfer in the presence of convective boundary conditions, PLoS One, 11, 6, (2016)
[22] Labroopulu, F.; Dorrepael, J. M.; Chandna, O. P., Oblique flow impinging on a wall with suction or blowing, Acta Mech., 115, 15-25, (1996) · Zbl 0859.76016
[23] Attia, A. H., The effect of suction and injection on unsteady Couette flow with variable properties, Kragujevac J. Sci., 32, 17-24, (2010)
[24] Ahmed, S.; Khatun, H., Magnetohydrodynamic oscillatory flow in a planer porous channel with suction and injection, Int. J. Eng. Technol., 11, 1024-1029, (2013)
[25] Rundora, L.; Makinde, O. D., Effect of suction/injection on unsteady reactive variable viscosity non-Newtonian fluid flow in a channel filled with porous medium and convective boundary conditions, J. Petrol Sci. Eng., 108, 328-335, (2013)
[26] Makinde, O. D.; Chinyoka, T., Analysis of unsteady flow of a variable viscosity reactive fluid in a slip with wall suction or injection, J. Petrol Sci. Eng., 94-95, 1-11, (2013)
[27] Merkin, J. H., On dual solutions occurring in mixed convection in a porous medium, J. Eng. Math., 20, 171-179, (1985) · Zbl 0597.76081
[28] Weidman, P. D.; Kubitschek, D. G.; Davis, A. M.J., The effect of transpiration on self-similar boundary layer flow over moving surfaces, Internat. J. Engrg. Sci., 44, 730-737, (2006) · Zbl 1213.76064
[29] Bhattacharyya, K., Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet, Int. Commu. Heat Mass Transf., 38, 917-922, (2011)
[30] Katagiri, M., Magnetohydrodynamic flow with suction or injection at the forward stagnation-point, J. Phys. Soc. Japan, 27, 1677-1685, (1969)
[31] Lok, Y. Y.; Amin, N.; Pop, I., Unsteady boundary layer flow of a micropolar fluid near a stagnation point with uniform suction or injection, J. Teknologi. (University Teknologi Malaysia), 46, 15-32, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.