Fluid-conveying Single-walled Carbon Nanotubes Research Articles

Lie symmetry analysis, invariant subspace method and q-homotopy analysis method for solving fractional system of single-walled carbon nanotube

In this paper, we investigate the time fractional system of fluid-conveying single-walled carbon nanotube (SWCNT), the generalization of SWCNT system which plays important roles in many applied fields. The corresponding Lie symmetries admitted by this fractional system in Riemann–Liouville sense are obtained and symmetry reductions are performed. In addition, based on the above symmetries, the conservation laws are derived using new Noether theorem. Furthermore, analytical solution and numerical series solution to the initial value problem of time fractional SWCNT system in Caputo sense are constructed by applying invariant subspace method and q-homotopy analysis method, respectively.

Instabilities of SWCNT conveying laminar, incompressible and viscous fluid flow

PurposeThis study aims to study the transverse vibration and instabilities of the fluid-conveying single-walled carbon nanotubes (CNTs). To this purpose, the Euler–Bernoulli beam model is used. Also, the surface effects, small-size effects of the both fluid and structure and two different elastic mediums viscoelastic and Pasternak elastic are investigated.Design/methodology/approachTo consider the nano-scale for the CNT, the strain-inertia gradient theory is used and to solve the governing equation of motion for the system, the Galerkin’s method is used. The effect of the flow velocity, aspect ratio, characteristic lengths of the mentioned theory, effects of Knudsen number and effects of the Winkler, the Pasternak elastic and the viscoelastic medium on the frequencies and stabilities of the system are studied. The effects of the above parameters on the vibrational behavior are investigated both separately and simultaneously.FindingsThe results show that the critical flow velocity value is increased as the aspect ratio, characteristic lengths, Winkler modulus, shear and damping factors increase. Also, the critical flow velocity is increased by considering the surface effects. In addition, the consequence of increase in the nano-flow-size effects (Knudsen number) is decreasing the critical flow velocity. Moreover, it can be observed that the effect of the shear factor on increasing the critical flow velocity is different from the rest of parameters.Originality/valueUse of Timoshenko and modified couple stress theories and taking into account Von-Karman expressions for investigating the nonlinear vibrations of triple-walled CNTs buried within Pasternak foundation.

Critical velocities in fluid-conveying single-walled carbon nanotubes embedded in an elastic foundation

The problem of stability of fluid-conveying carbon nanotubes embedded in an elastic medium is investigated in this paper. A nonlocal continuum mechanics formulation, which takes the small length scale effects into consideration, is utilized to derive the governing fourth-order partial differential equations. The Fourier series method is used for the case of the pinned–pinned boundary condition of the tube. The Galerkin technique is utilized to find a solution of the governing equation for the case of the clamped–clamped boundary. Closed-form expressions for the critical flow velocity are obtained for different values of the Winkler and Pasternak foundation stiffness parameters. Moreover, new and interesting results are also reported for varying values of the nonlocal length parameter. It is observed that the nonlocal length parameter along with the Winkler and Pasternak foundation stiffness parameters exert considerable effects on the critical velocities of the fluid flow in nanotubes.

Wave propagation in fluid-conveying viscoelastic carbon nanotubes under longitudinal magnetic field with thermal and surface effect via nonlocal strain gradient theory

Based on nonlocal strain gradient theory, wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes (SWCNTs) is studied in this paper. With consideration of thermal effect and surface effect, wave equation is derived for fluid-conveying viscoelastic SWCNTs under longitudinal magnetic field utilizing Euler–Bernoulli beam theory. The closed-form expressions are derived for the frequency and phase velocity of the wave motion. The influences of fluid flow velocity, structural damping coefficient, temperature change, magnetic flux and surface effect are discussed in detail. SWCNTs’ viscoelasticity reduces the wave frequency of the system and the influence gets remarkable with the increase of wave number. The fluid in SWCNTs decreases the frequency of wave propagation to a certain extent. The frequency (phase velocity) gets larger due to the existence of surface effect, especially when the diameters of SWCNTs and the wave number decrease. The wave frequency increases with the increase of the longitudinal magnetic field, while decreases with the increase of the temperature change. The results may be helpful for better understanding the potential applications of SWCNTs in nanotechnology.

Critical Velocities in Fluid-Conveying Single-Walled Carbon Nanotubes Embedded in an Elastic Foundation

Critical Velocities in Fluid-Conveying Single-Walled Carbon Nanotubes Embedded in an Elastic Foundation

Vibrations of fluid-conveying inclined single-walled carbon nanotubes acted upon by a longitudinal magnetic field

This work deals with the influence of the longitudinal magnetic field on vibrations of inclined single-walled carbon nanotubes (SWCNTs) subjected to an inside fluid flow. Using an equivalent continuum structure model for the SWCNT and a plug-like model for the moving inside fluid flow, the nonlocal longitudinal and transverse equations of motion of magnetically affected SWCNTs are obtained in the context of small deformations. By application of the assumed-mode methodology, the displacements are discretized in terms of vibration mode shapes, and by exploiting generalized Newmark-β scheme, their corresponding time-dependent parameters are determined at each time. In the presence of the longitudinal magnetic field, the effects of the small-scale parameter, fluid flow velocity, and inclination angle on both longitudinal and transverse vibrations of SWCNTs are addressed. The obtained results reveal that the longitudinal magnetic field has fairly no effect on the longitudinal dynamic behavior of the nanostructure. However, maximum values of both transverse displacement and nonlocal bending moment of the fluid-conveying SWCNT would reduce as the strength of the magnetic field grows. Such a fact becomes more highlighted for high levels of the fluid flow velocity. The obtained results indicate that the longitudinal magnetic field can be exploited as an efficient way to control transverse vibrations of SWCNTs conveying fluids.

Wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes with surface and nonlocal effects

In this paper, the transverse wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes is investigated based on nonlocal elasticity theory with consideration of surface effect. The governing equation is formulated utilizing nonlocal Euler-Bernoulli beam theory and Kelvin-Voigt model. Explicit wave dispersion relation is developed and wave phase velocities and frequencies are obtained. The effect of the fluid flow velocity, structural damping, surface effect, small scale effects and tube diameter on the wave propagation properties are discussed with different wave numbers. The wave frequency increases with the increase of fluid flow velocity, but decreases with the increases of tube diameter and wave number. The effect of surface elasticity and residual surface tension is more significant for small wave number and tube diameter. For larger values of wave number and nonlocal parameters, the real part of frequency ratio raises.

Wave dispersion of carbon nanotubes conveying fluid supported on linear viscoelastic two-parameter foundation including thermal and small-scale effects

In this paper, for the first time, a nonlocal Timoshenko beam model is employed for studying the wave dispersion of a fluid-conveying single-walled carbon nanotube on Viscoelastic Pasternak foundation under high and low temperature change. In addition, the phase and group velocity for the nanotube are discussed, respectively. The influences of Winkler and Pasternak modulus, homogenous temperature change, steady flow velocity and damping factor of viscoelastic foundation on wave dispersion of carbon nanotubes are investigated. It was observed that the characteristic of the wave for carbon nanotubes conveying fluid is the normal dispersion. Moreover, implying viscoelastic foundation leads to increasing the wave frequencies.

Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory

The governing equation of wave motion of fluid-conveying viscoelastic single-walled carbon nanotubes is formulated on the basis of the nonlocal strain gradient theory and the Kelvin–Voigt viscoelastic model. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that, the effects of nonlocal parameters and small scale material parameters on the dispersion relation between the phase velocity and the wave number are significant at high wave numbers, however, may be ignored at low wave numbers. The upstream phase velocities decrease as increasing flow velocity, whereas the downstream phase velocities firstly increase as increasing flow velocity and then decrease as increasing flow velocity. The effect of damping coefficient on the phase velocity of both upstream and downstream waves is negligible at low wave numbers, however, is remarkable at high wave numbers.

Longitudinal magnetic field effect on wave propagation of fluid-conveyed SWCNT using Knudsen number and surface considerations

In this study, we considered the effects of nonlocal wave propagation on the interactions between single-walled carbon nanotubes (SWCNTs) and a viscous fluid under a longitudinal magnetic field based on the Euler–Bernoulli beam model. Carbon nanotubes were enclosed by a viscoelastic medium, which was simulated as a visco-Pasternak foundation. The longitudinal magnetic field was considered to be a longitudinal Lorentz force according to Maxwell's relations. In addition, the surface effect and Knudsen-dependent flow velocity were examined based on nonlocal elasticity theory. Nonlocal elasticity theory was utilized to derive the corresponding higher-order equations of motion based on Hamilton's principle, including the small-scale effects. The numerical results demonstrated that the longitudinal magnetic field, Knudsen number, small-scale parameters, viscoelastic medium, aspect ratio, fluid viscosity, and velocity of the fluid had significant effects on the wave propagation behavior of the fluid-conveying SWCNTs. In addition, increasing the longitudinal magnetic field caused the wave velocity to increase for different values of nonlocality. The results of this study may be beneficial for the fabrication of smart nano-structures that can be employed to transport fluidic drug to diseased areas, where a longitudinal magnetic field may help the viscous fluid to flow in a suitable stream.

Nonlinear vibration of fluid-conveying single-walled carbon nanotubes under harmonic excitation

In this paper, the nonlinear vibration of a single-walled carbon nanotube conveying fluid is investigated utilizing a multidimensional Lindstedt–Poincaré method. Considering the geometric large deformation of the single-walled carbon nanotube and external harmonic excitation force, based on nonlocal elastic theory and Euler–Bernoulli beam theory, the nonlinear vibration equation of a fluid-conveying single-walled carbon nanotube is established. Analyzing the equation through the multidimensional Lindstedt–Poincaré method, and from the solvability condition of the nonlinear vibration equation, the cubic algebraic equation which indicates the amplitude–frequency relation is obtained. Based on the root discriminant of the cubic equation, the first order primary response of the pinned–pinned carbon nanotube is discussed. The relations among internal resonance, the amplitude and frequency of the external excitation force are analyzed in detail. When the external excite force frequency is around the first mode natural frequency, the first mode primary resonance occurs. If simultaneously the first two modes natural frequency ratio is around 3, internal resonance occurs and the internal resonance region depends on the amplitude of external excitation force.

Terahertz wave propagation in a fluid-conveying single-walled carbon nanotube with initial stress subjected to temperature and magnetic fields

This paper studies terahertz wave propagation in a fluid-conveying single-walled carbon nanotube (SWCNT) under temperature and magnetic fields. The SWCNT is modelled as a Timoshenko beam based on the theory of nonlocal elasticity, where the nanoscale effects are only included in bending moment and shear force through a nonlocal parameter. The governing equations of motion are derived based on nonlocal Timoshenko beam theory. A wave analysis is carried out to get the equations of the dispersion characteristics of wave propagation. Numerical results confirm the validity of the present model by comparing the results in reduced cases with those reported in the published literature. The dispersion curves of wave propagation show that the initial stress plays a very important role on the shear and flexural frequencies of a fluid-conveying SWCNT. Meanwhile, the influences of the nonlocal parameter, fluid velocity, flow density, temperature change and magnetic field on the critical stress of a fluid-conveying SWCNT are discussed. This study may be useful for the design of smart nanodevices for the delivery of drugs to cells, carrying gases, and other applications of nanobeam devices.

Chaos in embedded fluid-conveying single-walled carbon nanotube under transverse harmonic load series

Stability and precision are important characteristics of the nanosyringe, a typical targeted drug delivery device. These characteristics of the nanosyringe are affected by the oscillating properties, especially by the chaotic behaviour resulting from the fluid–structure interaction effect of the fluid-conveying carbon nanotube. In this paper, the chaos in an embedded fluid-conveying single-walled carbon nanotube under transverse harmonic load series is investigated by the generalized multi-symplectic method which has been proved to be an accurate, efficient and long-time integration of infinite-dimensional nonconservative Hamiltonian systems. Considering the geometrical nonlinearity, the elastic medium restraint, the transverse harmonic load series as well as the fluid-conveying, the transverse oscillating model of the nanotube is presented firstly. And then, a generalized multi-symplectic scheme is constructed for the oscillating model to simulating the transverse oscillation of the nanotube. In the numerical experiments on the chaos in the nanotube, the following factors are considered: the length of the nanotube, the amplitude and the number of the transverse load series, the frequency and the initial phase of the transverse load series. From the numerical results, it can be found that, with the increase in the length of the nanotube and the increase in the number of the transverse load series, the chaotic region respected to the flow velocity decreases obviously, which implies that the stability and the precision of the nanotube are improved. These conclusions give some advice on the design of the nanosyringe, and the way that improves the stability and the precision of the nanosyringe in the process of work.

Thermal–mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

Based on the theories of thermal elasticity mechanics and nonlocal elasticity, an elastic Bernoulli–Euler beam model is developed for thermal–mechanical vibration and buckling instability of a single-walled carbon nanotube (SWCNT) conveying fluid and resting on an elastic medium. The finite element method is adopted to obtain the numerical solutions to the model. The effects of temperature change, nonlocal parameter and elastic medium constant on the vibration frequency and buckling instability of SWCNT conveying fluid are investigated. It can be concluded that at low or room temperature, the fundamental natural frequency and critical flow velocity for the SWCNT increase as the temperature change increases, on the other hand, while at high temperature the fundamental natural frequency and critical flow velocity decrease as the temperature change increases. The fundamental natural frequency for the SWCNT decreases as the nonlocal parameter increases, both the fundamental natural frequency and critical flow velocity increase as elastic medium constant increases.

Wave propagation of fluid-conveying single-walled carbon nanotubes via gradient elasticity theory

This paper initiates a theoretical analysis of wave propagation of fluid-conveying single-walled carbon nanotubes based on strain gradient elasticity theory with consideration of both inertia and strain gradients, in which two small-scale parameters are accounted for. For comparison purpose, the stress gradient theory for fluid-conveying carbon nanotubes is also discussed. Both theories are formulated using either the Euler–Bernoulli or the Timoshenko beam assumptions. It is found that the results predicted by these beam models are quite different. From a continuum-based point-of-view, the combined strain/inertia gradient Timoshenko beam model and its conclusion regarding wave propagation may be more reliable. Results show that the effect of internal fluid flow on the phase velocity of upstream-travelling wave is significant when the wave number is relatively low. However, this effect may be ignored when the wave number is sufficiently high. Moreover, the two small-scale parameters related to the inertia and strain gradients are shown to significantly affect the phase velocity at higher wave numbers. The present theoretical study highlights the significance of the effects of fluid flow and small scale related to inertia gradients on wave propagation in carbon nanotubes conveying fluid.

Response to “Comment on ‘Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory’ ” [J. Appl. Phys. 103, 024302 (2008)

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Free vibration of a single-walled carbon nanotube containing a fluid flow using the Timoshenko beam model

The flexural vibration of the fluid-conveying single-walled carbon nanotube (SWCNT) is derived by the Timoshenko beam model, including rotary inertia and transverse shear deformation. The effects of the flow velocity and the aspect ratio of length to diameter on the vibration frequency and mode shape of the SWCNT are analyzed. Results show that the effects of rotary inertia and transverse shear deformation result in a reduction of the vibration frequencies, especially for higher modes of vibration and short nanotubes. The frequency is also compared with the previous study based on Euler beam model. In addition, if the ratio of length to diameter increased to 60, the influence of the shear deformation and rotary inertia on the mode shape and the resonant frequencies can be neglected. However, the influence is very obvious when the ratio decreased to 20. As the flow velocity of the fluid increases in the vicinity of 2 π, the SWCNT reveals the divergence instability. It regains stability when the flow velocity reaches about 9. As the velocity increases further, the SWCNT undergoes a coupled-mode flutter and results in a larger amplitude.

The thermal effect on vibration and instability of carbon nanotubes conveying fluid

Based on the theory of thermal elasticity mechanics, an elastic Bernoulli–Euler beam model is developed for vibration and instability analysis of fluid-conveying single-walled carbon nanotubes (SWNTs) considering the thermal effect. Results are demonstrated for the dependence of natural frequencies on the flow velocity as well as temperature change. The influence of temperature change on the critical flow velocity at which buckling instability occurs is investigated. It is concluded that the effect of temperature change on the instability of SWNTs conveying fluid is significant.

Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory

The effects of flow velocity on the vibration frequency and mode shape of the fluid-conveying single-walled carbon nanotube are analyzed using nonlocal elastic theory. Results show that the frequency and mode shape are significantly influenced by the nonlocal parameter e0a/L. Increasing the nonlocal parameter decreases the real component of frequency and the decrease is more obvious for a lower flow velocity and a higher-order mode. In addition, a higher mode shape is observed with increasing the value of e0a/L. When a critical flow velocity is reached, the combination of first and second modes takes place. The mode shape for the combination is large relative to mode 3 due to the coupled frequency effect, especially including negative imaginary frequency. Furthermore, the mode shape of the combination increases as the nonlocal effect increases.

Free vibration analysis of fluid-conveying single-walled carbon nanotubes

The effect of fluid flow on the free vibration and instability of fluid-conveying single-walled carbon nanotubes is studied. The possibility of developing a technique to measure the mass flow rate of fluid is examined. Atomistic simulations and the continuum beam model are used. Simulations are performed to quantify the inertial, stiffness, Coriolis, and centrifugal forces generated by flow during the free vibration. A numerical expression is developed to measure the mass flow rate of the fluid velocities up to 40% of the critical flow velocity. This observation is useful to quantify the mass flow measurement of fluid conveying single-walled carbon nanotubes.