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Recently, the subject of wavy geometries has attracted significant research attention, and the prominence of this issue is due to its application in many engineering systems as a means of enhancing transport performance. Different types of attempts had been made in the form of review articles or chapters in books to highlight the formulation and methods of solution for fluid flow and heat and mass transfer for various types of engineering problems with wavy boundaries under different physical effects and conditions. The flow and heat transfer within wavy surface is quite a new topic that considers the natural and mixed convection flows of fluids through fluid saturated porous media with wavy surface. Many of the engineering applications are within this scientific area. Example: cooling systems for micro-electronic devices, heat exchangers, underground cable systems, solar collectors etc. The main strength is in the development of the basic formulation of the fluid flow and heat transfer of the wavy surfaces, channel tubes and cavities having wavy wall in the presence or absence of porous media. At present for the beneficial result of heat transfer enhancement which is one of the hot topic in the engineering field, wavy wall is a passive system that allows a possible combination to realize the increase in thermal energy transfer and changes in the fluid flow. It is important to note that the wavy passage does not provide any significant heat transfer when the flow is steady. Therefore to made it unsteady either through some external forcing or through natural transitioning to an unsteady state results in an increases in heat exchange layer fluid through shear layer destabilization.
In many engineering systems as a means for enhancing the transport performance wavy geometries as a part were used. Therefore, to know the knowledge of fluid flow and heat transfer through wavy surfaces becomes important in this context. Solar collectors, condensers in refrigerators, cavity wall insulating systems, grain storage containers, and industrial heat radiators, for example, are a few of the many applications where wavy surfaces are encountered to transfer small- or large-scale heat. The focus on the area of flow and heat transfer from wavy surfaces in complex enclosures like square, trapezoidal, and rectangular spaces has been intensifying over the years due to the increasing interest of researchers from applied mathematics, and mechanical and chemical engineering, as well as from biomechanics and engineering mechanics. The accomplishment due to complex circulatory flows and boundary layer separation made the wavy surface as yet another special surface that can be used to promote heat transfer. It has a wide range of applications which includes oxygenator (transfers oxygen into blood and remove carbon dioxide from blood), small and large electrical, electronic devices to make corrugated surfaces for compactness along the tube banks in fossil fuel power plants.
The compressible boundary-layer flows over a wavy walls finds its application in various areas such as film vaporization in combustion chambers, rocket boosters’ transpiration cooling of re-entry vehicles, cross hatching on ablative surfaces etc. Flows over wavy walls provides the mean flow for the stability analysis Rayleigh problem which is generated through the vorticity in a fluid by a solid surfaces, naturally leads to the study of viscous boundary layers. But a standard example of in viscid fluid used to illustrate the effect of boundary perturbations is nothing but the wavy wall problems
The primary task in ?uid dynamics is to ?nd the velocity ?eld depicting the ?ow in a given area. To do this, one uses the basic equations of ?uid ?ow
• Conservation of mass (the continuity equation)
• Conservation of momentum (the Cauchy equation) at the level of ?uid elements
In any domain, the ?ow conditions must be explained subject to a set of conditions that demonstrate the domain boundary. If the ?ow leads to compression of the ?uid, we should likewise think about thermodynamics:
• Conservation of energy (expecting at first that the ?ow stays incompressible)
Governing equations for flow from wavy surfaces are not truly different from those that are utilized when dealing with flat and smooth surfaces. The main difference lies in the utilization of an additional equation which is required for describing the profile of the wavy surface.
Amid the invesitigations of flow and heat transfer past wavy surface simple coordinate transformations are used as a part of request to change the complex wavy geometry into a basic smooth one for which the governing equations can be solved by well-known methods. Despite the fact that this strategy achieves impressive about considerable simplifications the effort required to solve the transformed equations numerically are frequently just about the same as those of the original conditions. In any case the benefit of the transformation lies in the fact that the profile of the wavy surface can be changed during the numerical calculations without resorting to revariations in the governing equations.
Perturbation technique
A considerable lot of the issues Physicists, engineers and applied mathematicians involve such difficulties as nonlinear governing equations, variable coefficients and nonlinear boundary conditions at complex known or unknown boundaries that preclude solving them exactly. Therefore, solutions are approximated using numerical methods, analytical techniques and combinations of both. Preeminent among the analytic techniques are the systematic strategies of perturbations
Perturbation methods, otherwise as asymptotic expansions regarding of a small or a large parameter or coordinate, permit the improvement of complex numerical issues. Use of perturbation theory will enable approximate solutions to be determined for problems which cannot be solved by conventional analytical methods. Second order ordinary linear differential equations are understood by engineers and researchers routinely. Anyway as a rule, genuine circumstances can require substantially more difficult mathematical models, such as non-linear differential equations. Numerical strategies used on a PC of today are equipped for solving extremely complex mathematical problems; however, they are not perfect. The numerical strategies of today can still run into a multitude of problems ranging from diverging solutions to tracking wrong solutions. Numerical techniques on a computer do not provide much understanding to the specialists or researchers running them. Perturbation theory can offer an elective way to deal with tackling certain sorts of issues. Solving problems systematically often helps an engineer or scientist to comprehend a physical problem better, and may help to enhance future methods and outlines used to solve their problems.
Perturbation is a mathematical method for finding an approximate solution which starts from the exact solution of a related problem. The main feature of this method is that it breaks the problem into solvable part and perturbed part which is applicable for the problems that cannot be solved directly so that a small term have been included to the mathematical description of the problem.
A few scientists have considered numerous outcomes regarding wavy wall. Lekoudis et al 1 influenced a linear analysis of compressible boundary layer flows over a wavy wall. Shankar et al 2 considered the Rayleigh issues for a wavy wall. At low Reynolds numbers the waviness of the wall rapidly stops to be of significance as the fluid is hauled along by the wall. At high Reynolds number on the other hand gain the impact of viscosity are shown to be confined to a narrow layer near to the wall and the know potential solution rises in time. An analyses of the effect of waviness of one of the walls on the flow and heat transfer qualities of an incompressible viscous fluid restricted between two long vertical walls and set in motion by a distinction in the wall temperatures was made by vajravelu and sastri 3. Rao and sastri 4 examined crafted by vajravelu and sastri 5 for viscous heating effects when the fluid properties are constants and variables. The free convective magneto hydrodynamics flow and heat transfer in a viscous incompressible fluid confined between a long vertical wavy and a parallel flat wall was studied by Das and Ahmed 6. Soret and Dufour consequences for a two dimensional free convective magneto hydrodynamic flow of a viscous incompressible and electrically conducting fluid through a channel bounded by a long vertical wavy wall and a parallel flat wall was studied by Nazibuddin Ahmed, sarma and Deka 7.


1 Spyridon G. Lekoudis, Ali H. Nayfeh and William S. Saric, “Compressible boundary layers over wavy walls” Physics of Fluids, 19, 514 (1976).
2 Shankar P.N, Sinha U.N, “The Rayleigh problem for a wavy wall”, J. Fluid Mech, 77, Part 2, 243-256 (1976).
3 Vajravelu K, Sastri K.S, “Free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall”, J. Fluid Mech, 86, part 2, 365-383 (1978).
4 Rao C.N.B and Sastri K.S, “The response of skin friction, wall heat transfer and pressure drop to wall waviness in the presences of buoyancy”, Int. J. Math and Math. Sci, 5, No 3, 585-594 (1982).
5 Vajravelu K, Sastri K.S, “Natural convective heat transfer in vertical wavy channels”, Int. J. Heat Mass transfer, 23, 408-411 (1980).
6 Das U.N, Ahmed N, “Free convective MHD flow and heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall”, Indian. J. Pure appl. Math, 23(4), 295-304 (1992).
7 Nazibuddin Ahmed, Kalpana sarma and Hiren Deka, “Soret and Dufour effect on an MHD free convective flow through a channel bounded by a wavy wall and a parallel flat wall”, Turkish journal of Physics, 38, 50-63 (2014).

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