A Combined Analytic and Numerical Procedure for Stagnation-Point Flow

Okey Oseloka Onyejekwe

doi:doi:10.11648/j.acm.20170602.12

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A Combined Analytic and Numerical Procedure for Stagnation-Point Flow

Okey Oseloka Onyejekwe

Published in Applied and Computational Mathematics (Volume 6, Issue 2)

Received: 8 January 2017 Accepted: 7 February 2017 Published: 14 March 2017

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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.

Published in	Applied and Computational Mathematics (Volume 6, Issue 2)
DOI	10.11648/j.acm.20170602.12
Page(s)	75-82
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Navier-Stokes Equation, Streamfunction, CFL Number, Fourth-Order Equation, Quasi-Linear Third-Order Differential Equation, Crank-Nicolson, Newton-Linearization

References

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[3]	Anderson, J. D. Fundamentals of Aerodynamics 3^rd 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).
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[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).
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[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).
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Okey Oseloka Onyejekwe. (2017). A Combined Analytic and Numerical Procedure for Stagnation-Point Flow. Applied and Computational Mathematics, 6(2), 75-82. https://doi.org/10.11648/j.acm.20170602.12

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Okey Oseloka Onyejekwe. A Combined Analytic and Numerical Procedure for Stagnation-Point Flow. Appl. Comput. Math. 2017, 6(2), 75-82. doi: 10.11648/j.acm.20170602.12

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Okey Oseloka Onyejekwe. A Combined Analytic and Numerical Procedure for Stagnation-Point Flow. Appl Comput Math. 2017;6(2):75-82. doi: 10.11648/j.acm.20170602.12

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@article{10.11648/j.acm.20170602.12,
  author = {Okey Oseloka Onyejekwe},
  title = {A Combined Analytic and Numerical Procedure for Stagnation-Point Flow},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {2},
  pages = {75-82},
  doi = {10.11648/j.acm.20170602.12},
  url = {https://doi.org/10.11648/j.acm.20170602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170602.12},
  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.},
 year = {2017}
}

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TY  - JOUR
T1  - A Combined Analytic and Numerical Procedure for Stagnation-Point Flow
AU  - Okey Oseloka Onyejekwe
Y1  - 2017/03/14
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170602.12
DO  - 10.11648/j.acm.20170602.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 75
EP  - 82
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170602.12
AB  - 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.
VL  - 6
IS  - 2
ER  -

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Author Information

Okey Oseloka Onyejekwe

Computational Science Program, Faculty of Natural Science, Addis Ababa University, Arat-Kilo Campus, Addis Ababa, Ethiopia

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Okey Oseloka Onyejekwe. (2017). A Combined Analytic and Numerical Procedure for Stagnation-Point Flow. Applied and Computational Mathematics, 6(2), 75-82. https://doi.org/10.11648/j.acm.20170602.12

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ACS Style

Okey Oseloka Onyejekwe. A Combined Analytic and Numerical Procedure for Stagnation-Point Flow. Appl. Comput. Math. 2017, 6(2), 75-82. doi: 10.11648/j.acm.20170602.12

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AMA Style

Okey Oseloka Onyejekwe. A Combined Analytic and Numerical Procedure for Stagnation-Point Flow. Appl Comput Math. 2017;6(2):75-82. doi: 10.11648/j.acm.20170602.12

Copy | Download

@article{10.11648/j.acm.20170602.12,
  author = {Okey Oseloka Onyejekwe},
  title = {A Combined Analytic and Numerical Procedure for Stagnation-Point Flow},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {2},
  pages = {75-82},
  doi = {10.11648/j.acm.20170602.12},
  url = {https://doi.org/10.11648/j.acm.20170602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170602.12},
  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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - A Combined Analytic and Numerical Procedure for Stagnation-Point Flow
AU  - Okey Oseloka Onyejekwe
Y1  - 2017/03/14
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170602.12
DO  - 10.11648/j.acm.20170602.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 75
EP  - 82
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170602.12
AB  - 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.
VL  - 6
IS  - 2
ER  -

Copy | Download