**6. References**


[14] Zevin A A, Filonenko L A (2007) Qualitative Study of Oscillations of a Pendulum with Periodically Varying Length and a Mathematical Model of Swing. Journal of applied mathematics and mechanics 71: 989-1003.

30 Will-be-set-by-IN-TECH

*f*<sup>6</sup> + *f*1*u*<sup>5</sup> + *f*2*u*<sup>4</sup> + *f*3*u*<sup>3</sup> + *f*4*u*<sup>2</sup> + *f*5*u*<sup>1</sup> + *f*1*u*1,4 + *f*1*u*2,3 + *f*2*u*1,3 + *f*3*u*1,2

<sup>2</sup> *<sup>f</sup>*2*u*1,1,2 <sup>+</sup>

1

<sup>6</sup> *<sup>f</sup>*3*u*1,1,1 <sup>+</sup>

1

*<sup>m</sup>*. Functions *Uk* = *Uk*(*X*) can be chosen

, (128)

<sup>6</sup> *<sup>f</sup>*1*u*1,1,1,2

1

arbitrarily, so for convenience we set *Uk*(*X*) ≡ 0. Thus, knowing the functions *Fi* from expressions (120)-(128) we can write averaged system (118), which is simpler than the original system (117). If we solve averaged system (118) we are able to write the approximate solution (119) of system (117) substituting slow variables *X*(*t*) into the functions *ui*(*X*, *t*) obtained from (121)-(127). Since the functions *ui*(*X*, *t*) are periodic with respect to time *t* the behavior of slow variables determines the behavior of the approximate solution. It means that we can study stability of the approximate solutions by stability of the regular solutions of averaged system

[1] Seyranian A P (2004) The Swing: Parametric Resonance. Journal of applied mathematics

[2] Seyranian A P, Belyakov A O (2008) Swing Dynamics. Doklady physics 53(7): 388-394. [3] Belyakov A O, Seyranian A P, Luongo A (2009) Dynamics of the Pendulum with

[4] Belyakov A O, Seyranian A P (2010) On Nonlinear Dynamics of the Pendulum with Periodically Varying Length. Mechanics of machines, BulKToMM 18(87): 21-28. [5] Belyakov A O (2011) On Rotational Solutions for Elliptically Excited Pendulum. Physics

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[13] Pinsky M F, Zevin A A (1999) Oscillations of a Pendulum with a Periodically Varying Length and a Model of Swing. International journal of non-linear mechanics 34: 105-109.

*F*<sup>6</sup> = 

(118).

**6. References**

99-104.

712-715.

+ 1 <sup>2</sup> *<sup>f</sup>*2*u*2,2 <sup>+</sup>

+ 1 1 <sup>2</sup> *<sup>f</sup>*4*u*1,1 <sup>+</sup>

1

Periodically Varying Length. Physica D 238: 1589-1597.

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Non-Linear Oscillations. New York: Gordon and Breach.

Concepts, Paradoxes and Mistakes. Moscow: Nauka.

Schwingungensproblemen. Stuttgart: J.Teubner.

Linear Systems. Moscow: Mashinostroenie.

<sup>24</sup> *<sup>f</sup>*2*u*1,1,1,1 <sup>+</sup>

where we similarly denote *ukFm* = ∑*<sup>n</sup>*

and mechanics 68: 757-764.

letters A 375: 2524-2530.

1

<sup>120</sup> *<sup>f</sup>*1*u*1,1,1,1,1

<sup>2</sup> *<sup>f</sup>*1*u*1,2,2 <sup>+</sup>

*i*=1 *∂uk ∂xi Fi*

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© 2012 Dumitrache et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Mathematical Modelling and Numerical** 

Jets are frequently observed to adhere to and to flow around nearby solid boundaries. This general class of phenomena, which may be observed in both liquid and gaseous jets, are known as the Coanda effect. Flows deflected by a curved surface have caused great interest in last fifty years [1-4]. A major interest in the study of this phenomenon is caused by the possibility of using this effect to aircrafts with short takeoff and landing, for fluidic

Flow control offers a multitude of opportunities for improving not only the aerodynamic performance, but also the safety and environmental impact of flight vehicles. Circulation control (CC) is one type of flow control which is currently receiving considerable attention. Such flow control is usually implemented by tangentially injecting a jet sheet over a rounded wing trailing edge. The jet sheet remains attached further along the curved surface of the wing due to the Coanda effect (i.e., a balance of pressure and centrifugal forces). This results in the effective camber of the wing being increased, producing lift augmentation.

At the beginning of the chapter we achieve an analytic solution that approximates a twodimensional Coanda flow. The validity of the results is limited to cases *b R*/ 1, since in the tangential component of the momentum equation, the curvature was neglected ( *y* 1 ).

In many applications that use boundary layer control by tangential blowing, the solid surface downstream of the blowing slot is strongly curved and, in this case, the prediction of the jet involves both separation and a more accurate knowledge of the flow (radial and

After the analytical approach, using the FLUENT code both external and internal flows are analyzed, with emphasis on the Coanda effect, in order to determine its advantages and

and reproduction in any medium, provided the original work is properly cited.

tangential pressure - velocity profiles) which can be done by CFD methods.

limitations. Finally, we analyze the situations when bifurcations of the flow occur.

**Investigations on the Coanda Effect** 

A. Dumitrache, F. Frunzulica and T.C. Ionescu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50403

**1. Introduction** 

vectoring.
