## Abstract

In this present study, the nonlinear thermal-magneto-mechanical stability and vibration of branched nanotube conveying nano-magnetic fluid embedded in linear and nonlinear elastic foundations are analyzed. The governing equations are established via Euler–Bernoulli theory, Hamilton’s principle, and the nonlocal theory of elasticity. The fluid flow and thermal behaviors of the nanofluid are described using modified Navier–Stokes and conservation of energy equations. With the aid of the Galerkin decomposition technique and differential transformation method (DTM), the coupled thermos-fluidic-vibration equation is solved analytically. The analytical solutions as presented in this study match with an existing experimental result and as such used to explore the influences of nonlocal parameters, downstream or branch angle, temperature, magnetic effect, fluid velocity, foundation parameters, and end conditions on vibrations of the nanotube. The results indicate that decreasing temperature change and augmenting the nanotube branch angle decreases the stability for the prebifurcation domain but increases for the post-bifurcation region. Furthermore, the magnetic term possesses a damping or an attenuating impact on the nanotube vibration response at any mode and for any boundary condition considered. It is anticipated that the outcome of this present study will find applications in the strategic optimization of designed nano-devices under thermo-mechanical flow-induced vibration.

## 1 Introduction

Immediately Iijima [1] flagged up carbon nanotube (CNT) discovery, studies related to geomorphological characteristics have been examined [2–4]. Establishments have been made that such structures have merits when applied to nano-devices such as transistors, sensors, and diodes. The structural chirality and applications of a Y-branched nanotube make it outstanding for three terminal applications [5–9]. The useful research works of Zhaoo et al. [10] and Kam and Dai [11] illustrated how nanotubes may be employed in the treatment of cancer and as scaffoldings for treating broken bones. Interesting analytical models have been presented for exploring the properties of these nanotubes [12–17] by employing the higher-order continuum theories, such as partial nonlocal elasticity, exact nonlocal elasticity, modified couple stress, strain gradient elasticity, and surface elasticity theory [18–21]. Lee and Chang [22] analyzed the influences of the velocity of flow in the SWCNTs on frequencies and mode shapes of vibration. In their work, the importance of the nonlocal term was established. Similar works have also been described with the consideration of slip and the use of Timoshenko beam’s theory [23–27]. The nonlinear governing equations generated by Asgharifard and Haeri derived used in application to surface effect treatment of graded nanobeams. A closed-form free vibration model for envisaging the surface integrity of fluid-CNTs was established by Wang [28]. He discovered that the surface effect is momentous for tubes with small thicknesses or with large aspect ratios. Different studies on the foundations where CNTs can rest or be embedded in have been reported. In fact, these foundations have been modeled to either behave in a linear or nonlinear manner [5,29–33]. Yinusa and Sobamowo [34] used a closed-form integral transform method to analyze the vibration of an SWCNT under pressurized function and thermal conditions. The vibrational frequencies obtained from the eigenvalues in their study were used for establishing stability criteria. They proceeded to ascertain dynamic responses using the deflection solution of the nanotube investigated. The dynamic response solution was obtained up to the fourth mode with proper verification. They resolved that increasing foundation and pressure terms increase nanotube’s frequency while that of the mass possesses an antonymous effect. Ouyang used carbon nanotubes and graphene for the optimization of solar and fuel cells [35]. In the study, they adopted the tremendous properties associated with these nano-materials to provide a short review on how graphene, carbon nanotubes, and other nano-devices may find application in DSSCs PSCs, OSCs, and fuel cells. Gou et al. [36] recently presented a study on how additive manufacturing or 3D printing may be used in the generation of nano-material-based composites like graphene. They employed fusion-based techniques with a selectively acquired laser sintering to outline the process efficiently and effectively. Shuang et al. [37] analytically obtained a stress field solution for a finite nano-circular inhomogeneous matrix and concluded that the obtained matrix inhomogeneity can be influenced by the size, elastic properties, and matrix type. Furthermore, Xie et al. offered a novel tactic for fabricating aligned nanotubes while Smith et al. presented different means of synthesizing graphene oxide in the reduced form [38,39]. Meanwhile, the widespread applications of nanotubes have been justified via different experimental and mathematical modeling research works [40–42]. Motivated by the aforementioned deliberations, this present study scrutinizes the nonlinear thermal-magneto-mechanical vibrations as well as stability responses of branched nanotube conveying nano-magnetic fluid (NMF) and embedded in linear and nonlinear Winkler and Pasternak elastic foundations using Galerkin decomposition procedure and differential transform method. Parametric studies were carried out using the analytical solution developed. Subsequently, the effects of model parameters on the thermal-fluidic and thermal-magneto-mechanical vibration of the carbon nanotube are investigated and the results are presented graphically.

## 2 Models Development for Nanotube’s Vibration

Modeling the thermal-mechanical behavior of fluid-conveying structures generates nonlinear equations whose closed-form solutions are difficult to realize. In order to generate these solutions symbolically, semi or approximate analytical approaches are employed with accuracies depending to a large extent on the number of terms used in the obtained series solutions. Here, governing mathematical models for the nonlinear vibrations are developed, analyzed, and validated.

### 2.1 Basic Formulation.

Consider a branched nano-magnetic fluid conveying SWCNT subjected to an external pretension, global pressure, slip boundary conditions, and two-dimensional magnetic field as shown in Fig. 1. The different end shapes realized due to variation in the center angle of the downstream are provided in Table 1.

### 2.2 Interactions Between the Nano-Magnetic Fluid and the Surrounding Two-Dimensional Magnetic Field.

This interaction may be modeled using Maxwell’s equation as

The applied Lorentz force on the CNT can be written as

The effective force on CNT from the magnetic field will be from the Lorentz force and the generated moment expressed as

*a*)

*b*)

### 2.3 Influence of Slip Conditions on Fluid Velocity Profile in the SWCNT.

To solve the problem of NMF flows, the modified Navier–Stokes equation for two-component mixtures which includes two mass equations, one momentum equation, and one energy equation may be invoked as shown below:

The subsequent assumptions are used for the thermos-fluidic model development

The gradient of nanoparticles volumetric fraction is independent of time.

Neglecting the diffusion mass flux term

Fluid is Newtonian

Incompressible fluid flow

Uniform magnetic field

Low dilute mixture

Neglecting the transient and inertia terms.

*a*)

*b*)

*a*)

*b*)

*a*)

*b*)

*U*

_{PI}=

*δ*and substitute into (19

*b*) to obtain

*A*and replacing

*μ*

_{eff}=

*μ*

_{b}/(1 +

*a*Kn) and $\xi =\sigma Bl2$,

*λ*, there is a need to recall the two basic boundary conditions related to slip and nonslip as

### 2.4 Influence of Nano-Magnetic Fluid at Nanotube Junction.

*Q*

_{2}=

*Q*

_{3}and

*Q*

_{2}=

*Q*

_{1},

*U*

_{2}/

*U*

_{1}= cos

*ϕ*, and

*U*

_{i}/

*U*= Γ, axial force becomes

### 2.5 Influence of Foundation on the SWCNT.

The reaction from the foundation which directly influences the dynamics of the SWCNT may be expressed as any of the following based on where the CNT is to be embedded:

### 2.6 Derivation of the Equations of Motion.

*variation of kinetic energy*of the SWCNT, we have

*a*)

*b*)

*c*)

*d*)

*Influence of the two-dimensional magnetic field*:

*Influence of the nano-magnetic fluid*:

*Foundation effect*:

*δw*to zero generates an equation of motion as

The expressions for *α*_{1}, *α*_{2}, *α*_{3}…*α*_{16} are shown in the Appendix.

## 3 Temporal Solution by Differential Transform Method (DTM)

The nonlinear equation in Eq. (78) is to be solved via DTM. The definition and operational principles of DTM are already established in our previous work [41]. Using DTM, Eq. (78) transforms into

*p*= 0, the lowest that can be obtained from Eq. (80) is

*U*[2], it implies that the initial conditions will also need to be transformed to obtain

*U*[0] and

*U*[1] for the subsequent result. Applying DTM to the initial conditions, we have

Similarly, the deflection for other support using a similar table will be as below:

*clamped–pinned supported nanobeam*:

*clamped–clamped supported nanobeam*:

*clamped–free (cantilever) supported nanobeam*:

The above solutions may be employed for practical applications. However, they are in truncated series. This truncated series is periodic in a small region. To capture a large range and increase the accuracies of the solutions, after-treatment techniques (SAT and CAT) are applied. The applications of the technique can be found in our previous works [42,43].

## 4 Result and Discussion

Concentrating on the physics of the problem and performing proper parametric studies, the following results are obtained.

### 4.1 Effect of Modal Number and End Conditions on Nanotube’s Mode Shape.

Figures 2(a)–2(c) depict the effects of mode number and the boundary conditions associated with the spatial on the nonlinear mode shape of the nanotube. The figures display nanotube’s deflection along the dimensionless beam length for the first five mode shapes. Assessment of the plots illustrates how an increase in mode number reduces nanotube’s stability for the same beam length. These occur because the trial function is contingent on the mode number. The supports associated with Figs. 2(a)–2(c) present responses with mode shape starting and ending at the same base point for any modal number considered.

### 4.2 Developed Equivalent Trial Function for Hyperbolic–Trigonometric Function.

The comparison or trial functions that are normally used to represent the spatial part of the deflection are in hyperbolic–trigonometric form. As a result, differentiating, integrating as well as obtaining the roots of these terms become tedious, rigorous, and time-consuming, hence the need for an equivalent polynomial as depicted in Table 2. The verification of the equivalent polynomial functions for the different boundary conditions is as shown below:

Figures 3(a)–3(d) depict the validity of the polynomial functions as there is a good agreement between the polynomial functions and the complex hyper-trigonometric functions. Hence, in this work, the polynomial functions will be used instead of the hyperbolic–trigonometric functions to represent the spatial part.

### 4.3 Impact of Boundary Condition on Nonlinear Dimensionless Amplitude-Dimensionless Frequency Response of the Nanotube.

Figure 4 depicts the impact of the end condition on the frequency ratio of the nanotube with the dimensionless maximum amplitude of the system. The result shows that a nanotube with clamped–free (cantilever) supports has the highest frequency ratio while clamped–clamped gives the lowest. The highest frequency ratio associated with the clamped–free support is due to the other end being free and because of the lower stiffness of the tube than the other boundary conditions. This makes the clamped–free beam deviate from linearity faster than the other boundary conditions.

### 4.4 Influence of Branch Angle on Nanotube’s Stability.

Figures 5(a)–5(f) describe the effect of the downstream angle which results into different end shape nanotubes on the curve of dimensionless frequency against dimensionless flow velocity. The figures show that frequency first decreases parabolically for increasing velocity for both linear and nonlinear foundations which implies that the nanotube is stable in this domain. After that, a critical dimensionless velocity *U*_{c} is reached at which the nanotube immediately becomes unstable due to bifurcation. The critical dimensionless velocity is in the range of 2.7–3.2 for the linear analysis and 4.2–5.15 for the nonlinear case. This implies that the stability region for the nonlinear case is more than that of the linear. The bifurcation region for the nonlinear analysis is shorter than that of the linear, this means that the nanotube subjected to nonlinear analysis restabilizes faster than the linear. As the flow velocity is increased above the critical dimensionless velocity, the system continues its instability because the frequency is zero throughout the region. Subsequently, a velocity *U*_{s} is attained where the system regains its stability for a while and after that, a continuous increase in velocity will result in divergence and may even cause flutter.

### 4.5 Influence of Mass Ratio Term on Nanotube’s Stability.

Figures 6 and 7 depict the effect of *β*_{f} on frequency and the dimensionless damping frequency of nanotubes for varying dimensionless flow velocity. For Fig. 6, the effect is obvious at a higher value of dimensionless flow velocity. However, the nanotube’s stability and velocity of flow are decreased as *β*_{f} increases at this region. Figure 7 illustrates a means of obtaining damping values. It is shown that the critical value of *β*_{f} is 0.40 because an incessant increase in *β*_{f} above this limit results in an effect analogous to when values lower than *β*_{f} = 0.4 were used.

### 4.6 Impact of Foundation Coefficient on Nanotube’s Stability.

Figures 8(a)–8(d) portray the impact of the foundation on the nanotube’s stability. It is shown from the plots that an increase in Pasternak and Winkler foundation coefficients results in a corresponding increase in nanotube’s stiffness. Hence, the system’s frequency becomes higher. It is also important to know that the nonlinear Pasternak and Winkler constants generate higher stability than the corresponding linear Winkler and Pasternak constants.

### 4.7 Influence of Fluid Velocity on Frequency Ratio-Amplitude Plots of Different End-Shaped Nanotube Embedded in Foundations.

Figures 9(a)–9(h) and 10(a)–10(h) describe the impact of fluid velocity on stability curves of the different end shape nanotubes embedded in Pasternak and Winkler foundations. It is realized that an augmentation in flow velocity augments the frequency ratio and the plot shifts away from linearity. Analyzing different end shape nanotubes by varying the downstream angle shows that there is a limit to this angle. The limits are observed to be 79 deg for Winkler and 83 deg for the Pasternak foundations. When the magnetic effect is considered, the defect is immediately annulled and a T-shape nanotube can then be generated without instability at moderate flow velocity. Figures 11(a) and 11(b) show the influences of end shapes on stability curves without and with magnetic fields.

### 4.9 Effect of Tension on Nanotube’s Stability Curve.

Figure 12 portrays the effect of axial tension on the nanotube’s stability curve. It is noticed that an intensification in the tension term possesses analogous characteristics as flow velocity as it upsurges the frequency ratios and the plots automatically adjust from linearity. This demonstrates that the axial tension and flow velocity may be employed for adjusting and controlling nanotube’s nonlinearity.

### 4.10 Impact of Nonlocal Terms on Nanotube’s Stability for Pre- and Post-Buckling.

Figures 13(a) and 13(b) describe the effect of nonlocal terms on the nanotube’s stability for prebuckling analysis while Fig. 14 depicts for the pre- and post-buckling analysis. From the stability and dynamic analyses of the nanotube, it is practical that as the nano-term increases, the dimensionless frequency and dimensionless velocity decrease. The highest frequency is attained with nano-term of zero magnitude because the result obtained at this value represents the classical Euler–Bernoulli model. A critical look at Fig. 14 shows that for the post-buckling region, the classical Euler–Bernoulli model becomes unstable after bifurcation and did not regain its stability. Whereas when the nano-parameter is incorporated, the system regains its stability at an average flow velocity of 2.00, hence the need for nonlocal analysis.

### 4.11 Thermal Effects on Nanotube’s Frequency Without and With Magnetic Effect.

Figures 15(a)–15(d) show the temperature change influences on the frequency of nanotubes embedded in foundations with and without magnetic effect. Figures 16(a)–16(d) illustrate the influences of change in temperature on nanotube’s frequency embedded in Pasternak and Winkler foundations with and without magnetic effect. It is shown that dimensionless frequency and dimensionless velocity increase with increasing temperature change. When the magnetic effect is considered, the nanotube’s stability increases because the system’s frequency reduces for the same flow velocity range as that when there is no magnetic effect.

### 4.12 Influence of Shear Modulus on Nanotube’s Stability.

Figures 17(a) and 17(b) illustrate the influences of shear modulus on the frequency and stability of the nanotube with and without magnetic properties. From the plots obtained, it is noticed that as the shear modulus term increases, the dimensionless frequency and flow velocity increase for the prebifurcation analysis. However, when the magnetic property is introduced, the dimensionless frequency reduces while the flow velocity range is maintained. This implies an augmentation in the system’s stability.

### 4.13 Impact of Knudsen Number on Nanotube’s Stability.

Figure 18 shows the impact of Kn on the nanotube’s stability. Dimensionless natural frequencies against average flow velocity for different Knudsen numbers are plotted in Fig. 18. It is seen that *U _{c}* reduces as Kn increases. Hence, nanotube’s frequencies are meaningfully affected by the Kn number hence the need to introduce it in the correction of the flow field due to the existence of the magnetic term. Also, the small-scale influence of the flow field on the stability of the nanotube cannot be ignored.

### 4.14 Frequency Against Fluid Velocity for Different Modes.

Figures 19(a)–19(f) depict the effect of *β*_{f} on the nanotube’s stability for varying dimensionless flow velocity, different boundary conditions, and for different modes. For Figs. 19(a)–19(c), the impact of mass ratio is obvious at a higher value of dimensionless flow velocity for the considered first two modes. Nevertheless, the system’s stability and flow velocity decrease as *β*_{f} increases at this region.

### 4.15 Linear and Nonlinear Dynamic Responses of the Carbon Nanotube.

Figure 20 depicts the assessment of linear and nonlinear dynamic responses of the carbon nanotube. The deviation from linearity is due to the nonlinear term present in the Duffing equation used in the determination of the nonlinear frequency and frequency ratio. Furthermore, this deviation from linearity is found to increase as the maximum dynamic vibration increases.

### 4.16. Effects of Magnetic Field on Nanotube’s Dynamic Responses.

Figures 21(a)–21(d) and 22(a)–22(d) depict the influences of magnetic terms on nanotube’s dynamic responses for mode 1 and mode 2. The responses obtained for mode 2 were found to be out of phase by 180 deg with those of mode 1. Correspondingly system’s dynamic responses were observed to decay as the magnetic effect increased. This implies that the magnetic property has a hindering impact on the nanotube’s dynamic response.

### 4.17. Effect of Change in Temperature on Nanotube’s Dynamic Response.

Figures 23(a)–23(d) depict the effect of change in temperature on the nanotube’s dynamic response. As shown in Fig. 23(a), the change in temperature did not have an evident influence on the dynamic response of the system except when it is high. However, when the magnetic effect is included, the responses begin to dampen out as the magnetic term increases.

### 4.18. Effect of Foundation Parameter on Deflection for Low and High Temperature Change.

Figures 24(a) and 24(b) depict the effect of nonlinear Winkler, linear Winkler, and Pasternak foundation parameters on the nanotube’s deflection for low- and high-temperature change. For the two cases of temperature change, the Pasternak foundation parameter is observed to give a better attenuation than the Winkler foundation parameter which means it will find its application in the accommodation of higher modes.

### 4.19 Model Validation.

Figure 25 depicts the contrast of experimental outcomes obtained by Dodds and Runyans [40] with the results from the present study solution. It has been discussed earlier that when the nano-term (e_{o}a) = 0, the system and all possible results obtained are said to be governed by the classical Euler–Bernoulli model. With this classical approach, Dodds and Runyans [40] performed and presented experimental studies to determine the influence of high flow velocity on bending vibration as well as on the divergence of a pinned–pinned support pipe. This experimental result was employed in validating the analytical approach used in this study. As can be seen from Fig. 25, the analytical solutions presented in this study match with the experimental results reported by Dodds and Runyans [40], hence the validation of this work.

## 5 Conclusion

In this work, nonlinear thermal-mechanical vibration and stability analyses of branched nanotubes conveying nano-magnetic viscous fluid embedded in linear and nonlinear foundations under the magnetic influence have been investigated. The coupled thermal-fluidic or thermal-magneto-mechanical vibration equations are solved using Galerkin decomposition techniques and DTM with the after-treatment technique. After parametric studies, the following conclusions are established:

The downstream angle significantly affects the nanotube’s stability. As the angle of the nanotube is increased from 0 to π/2, the system’s stability decreased.

The magnetic field term has an attenuating impact on the nanotube’s dynamic responses.

When Kn or hydrodynamic slip parameter is increased, the critical fluid velocity and system’s frequency decrease.

The frequency and flow velocity are meaningfully influenced by nonlocal terms. Also, shear modulus and foundation parameters increase the system’s natural frequency.

The fluid–structure mass ratio became significant at the post-bifurcation region while the frequency and velocity increased with increasing temperature change.

Alteration of nonlinear flow-induced vibration frequencies from the linear equivalents is momentous as amplitude and flow velocity increase.

When the axial pretension on the nanotube is increased, the system’s stability decreased as it deviates rapidly from linearity.

## Conflict of Interest

There are no conflicts of interest. All procedures performed for studies involving human participants were in accordance with the ethical standards stated in the 1964 Declaration of Helsinki and its later amendments or comparable ethical standards. Informed consent was obtained from all participants. Documentation provided upon request. Informed consent was obtained for all individuals. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*w*=deflection of the SWCNT

- $t*$ =
dimensionless time

- $u~$ =
axial displacement component in the middle surface

- $w~$ =
transverse displacement component in the middle surface

- $w\u02d9$ =
derivative with respect to time

*w*″ =derivative with respect to spatial variable

*B*=2D magnetic field

*G*=shear modulus

*K*=spring stiffness

*M*=mass

*T*=axial pretension

*m*_{j}=mass of the junction

*I*_{0}=Bessel term

*M*_{f}=mass of fluid

*R*_{i}=radius of CNT

*I*^{CNT}=moment of area of SWCNT

- $qBCNT$ =
Lorentz force on SWCNT

- $Atf$ =
area of fluid in tube

- $MxCNT$ =
moment on SWCNT

- Kn =
Knudsen number

*U*(*r*) =fluid velocity distribution

- (
*e*_{o}*a*) = nonlocal parameter

*ρ*_{f}=fluid density,

- $\rho jf$ =
density of fluid in junction

- $\rho tf$ =
density of fluid in tube

*σ*_{v}=tangential momentum

- $\u2207$ =
gradient