Erscheint 12 Ausgaben pro Jahr
ISSN Druckformat: 1091-028X
ISSN Online: 1934-0508
Indexed in
SIMILARITY SOLUTIONS OF THE UNSTEADY BOUNDARY LAYER FLOW PAST A PERMEABLE WEDGE EMBEDDED IN A POROUS MEDIUM
ABSTRAKT
We investigate the unsteady two-dimensional laminar boundary layer flow of a viscous and incompressible fluid over a constant and permeable wedge inserted into a porous medium. The outer freestream velocity is assumed to be proportional to a power of distance along the wedge surface, i.e., xm, where x is the distance from the leading edge and m is a constant. The model is described by the unsteady Falkner-Skan equation and solved analytically when the unsteady parameter equals 2 and otherwise numerically using the Keller-box method, for the wall shear stresses and mean velocity profiles. The system is also solved asymptotically far away from the wedge surface to compliment the numerical results, and asymptotic solutions produce oscillatory-type velocity profiles. Results show that the flow region is divided into near- and far-field regions. The effects of suction are to reduce the horizontal flow velocity near the viscous region, whereas, for the case of injection, these can extend far away from the wedge surface. In addition, our results show that boundary layer thickness decreases for an accelerated flow; whereas, there is a boundary layer separation for strong decelerated flow. The dynamics behind these results are discussed.
-
Abramowitz, M. and Stegun, I., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Ed., Mineola, NY: Dover, 1970.
-
Cebeci, T. and Bradshaw, P., Momentum Transfer in Boundary-Layers, New York: Mc-Graw Hill, 1977.
-
Chang, W.D., TheNonuniqueness oftheFlowof a Viscoelastic Fluid over a Stretching Sheet, Q. Appl. Math., vol. 47, pp. 365-366, 1989.
-
Cheng, P., Combined Free and Forced Convection Flow about Inclined Surfaces in Porous Media, Int. J. Heat Mass Transf., vol. 20, pp. 807-814, 1977.
-
Chinyoka, T. and Makinde, O.D., Analysis of Non-Newtonian Flow with Reacting Species in a Channel Filled with a Saturated Porous Medium, J. Petr. Sci. Eng., vol. 121, pp. 1-8,2014.
-
Civan, F. and Tiab, D., Effect of External Boundary Conditions on Steady and Semi-Steady Radial Flow Equations based on Darcy, Forchheimer, Brinkman and Capillary-Orifice Models, Proc. of SPE Annual Tech. Conf., Dallas, SPE 22923, October 6-9, 1991.
-
Craven, A.H. and Peletier, L.A., Reverse Flow Solutions of Falkner-Skan Equation for A > 1, Mathematika, vol. 19, no. 1, pp. 135-138, 1972.
-
Das, S., Guchhaiy, S.K., Jana, R.N., and Makinde, O.D., Hall Effects on an Unsteady Magneto-Convection and Radiative Heat Transfer past a Porous Plate, Alexandria Eng. J., vol. 55, pp. 1321-1331, 2016.
-
Dhanak, M.R. and Duck, P.W., The Effects of Free Stream Pressure Gradient on a Corner Boundary-Layer, Proc. R. Soc. Lond. A, vol. 453, pp. 1793-1815,1997.
-
Duck, P.W., Stow, S.R., and Dhanak, M.R., Boundary-Layer Flow along a Ridge: Alternatives to the Falkner-Skan Solutions, Phil. Trans. R. Soc. Lond. A, vol. 358, pp. 3075-3090, 2000.
-
Fang, T., Flow and Mass Transfer for an Unsteady Stagnation-Point Flow over a Moving Wall Considering Blowing Effects, ASME: J. Fluids Eng., vol. 136, no. 7, p. 071103,2014.
-
Firoozabadi, A. and Katz, D.L., An Analysis of High-Velocity Gas Flow through Porous Media, J. Petroleum. Tech., vol. 32, no. 2, pp. 211-216,1979.
-
Guedda, M., Aly, E.H., and Ouahsine, A., Analytical and ChPDM Analysis of MHD Mixed Convection over a Vertical at Plate Embedded in a Porous Medium Filled with Water at 4 C, Appl. Math. Model., vol. 35, pp. 5182-5197,2011.
-
Keller, H.B., Numerical Methods in Boundary-Layer Theory, Ann. Rev. FluidMech., pp. 417-428, 1978.
-
Khan, W.A., Culham, R., and Makinde, O.D., Hydromagnetic Blasius Flow of Power-Law Nanofluids over a Convectively Heated Vertical Plate, Canadian J. Chem. Eng., vol. 93, pp. 1830-1837,2015.
-
Khan, W.A., Jashim Uddin, M., and Ismail, A.I.M., Effects of Melting and Thermal Dispersion on Unsteady Mixed Convection with Heat and Mass Transfer in Non-Darcy Porous Medium, J. Porous Media, vol. 17, no. 3, pp. 211-223, 2014.
-
Klemp, J.B. and Acrivos, A., A Moving-Wall Boundary-Layer with Reverse Flow, J. Fluid Mech., vol. 76, no. 2, pp. 363-381, 1976.
-
Kolomenskiy, D. and Moffatt, H.K., Similarity Solutions for Unsteady Stagnation Point Flow, J. Fluid Mech., vol. 711, pp. 394-410,2012.
-
Kudenatti, R.B., A New Exact Solution for Boundary-Layer Flow over a Stretching Plate, Int. J. Non-Linear Mech., vol. 47, pp. 727-733,2012.
-
Kudenatti, R.B., Kirsur, S.R., Naragund, L.N., and Bujurke, N.M., MHD Boundary-Layer Flow over a Non-Linear Stretching Boundary with Suction and Injection, Int. J. Non-Linear Mech., vol. 50, pp. 58-67, 2013.
-
Kudenatti, R.B., Kirsur, S.R., Nargund, A.L., and Bujurke, N.M., Similarity Solution of the MHD Boundary-Layer Flow past a Constant Wedge within Porous Media, Math. Prob. Eng., vol. 2017, pp. 1-11, 2017.
-
Ludlow, D.K., Clarkson, P.A., and Bassom, A.P., New Similarity Solutions of the Unsteady Incompressible Boundary-Layer Equations, Q. J. Mech. Appl. Math., vol. 53, no. 2, pp. 175-206,2000.
-
Ma, P.K.H. and Hui, W.H., Similarity Solutions of the Two Dimensional Unsteady Boundary-Layer Equations, J. Fluid Mech., vol. 216, pp. 537-559, 1990.
-
Makinde, O.D. and Mishra, S.R., Chemically Reacting MHD Mixed Convection Variable Viscosity Blasius Flow Embedded in Aporous Medium, Defect Diffus. Forum, vol. 374, pp. 83-91, 2017.
-
Mehmood, A., Beg, A.O., and Ali, A., Suction and Blowing Effects on Unsteady Flow and Heat Transfer through Porous Media with Variable Viscosity, J. Porous Media, vol. 15, no. 3, pp. 293-302, 2012.
-
Merkin, J.H., Mixed Convection Boundary-Layer Flow on a Vertical Surface in a Saturated Porous Medium, J. Eng. Math., vol. 14, pp. 301-313, 1980.
-
Merkin, J.H., Mixed Convection in a Falkner-Skan System, J. Eng. Math., vol. 100,no. 1,pp. 167-185,2016.
-
Nield, D.A., Modelling Fluid Flow and Heat Transfer in a Saturated Porous Medium, J. Appl. Math. Dec. Sci., vol. 4, no. 2, pp. 165-173,2000.
-
Nield, D.A. and Bejan, A., Convection Porous Media, 4th Ed., New York: Springer, 2013.
-
Oskam, B. and Veldman, A.E.P., Branching of the Falkner-Skan Solutions for A < 0, J. Eng. Math., vol. 16, no. 4, pp. 295-308, 1982.
-
Ma, P.K.H. and Hui, W.H., Similarity Solutions of the Two Dimensional Unsteady Boundary-Layer Equations, J. Fluid Mech., vol. 216, pp. 537-559, 1990.
-
Rajagopal, K.R., Szeri, A.Z., and Troy, W., An Existence Theorem for the Flow of a Non-Newtonian Fluid past an Infinite Porous Plate, Int. J. Non-Linear Mech., vol. 21, no. 4, pp. 279-289, 1986.
-
Riley, N. and Weidman, P.D., Multiple Solutions of the Falkner-Skan Equation for Flow past a Stretching Boundary, SIAMJ. Appl.
-
Sattar, M.A., A Local Similarity Transformation for the Unsteady Two Dimensional Hydrodynamic Boundary-Layer Equations of a Flow past a Wedge, Int. J. Appl. Math. Mech., vol. 7, no. 1,pp. 15-28,2011.
-
Schlichting, H. and Gersten, K., Boundary-Layer Theory, 8th Ed., New York: Springer, 2000.
-
Szeri, A.Z., Fluid Film Lubrication, 2nd Ed., Cambridge, UK: Cambridge University Press, 2010.
-
Vafai, K. and Tien, C.L., Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media, Int. J. Heat Mass Transf., vol. 24, pp. 195-203, 1981.
-
Volchkov, E.P., Makarov, M.S., and Sakhnov, A.Yu., Boundary-Layer with Asymptotic Favourable Pressure Gradient, Int. J. Heat Mass Transf., vol. 53, pp. 2837-2843, 2010.
-
Yuan, S.W., Foundation of Fluid Mechanics, London: Prentice-Hall International Inc., 1970.
-
Kalpana G., Madhura K.R., Kudenatti Ramesh B., Numerical study on the combined effects of Brownian motion and thermophoresis on an unsteady magnetohydrodynamics nanofluid boundary layer flow, Mathematics and Computers in Simulation, 200, 2022. Crossref
-
Kalpana G., Madhura K.R., Kudenatti Ramesh B., Magnetohydrodynamic boundary layer flow of hybrid nanofluid with the thermophoresis and Brownian motion in an irregular channel: A numerical approach, Engineering Science and Technology, an International Journal, 32, 2022. Crossref
-
Bai Yu, Wan Sa, Zhang Yan, Wang Xin, Unsteady Falkner-Skan flow of fractional Maxwell fluid towards a stretched wedge with buoyancy effects, Physica Scripta, 98, 1, 2023. Crossref