Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T15:54:07.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of Double Side Approach Method to the Solution of Fully Developed Laminar Flow in Duct Problems

Published online by Cambridge University Press:  01 May 2013

H.-W. Tang ,
Y.-T. Yang  and
C.-K. Chen
H.-W. Tang
Affiliation:
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
Y.-T. Yang
Affiliation:
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
C.-K. Chen*
Affiliation:
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
*Corresponding author ([email protected])
Get access

Abstract

The double side approach method combines the method of weighted residuals (MWR) with mathematical programming to solve the differential equations. Once the differential equation is proved to satisfy the maximum principle, collocation method and mathematical programming are used to transfer the problem into a bilateral inequality. By utilizing Genetic Algorithms optimization method, the maximum and minimum solutions which satisfy the inequality can be found. Adopting this method, quite less computer memory and time are needed than those required for finite element method.

In this paper, the incompressible-Newtonian, fully-developed, steady-state laminar flow in equilateral triangular, rectangular, elliptical and super-elliptical ducts is studied. Based on the maximum principle of differential equations, the monotonicity of the Laplace operator can be proved and the double side approach method can be applied. Different kinds of trial functions are constructed to meet the no-slip boundary condition, and it was demonstrated that the results are in great agreement with the analytical solutions or the formerly presented works. The efficiency, accuracy, and simplicity of the double side approach method are fully illustrated in the present study to indicate that the method is powerful for solving boundary value problems.

Keywords

Double side approach methodFully developed laminar flowGenetic algorithms
Type
Articles
Information
Journal of Mechanics , Volume 29 , Issue 3 , September 2013 , pp. 471 - 479
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Finlayson, B. A. and Scriven, L. E., “The Method of Weighted Residuals-A Review,” Applied Mechanics Reviews, 19, pp. 735–748 (1966).Google Scholar
2.Chen, C. K., Lin, C. L. and Chen, C. L., “Error Bound Estimate of Weighted Residuals Method Using Genetic Algorithms,” Applied Mathematics Computers, 81, pp. 207–219 (1997).Google Scholar
3.Xing, S. Y., Li, Z. Q. and Zhu, B. A., “Two Methods of Mathematical Programming and Weighted Residuals,” Journal of Xi'an Highway University, 17, pp. 61–65 (1997).Google Scholar
4.Appl, F. C. and Hung, H. M., “A Principle for Convergent Upper and Lower Bounds,” International Journal of Mechanical Sciences, 6, pp. 289–381 (1964).Google Scholar
5.Goldberg, D. E., Genetic Algorithms in Search. Optimization and Machine Learning, Addison-Wesley, Reading, Ma (1989).Google Scholar
6.Davis, L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York (1991).Google Scholar
7.Holland, J. H., Adaptation in Natural and Artificial System, MIT Press, Cambridge, MA (1992).CrossRefGoogle Scholar
8.Srinivas, M. and Patmaik, L. M., “Genetic Algorithms-A Survey,” Computer, 27, pp. 17–26 (1994).Google Scholar
9.Jenkis, W. M., “Towards Structural Optimization Via the Genetic Algorithm,” Computers & Structures, 40, pp. 1321–1327 (1991).Google Scholar
10.Christopher, L. H. and Donald, E. B., “A Parallel Heuristic for Quadratic Assignment Problems,” Computer Operations Research, 18, pp. 275–289 (1991).Google Scholar
11.Reeves, C. R., Modern Heuristic Technique for Combinational Problems, Halsted Press, New York (1993).Google Scholar
12.Wang, C. C., Lin, T. W. and Hu, H. P., “Error Analysis of Approximate Solutions of Non-Linear Burger's Equation,” Applied Mathematics and Computation, 212, pp. 387–395 (2009).Google Scholar
13.Peng, H. S., Chen, C. L. and Li, G. S., “Application of Residual Correction Method to Laser Heating Process,” International Journal of Heat and Mass Transfer, 55, pp. 316–324 (2012).Google Scholar
14.Wang, C. C., Huang, J. H. and Yang, D. J., “Cubic Spline Difference Method for Heat Conduction,” International Communications in Heat and Mass Transfer, 39, pp. 224–230 (2012).Google Scholar
15.Dryden, H. L., Murnaghan, F. D. and Bateman, H., Hydrodynamics, Dover, New York (1956).Google Scholar
16.Shah, R. K. and London, A. L., Laminar Flow Forced Convection in Ducts, Academic Press, New York (1978).Google Scholar
17.Spiga, M. and Morini, G. L., “A Symmetric Solution for Velocity Profile in Laminar Flow Through Rectangular Ducts,” International Communications in Heat and Mass Transfer, 21, pp. 469–475 (1994).Google Scholar
18.Wang, C. Y., “Heat Transfer and Flow Through a Super-Elliptic Duct-Effect of Corner Rounding,” Mechanics Research Communications, 36, pp. 509–514 (2009).CrossRefGoogle Scholar