Abstract

This chapter presents the applications of perturbation methods such as regular and homotopy perturbation methods to thermal, fluid flow and dynamic behaviors of engineering systems. The first example shows the utilization of regular perturbation method to thermal analysis of convective-radiative fin with end cooling and thermal contact resistance. The second example is concerned with the application of homotopy perturbation method to squeezing flow and heat transfer of Casson nanofluid between two parallel plates embedded in a porous medium under the influences of slip, Lorentz force, viscous dissipation and thermal radiation. Additionally, the dynamic behavior of piezoelectric nanobeam embedded in linear and nonlinear elastic foundations operating in a thermal-magnetic environment is analyzed using homotopy perturbation method which is presented in the third example. It is believed that the presentation in this chapter will enhance the understanding of these methods for the real world applications.

Highlights

  • The descriptions of the behaviors of the real world phenomena and systems through the use of mathematical models often involve developments of nonlinear equations which are difficult to solve exactly and analytically

  • Example 2: homotopy perturbation method to analysis of squeezing flow and heat transfer of Casson nanofluid between two parallel plates embedded in a porous medium under the influences of slip, Lorentz force, viscous dissipation and thermal radiation

  • Regular perturbation was used in the first example to developed approximate analytical solutions for thermal behavior of convective-radiative fin with end cooling and thermal contact resistance

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Summary

Introduction

The descriptions of the behaviors of the real world phenomena and systems through the use of mathematical models often involve developments of nonlinear equations which are difficult to solve exactly and analytically. In these methods, there are always search for particular value(s) that will satisfy the end boundary condition(s) This always necessitates the use of software and such could result in additional computational cost in the generation of solution to the problem. The quests involve applications of numerical schemes to determine the required value(s) that will satisfy the end boundary condition(s) This fact renders most of the approximate analytical methods to be taken as more of semi-analytical methods than total approximate analytical methods. In the limit of small parameter, perturbation method is widely used for solving many heat transfer, vibration, fluid mechanics and solid mechanics problems. It is capable of solving nonlinear, inhomogeneous and multidimensional problems with reasonable high level of accuracy. X 1⁄4 x , θ 1⁄4 T À Ta , Ra 1⁄4 gkβðTb À TaÞb , N 1⁄4 4σstbT3a , Ha 1⁄4 σB20u2 : (8)

Method of solution using regular perturbation method
Method of solution by homotopy perturbation method
The basic idea of homotopy perturbation method
ÀHa2 f 00 À f 00
A3ð1 À φÞ2:5 0 1
B À 1009008
B B þ 672672
Solution methodology
Conclusion
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