Archive
Special Issues
A Combined Analytic and Numerical Procedure for Stagnation-Point Flow
Applied and Computational Mathematics
Volume 6, Issue 2, April 2017, Pages: 75-82
Received: Jan. 8, 2017; Accepted: Feb. 7, 2017; Published: Mar. 14, 2017
Author
Okey Oseloka Onyejekwe, Computational Science Program, Faculty of Natural Science, Addis Ababa University, Arat-Kilo Campus, Addis Ababa, Ethiopia
Article Tools
Abstract
The two dimensional vector form of the Navier-Stokes equation is reduced to a fourth–order equation for the streamfunction. Boundary conditions arise from considerations of the no-slip constraint at the surface as well the interaction of viscous flow with potential-flow at the edge of the boundary layer. By employing a separation of variables technique and introducing certain dimensionless variables, the stream function equation is converted into its dimensionless analog with appropriate boundary conditions. The resulting quasi-linear third-order ordinary differential equation facilitates the numerical computation of the velocity and the pressure terms. This is achieved by solving the nonlinear two-point boundary-value problem with a time-marching method involving a Crank-Nicolson and Newton-linearization schemes until steady-state solution is obtained. The velocity, stream-function and pressure profiles are discussed with reference to various computation parameters and are found to be in good agreement with the physics of the problem. It was also found that there is no penalty in accuracy for a broad range of CFL numbers. However as the CFL number exceeds a certain threshold, the approach to convergence becomes erratic as indicated by the spurious results produced by the solution residuals.
Keywords
Navier-Stokes Equation, Streamfunction, CFL Number, Fourth-Order Equation, Quasi-Linear Third-Order Differential Equation, Crank-Nicolson, Newton-Linearization
Okey Oseloka Onyejekwe, A Combined Analytic and Numerical Procedure for Stagnation-Point Flow, Applied and Computational Mathematics. Vol. 6, No. 2, 2017, pp. 75-82. doi: 10.11648/j.acm.20170602.12
References
[1]
Agbaje, T. M., Mondal, S., Makukula, Z. G., Motsa, S. S. Sibanda, P. A new numerical approach to MHD stagnation point flow and heat transfer towards a stretching sheet Ain Shams Engineering Journal http://dx.doi.org/10.1016/j.asej.2015.10.015 (2015).
[2]
Anderson, J. D. Computational fluid dynamics: The basics with application McGraw-Hill N ew York, NY (1990).
[3]
Anderson, J. D. Fundamentals of Aerodynamics 3rd ed. McGraw Hill International edition (2001).
[4]
Ariel, P. D., A hybrid method for computing the flow of viscoelastic fluids, In. Jnl. Numer. Methods Fluids Vol. 14 757-774 (1992).
[5]
Ariel, P. D., Stagnation point flow-A free boundary value problem formulation Int. Jnl. Comput. Math. Vol. 49 123-131 (1993).
[6]
Beard, D. W., Walters K. Elastico-viscous boundary-layer flows I: 2-D flow near a stagnation point Proc. Camb. Phil. Soc. Vol. 60 667-674 (1964).
[7]
Davies M. H. A note on elastico boundary-layer flows, ZAMP Vol. 17 189-191 (1966).
[8]
Garg, V. K., Heat transfer due to stagnation point flow on a non-Newtonian fluid, Acta Mech. Vol. 104 159-171 (1994).
[9]
Ibrahim, W., Makinde, O. D., Double-diffusive mixed convection and MHD stagnation point flow of nanofluid over a stretching sheet. J. Nanofluids, Vol 4 28-37 (2015).
[10]
Ibrahim, W., Shankar, B., MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions J. Computational fluids Vo. 75 1-10 (2013).
[11]
Hiemenz, K. K., Die Grenzschicht an einem in den gleichformingen Flussigkeitsstrom eingetauchten graden kreiszylinder. Dingl polytech J. Vol. 326 321-324 (1911).
[12]
Ishak, A., Jafar, K., Nazar, R., Pop, I. MHD stagnation point flow over nonlinearly stretching sheet with chemical reaction and magnetic field, Physica A, Vol 388 3377-3383 (2009).
[13]
Makinde, O. D., Khan, W. A. Khan, Z. H. Buoyancy effects on MHD stagnation point flow and heat transfer on a nanofluid past a covectively heated stretching shrinking sheet. Int. J. Heat and Mass Transfer Vol. 62 526-533 (2013).
[14]
Makinde, O. D. Mishra, S. R. On Stagnation point flow of variable viscosity nanofluids past a stretching surface with radiative heat Int. Jnl. Comput. Math. DOI 10.1007/s40819-015-0111-1 (2015).
[15]
Massoudi, M., Ramezan, M. Heat transfer analysis of a viscoelastic fluid at a stagnation point, Mech. Res. Commun., Vol. 19 129-134 (1992).
[16]
Na, T. T. Computational methods in engineering boundary value problem, Academic Press, NY (1979).
[17]
Ng, K. Y. K. Solution of Navier-Stokes equations by goal programming, J. Comput CRC Press Boca Raton, London, NY (2000).
[18]
Rajeswar, G. K., Rathna, S. L. Flow past a particular class of non-Newtonian viscoelastic and visco-inelastic fluids near a stagnation point, Z. Agnew. Math. Physics Vol. 13 43-57 (1962).
[19]
Serth, R. W. Solution of viscoelastic boundary layer equation by orthogonal collocation. J. Engnr. Math. Vol. 8 89-92 (1974).
[20]
Teipel, I. Die raumliche staupunktstromung fur ein viscoelastisches, Fluid Rheol, Acta Vil 25 75-79 (1986).
[21]
Wilcox, D. C. Basic fluid mechanics 2nd. Ed. DCW industries (2000).
PUBLICATION SERVICES