THERMO-FLUIDIC PARAMETERS EFFECTS ON NONLINEAR VIBRATION OF FLUID- CONVEYING NANOTUBE RESTING ON ELASTIC FOUNDATIONS USING HOMOTOPY PERTURBATION METHOD

In this paper, effects of thermo-fluidic parameters on the nonlinear dynamic behaviours of single-walled carbon nanotube conveying fluid with slip boundary conditions and resting on linear and nonlinear elastic foundations under external applied tension and global pressure is studied using homotopy perturbation method. From the result, it is observed that increase in the Knudsen number, the slip parameter, leads to decrease in the frequency of vibration and the critical velocity while natural frequency and the critical fluid velocity increase as the in stretching effect increases. Also, as the Knudsen number increases, the bending stiffness of the nanotube decreases and in consequent, the critical continuum flow velocity decreases as the curves shift to the lowest frequency zone. As the change in temperature increases, the natural frequencies and the critical flow velocity of the structure increase for the low or room temperature while at high temperature, increase in temperature change, decreases the natural frequencies and the critical flow velocity of the structure. Further, it is established that the alteration of nonlinear flow-induced frequency from linear frequency is significant as the amplitude, flow velocity and axial tension increase. The developed analytical solutions can be used as starting points for better understanding of the relationship between the physical quantities of the problem.


INTRODUCTION
There have been increasing interests and rapid developments in the study of nanotube following the discovery of Iijima [1]. As part of the numerous applications, carbon nanotube (CNT) has been used for conveying/transporting fluid and the study of effects and the conditions of moving fluid on the overall mechanical behaviour of CNTs have been an area that has aroused significant and challenging research interests. Consequently,

Problem formulation based on nonlocal beam theory
Consider a single-walled carbon nanotube conveying hot fluid, subjected to stretching effects and resting on linear and nonlinear elastic foundations under external applied tension and global pressure as shown in Figure  1. Based on the Eringen's nonlocal elasticity theory [31][32][33][34][35] and Hamilton's principle, we arrived at the governing equation of motion for the single-walled carbon nanotube (SWCNT) as; The derivation of the governing equation (although, with some modifications to include effects of elastic foundation, axial tension, global pressure and temperature change in this paper) has been shown in the author's previous paper [36].
If the nanotube is slightly curved, then the governing equation for the nanotube becomes   ) 6  Where Zo(x) is the arbitrary initial rise function. For nanotube conveying fluid, the radius of the tube is assumed to be the characteristics length scale, Knudsen number is larger than 10 -2 . Therefore, the assumption of no-slip boundary conditions does not hold and modified model should be used. Therefore, we have Where Kn is the Knudsen number, σv is tangential moment accommodation coefficient which is considered to be 0.7 for most practical purposes [37,38].  2  4  3  2  4  2  1  3  ,  2 2  2  2  3   2  4 4 2 , 4 4 0

The initial and the boundary conditions
In this work, different boundary conditions are considered for the nanotube.

Clamped-Clamped (doubly clamped)
Where the trial/comparison function are given as; The applications of space function as given above for clamped-clamped will involve long calculations and expressions in finding M, G, K, C, and V, alternatively, a polynomial function of the form Equation (10) can be applied for this type of support system. 2 3 Orthogonal function should satisfy the equation Substitute Equation (11) Alternatively, a polynomial function of the form in Equation (17) can be applied for this type of support system. x X X X a        (17) On using orthogonal functions, 4 11.625 a  for the first mode Alternatively, a polynomial function of the form in Equation (21) can be applied for this type of support system.
Alternatively, a polynomial function of the form Equation (26) can be applied for this type of support system.

 
Also, with the aid of orthogonal functions, 4 0.6625 a  for the first mode

The Spatial and Temporal Decomposition Procedures
Using the Galerkin's decomposition procedure to separate the spatial and temporal parts of the lateral displacement functions as where () ut the generalized coordinate of the system and () x  is a trial/comparison function that will satisfy both the geometric and natural boundary conditions.
Applying one-parameter Galerkin's solution given in Equation (1) and Equation (2) For the slightly curved nanotube, M, G, K and C are the same but For the undamped clamped-clamped, clamped-simple and simple-simple supported structures, G=0 and Equation (9) reduces to which can be written as

Method of solution by homotopy perturbation method
Application of regular perturbation to the nonlinear Equation (32) breaks down at the time t of O(ε -2 ). Also, the traditional perturbation methods (regular and singular perturbation methods) are based on the existence of small parameter in the nonlinear equations and such are limited to analysis of weakly nonlinear equation. Unfortunately, the nonlinear equation as shown in Equation (32) does not have any small perturbation and it is strongly nonlinear. Therefore, in this work, homotopy perturbation method is used to solve the equation. The homotopy perturbation method eliminates the "the small parameter assumption" as carried in the traditional perturbation 2219 methods. It is a powerful method that gives acceptable analytical results with convenient convergence and stability [16][17][18][19][20].

The basic idea of homotopy perturbation method
In order to establish the basic idea behind homotopy perturbation method, consider a system of nonlinear differential equations given as, with the boundary conditions , 0, where A is a general differential operator, B is a boundary operator,   fr a known analytical function and  is the boundary of the domain  The operator A can be divided into two parts, which are L and N, where L is a linear operator, N is a non-linear operator. Equation (13) can be therefore rewritten as follows; By the homotopy technique, a homotopy     In the above Equations (36) and (37), is an embedding parameter, o u is an initial approximation of equation of Equation (33), which satisfies the boundary conditions. Also, from Equations (36) and (37), we will have, The changing process of p from zero to unity is just that of ur. This is referred to homotopy in topology. Using the embedding parameter p as a small parameter, the solution of Equations (36) and (37) can be assumed to be written as a power series in p as given in Equation (40) 2 12 ...
It should be pointed out that of all the values of p between 0 and 1, p=1 produces the best result. Therefore, setting 1 p  , results in the approximation solution of Equation (33)  12  1 lim The basic idea expressed above is a combination of homotopy and perturbation method. Hence, the method is called homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantages of the traditional perturbation techniques. The series Equation (41) is convergent for most cases.

Application of the homotopy perturbation method to the present problem
According to homotopy perturbation method (HPM), we can construct an homotopy for Equation (32) as Or equivalently, Supposing that the solution of Equation (33) can be expressed in a series in p : According to HPM, a constant can be expanded as a power series of the embedding parameter p .

Clamped-Clamped (doubly clamped)
The approximate analytical solution is   While for the slightly curved nanotube, M, G, K and C are the same as above but It should be noted that the definitions and the values M, K, C, and V are trial/shape function-dependent. Therefore, they are different for the different boundary conditions considered. Also, by extension, the values of α and β in the solutions are different for the different supports analyzed.

RESULTS AND DISCUSSION
Based on the shape functions defined in Equations (7), (14), (18), (22) and (23), the first five normalized mode shapes of the beams for clamped-clamped, simple-simple, clamped-simple and clamped-free supports are shown in Figure. 2-5. Also, the figures depict the deflections of the beams along the beams' span at five different buckled and mode shapes.
Following to the polynomial functions developed in this work in Equations (11), (17), (21) and (26) for the hyperbolic-trigonometric functions defined in Equations (7), (14), (18), (22) and (23), Figures 6-9 show the comparison of hyperbolic-trigonometric and the polynomial functions for the normalized mode shapes of the beams for clamped-clamped, simple-simple, clamped-simple and clamped-free supports. The figures depict the validity of the developed polynomial functions in this work as there are very good agreements between the hyperbolic-trigonometric and the developed polynomial functions.    Hyperbolic-Trigonometric and Polynomial functions Figure 10 illustrates the effects of boundary conditions on the nonlinear amplitude-frequency response curves of the nanotube. Also, the figure shows the variation of frequency ratio of the nanotube with the dimensionless maximum amplitude of the structure under different boundary conditions. From, the result, it shown that frequency ratio is highest in the beam which is clamped-free (cantilever) supported beam and lowest with clamped-clamped beam. The lowest frequency ratio of the clamped-clamped beam is due to high stiffness of the beam with this type of boundary conditions in comparison with other types of boundary conditions. Figure 10. Effects of boundary conditions on the nonlinear amplitude-frequency response curves of the nanotube Figure 11. Effects of axial tension on the nonlinear amplitude-frequency response curves of SWCNT Figure 11 shows effects of axial tension on the nonlinear amplitude-frequency response curves of pipe. It is observed that the increase of axial tension, the nonlinear vibration frequencies increases. It can be seen from the figure, in contrast to linear systems, the nonlinear frequency is a function of amplitude so that the larger the amplitude, the more pronounced the discrepancy between the linear and the nonlinear frequencies becomes.  Figure 14 illustrates the midpoint deflection time history for the nonlinear analysis of SWCBT when Kn=0.03 and U= 100 m/s while Figure 15 presents the midpoint deflection time history for the nonlinear analysis of SWCBT when Kn=0.03and U= 500 m/s.    In order to verify the model, Figures 18 and 19 show the comparison of the results of exact analytical solution and the results of the present study for the linear models while Figure 13 presents the comparison of numerical results models and the results of present work for the nonlinear models. The results show that good agreements are established between the solutions.

CONCLUSION
In this work, thermo-fluidic parameter effects on the nonlinear vibration of carbon nanotube conveying fluid under elastic foundations has been investigated using homotopy perturbation method. From the analysis it is established that increase in the Knudsen number, the slip parameter, leads to decrease in the frequency of vibration and the critical velocity while natural frequency and the critical fluid velocity increase as the in stretching effect increases. Also, as the Knudsen number increases, the bending stiffness of the nanotube decreases and in consequent, the critical continuum flow velocity decreases as the curves shift to the lowest frequency zone. As the change in temperature increases, the natural frequencies and the critical flow velocity of the structure increase for the low or room temperature while at high temperature, increase in temperature change, decreases the natural frequencies and the critical flow velocity of the structure. Further, it is established that the alteration of nonlinear flow-induced frequency from linear frequency is significant as the amplitude, flow velocity, and aspect ratio increase. The analytical solutions can serve as benchmarks for other methods of solutions of the problem. They can also provide a starting point for a better understanding of the relationship between the physical quantities of the problems.