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 [24]. 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 [59]. 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 [1217] 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 [1821]. 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 [2327]. 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 [5,2933]. 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 [4042]. 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.

Fig. 1
(a–g) Possible shapes of CNT
Fig. 1
(a–g) Possible shapes of CNT
Close modal
Table 1

Different end shapes realized due to variation in the center angle of the downstream

Downstream angle (ϕ deg)Shape of CNT
a0I-shaped CNT
b15V or 15 deg Y-shaped CNT
c30V or 30 deg Y-shaped CNT
d45L or 45 deg Y-shaped SWCNT
e60Y or 60 deg Y-shaped SWCNT
f75K or 75 deg Y-shaped SWCNT
g75 ˂ ϕ ˂ 90T-shaped SWCNT
Downstream angle (ϕ deg)Shape of CNT
a0I-shaped CNT
b15V or 15 deg Y-shaped CNT
c30V or 30 deg Y-shaped CNT
d45L or 45 deg Y-shaped SWCNT
e60Y or 60 deg Y-shaped SWCNT
f75K or 75 deg Y-shaped SWCNT
g75 ˂ ϕ ˂ 90T-shaped SWCNT

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

B(i,0,0)=(Bocosα)i
(1)
B(0,0,k)=(Bosinα)k
(2)
qBCNT=××(u~×Bo)×Boμp
(3)
while the two-dimensional magnetic field is defined as
B(i,0,k)=(Bocosα)i+(Bosinα)k

The applied Lorentz force on the CNT can be written as

q~BCNT=AqBCNTdA
(4)
The applied force’s moment will be
MxCNT=A(qBCNTz)dA
(5)

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

q~BTCNT=q~BCNT+MxCNT
(6)
making necessary substitutions,
q~BTCNT=2[Bo2cos2αμpwAdA]+2[Bo2sin2αμp2wAz2dA]
(7a)
q~BTCNT=[Bo2cos2αμp(AdA)]2w+[Bo2sin2αμp(Az2dA)]4w
(7b)
Introducing AdA=A and Az2dA=ICNT, Eq. (7) becomes
q~BTCNT=[Bo2cos2αAμp]2w[Bo2sin2αICNTμp]4w
(8)
If the magnetic effect in the transverse direction is neglected, the moment term will varnish and the total force on the SWCNT will become
q~BTCNT=[Bo2cos2αAμp]2w
(9)

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:

Continuity equation:
U=0
(10)
ψ˙+Uψ+εnpρnp=0
(11)
Momentum equation:
ρfU˙=P+μeff2U+(J×B)
(12)
Energy equation:
ρc(T˙+vT)=hεnp
(13)

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.

Applying the assumptions above, Eqs. (10)(13) becomes

Continuity equation:
U=0
(14)
Uψ=0
(15)
Momentum equation:
P+μeff2U+(J×B)=0
(16)
Energy equation:
ρc(vT)=hεnp
(17)
For the velocity solution, the momentum equation will be solved with slip conditions as expressed below:
P=μeff2U+(J×B)whereJ=σ(U×B)
(18a)
1μeffP=2U+σB2μeffU
(18b)
For the cylindrical coordinate chosen,
U=U(r)and2U=2U(r)r2+1rU(r)r
so Eq. (18) becomes
1μeffP=2U(r)r2+1rU(r)rσBl2μeffU(r)
(19a)
which on re-arranging yields
2U(r)r2+1rU(r)rσBl2μeffU(r)=1μeffP
(19b)
with slip boundary conditions:
r=0,U(r)r=0
(20a)
r=Ri,U(r)r=λU(r)
(20b)
where
λ=f(Kn,σ,Ri,b)
Kn number is introduced through imposed slip conditions to correct the velocity profile of the fluid due to the two-dimensional magnetic field. The solution to Eq. (19) will be
U(r)=UCF+UPI
(21)
For the complimentary part,
2U(r)r2+1rU(r)rσBl2μeffU(r)=0
(22)
Since the velocity profile obeys Bessel’s function when the magnetic field is effective, the solution to Eq. (22) will be of the form
UCF=AI0(σBl2μeffr)+CK0(σBl2μeffr)
(23)
For the particular integral, let UPI = δ and substitute into (19b) to obtain
1μeffP=2UPIr2+1rUPIrσBl2μeffUPI
(24)
which on simplifying gives
1μeffP=2r2(δ)+1rr(δ)σBl2μeff(δ)
(25)
and
δ=UPI=1σBl2P
(26)
Put (25) and (26) into (23) to get the generalized solution for the velocity profile as
U(r)=AI0(σBl2μeffr)+CK0(σBl2μeffr)1σBl2P
(27)
Using the specified slip conditions together with Eq. (19),
C=0,A=PσBl2(1I0(σBl2μeffRi)+1λσBl2μeffI1(σBl2μeffRi))
(28)
Substitute for A and replacing μeff = μb/(1 + aKn) and ξ=σBl2,
U(r)=Pξ(1I0((1+aKn)ξμbRi)+1λ(1+aKn)ξμbI1((1+aKn)ξμbRi))I0((1+aKn)ξμbr)Pξ
(29)
since P/ξ is a common term, grouping like terms will result into a model that can be used to determine the velocity profile for any boundary condition specified
U(r)=(I0((1+aKn)ξμbr)I0((1+aKn)ξμbRi)+1λ(1+aKn)ξμbI1((1+aKn)ξμbRi)1)Pξ
(30)
To determine 1/λ, there is a need to recall the two basic boundary conditions related to slip and nonslip as
Noslip:Ur=Ri=0
(31)
Slip:Ur=Ri=Ri(2σvσv)(Kn1bKn)(Ur)|r=Ri
(32)
which can also be written as
Ur|r=Ri=U[Ri(2σvσv)(Kn1bKn)]=λU
(33)
where
1λ=Ri(2σvσv)(Kn1bKn)
Put (27) into (28) to obtain the (nano-magnetic fluid) NMF velocity distribution in the SWCNT as
U(r)=(I0((1+aKn)ξμbr)I0((1+aKn)ξμbRi)+Ri(2σvσv)(Kn1bKn)(1+aKn)ξμbI1((1+aKn)ξμbRi)1)Pξ
(34)

2.4 Influence of Nano-Magnetic Fluid at Nanotube Junction.

Considering the control volume for fluid flow as depicted in Fig. 1. The forces applied to the fluid along the transverse and axial directions can be obtained by applying momentum balance at the inlet and exits of the control volume as analyzed below:
Fx=ρfQ1U1ρfQ2U2ρfQ3U3
(35)
Since the two exits have equal discharge, Q2 = Q3 and
Fx=ρfQ1U12ρfQ2U2
(36)
Also using the following relations, 2Q2 = Q1, U2/U1 = cos ϕ, and Ui/U = Γ, axial force becomes
Fx=ρfQ1U1ρfQ1U2Fx=ρfQ1(U1U2)Fx=ρfAfU1(U1U2)Fx=MfU12(1U2U1)Fx=Mf(ΓU)2(1cosϕ)
(37)
Similarly,
Fy=ρfQ2U2+ρfQ3U3
(38)
Fy=ρfQ2U1sinϕρfQ3U1sinϕ=0
(39)

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:

Linear Winkler:
qlw=kww
(40)
Nonlinear Winkler:
qnlw=kww+k3w3
(41)
Linear Pasternak:
qlp=kwwG2w+cw˙
(42)
Nonlinear Pasternak:
qnlp=kww+k3w3G2w+cw˙
(43)

2.6 Derivation of the Equations of Motion.

Several variational methods can be employed for deriving the governing differential equations of an elastic body. The principle of minimum potential energy, minimum complementary energy, and stationary Reissner energy can be used to formulate static problems. The variational principle valid for dynamics of systems of particles, rigid bodies, or deformable solids is called Hamilton’s principle where the variation of the function is carried out with respect to time. Consider an SWCNT conveying nano-magnetic fluid, subjected to a two-dimensional magnetic effect and supported with elastic foundations (Winkler and Pasternak foundations) under externally applied tension, global pressure, and slip boundary conditions as shown in Fig. 1. Using Eringen’s nonlocal theory, Euler–Bernoulli beam theory, and Hamilton’s variational principle, equation of motion and boundary conditions can be derived. Starting with Hamilton’s principle which could be enumerated as
0t(δuTδkTδvT)dt=0
(44)
The strain energy variation of the SWCNT is simply
δuT=12VtσxxδεxxdVt=120LAtσxxδεxxdxdAt
(45)
but εxx=zw, substitute into Eq. (42) and simplify to obtain
δuT=120LAtσxxδ(zw)dxdAt=120L[z(σxxAt)δw]dx=120L[Mxx2(δw)]dx
(46)
Therefore,
δuT=120L[Mxx2(δw)]dx
(47)
For the variation of kinetic energy of the SWCNT, we have
δkT=δkSWCNT+δkSWCNTf+δkJUNCTION+δkJUNCTIONf
(48)
where
δkSWCNT=12Vtρt[((δu~)t)2+((δw~)t)2]dV
(49a)
δkSWCNTf=120LAtρt[((δu~)t)2+((δw~)t)2]dxdAt
(49b)
δkJUNCTION=120Lmjδ(δ(xL)[(u~t)2+(w~t)2])dx
(49c)
δkJUNCTIONf=120Lmjfδ[(ΓU)2+(wt+ΓUwx)2]δ(xL)dx
(49d)

Substitute Eqs. (49a) and (49b) into Eq. (48),

δkT=(120LAtρt[((δu~)t)2+((δw~)t)2]dxdAt+120LAtfρtfδ[(ΓU)2+(wt+ΓUwx)2]dxdAtf+120Lmjδ(δ(xL)[(u~t)2+(w~t)2])dx120Lmjfδ[(ΓU)2+(wt+ΓUwx)2]δ(xL)dx)
(50)
Similarly, to obtain the variation in the total virtual work on SWCNT, we have
δvT=δvSWCNTvf+δvJUNCTIONvf+δvFIELDm+δvNMF+δvFOUNDATION
(51)
where
δkSWCNTvf=0L[μeffAtvf2(wt+ΓUwx)δw]dx
(52)
δvJUNCTIONvf=0L[(Mf(ΓU)2[1cosϕ]+PATKPEAαΔθ12v)2wx2]δwdx
(53)
If the effect of temperature change, pretension, global pressure, and Pasternak constant are ignored, Eq. (53) becomes
δvJUNCTIONvf=0L[Mf(ΓU)2(1cosϕ)2wx2]δwdx
(54)
Influence of the two-dimensional magnetic field:
δvFIELDm=0L([Bo2cos2αAtμp]2w[Bo2sin2αICNTμp]4w)δwdx
(55)
Influence of the nano-magnetic fluid:
δvNMF=12Vf(JBcosα)δwdVf
(56)
where
J=σ(U×B)=σ(UzBcosα)
and
Uz=wt+ΓUwx
Therefore,
δvNMF=0L[σBo2cos2αAf(wt+ΓUwx)]δwdx
(57)
Foundation effect:
δvFOUNDATION=0L(kww+k3w3G2w+cw˙)δwdx
(58)
Total virtual work on the SWCNT after grouping becomes
δvT=0L(μeffAtvf2(wt+ΓUwx)+(Mf(ΓU)2[1cosϕ]+PATKPEAαΔθ12v)2wx2+[Bo2cos2αAtμp]2w[Bo2sin2αICNTμp]4wσBo2cos2αAf(wt+ΓUwx)+kww+k3w3G2wx2+cwt)δwdx
(59)
Substitute Eqs. (47), (50), and (59) into Eq. (44), integrating by part and putting a coefficient of δw to zero generates an equation of motion as
2Mx+Bo2sin2αICNTμpwivμeffAfΓUw+(mjf(ΓU)2δ(xL)Bo2cos2αAtμp+Mf(ΓU)2cosϕG+PATKPEAαΔθ12v)wΓUw+(mt+Mf+[mj+mjf]δ(xL))w¨+(c+σBo2cos2αAf)w˙(ρfIf+ρfICNT)w¨μeffAfw˙+2ΓU(mt+mjfδ(xL))w˙++kww+k3w3=0
(60)
Based on the nonlocal Euler–Bernoulli beam theory, the nonlocal stress is
σijnl(x)=Vα(|xx|,τ)σijldV(x)
(61)
Expressing the nonlocal stress as a local one, we have
σxnlμ2σxnlx2=σxl=zEw
(62)
simplifying Eq. (62)
AtσxnlzdAtAtμ2(σxnlz)x2dAt=EwAtz2dAt
(63)
and
Mx(eoa)22Mxx2=EICNTw
(64)
Then
2Mx=2[(eoa)22Mxx2EICNTw]
(65)
After the differentiation, we have
2Mx=(eoa)26wx2EICNTwiv
(66)
which on re-arranging yields
2Mx=(eoa)2wviEICNTwiv
(67)
By making a necessary substitution, the equation becomes
(eoa)2wvi+(EICNT+Bo2sin2αICNTμp)wivμeffAfΓUw+(mjf(ΓU)2δ(xL)Bo2cos2αAtμp+Mf(ΓU)2cosϕG+PATKPEAαΔθ12v)wΓUw+(mt+Mf+[mj+mjf]δ(xL))w¨+(c+σBo2cos2αAf)w˙(ρfIf+ρfICNT)w¨μeffAfw˙+2ΓU(mt+mjfδ(xL))w˙+kww+k3w3=0
(68)
With vibration initial conditions of nanotube given as
w(x,0)=w0,w(x,0)x=0
(69)
For pinned–pinned supported nanotube,
w(0,t)=0,2w(0,t)2x=0,w(L,t)=0,2w(L,t)2x=0
(70)
For clamped–clamped supported nanotube,
w(0,t)=0,w(0,t)x=0,w(L,t)=0,w(L,t)x=0
(71)
For a clamped–pinned supported nanotube,
w(0,t)=0,w(0,t)x=0,w(L,t)=0,2w(L,t)x2=0
(72)
For a clamped–free (cantilever) supported nanotube,
w(0,t)=0,w(0,t)x=0,2w(L,t)x2=0,3w(L,t)x3=0
(73)
To obtain the equation in dimensionless form, we introduce the resulting dimensionless parameters which include
w*=xL,x*=xL,(eoa)*=(eoa)L2,t*=tL2EICNTmt+Mf,βT=mj+mjfL(mt+Mf)β=Mfmt+Mf,Mf*=mjfLMf,βjf=mjfLMf[(mt+Mf)],T*=TEA,P*=PE,Δθ*=αΔθ12vU*=ΓULMfEICNT,μeff*=μeffAfEICNTMf,c*=cL2EICNT(mt+Mf),σ=ρfIf+ρfICNTL2(mt+Mf)Bt=BoLAtEICNT(eoa),Bf=BoLσAfEICNTMf,G*=GL2EICNT,kw*=kwL4EICNT
(74)
Applying Eq. (74) in Eq. (68), the developed dimensionless equation of motion can be expressed as
α1w*vi+α2w*v+α3w¨*iv+α4w˙*iv+α5w*iv+α6w*+α7w˙*+α8w*+α9w˙*+α10w¨*+α11w˙*+α12w¨*+α13w˙*+α14w¨*+α15w*+α16w*3=0
(75)
Applying the Galerkin decomposition procedure to separate the spatial and temporal parts as
w*(x*,t*)=Ψ(x*)T(t*)
(76)
where T(t*) is the generalized coordinate and Ψ(x*) is a trial/comparison function that will satisfy the boundary conditions. Using the one-parameter Galerkin’s solution as shown below in Eq. (75),
0LR(x*,t*)Ψ(x*)dx
(77)
where
R(x*,t*)=α16w*x*6+α25w*x*5+α36w*x*4t*2+α45w*x*4t*+α54w*x*4+α63w*x*3+α74w*x*3t*+α82w*x*2+α93w*x*2t*+α104w*x*2t*2+α112w*x*t*+α123w*x*t*2+α13w*t*+α142w*t*2+α15w*+α16w*3

The expressions for α1, α2, α3α16 are shown in the  Appendix.

The Galerkin method yields the following Duffing equation for nonlinear analysis
MU¨(t*)+CU˙(t*)+KU(t*)+VU3(t*)=0
(78)
where
M=0L[α3Ψ4Ψx*4+α10Ψ2Ψx*2+α12ΨΨx*+α14Ψ2]dxC=0L[α4Ψ4Ψ*x*4+α7Ψ3Ψ*x*3+α9Ψ2Ψ*x*2+α11ΨΨ*x*+α13Ψ2]dxK=0L[α1Ψ6Ψ*x*6+α2Ψ5Ψ*x*5+α5Ψ4Ψ*x*4+α6Ψ3Ψ*x*3+α5Ψ2Ψ*x*2+α15Ψ2]dxandV=0L[α16Ψ4]dx

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

M[(p+1)(p+2)U[p+2]]+C[(p+1)U[p+1]]+KU[p]+V[npmnU[m]U[nm]U[pn]]=0
(79)
re-arranging and simplifying
U[p+2]=1(p+1)(p+2)M{KU[p]C[(p+1)U[p+1]]V[npmnU[m]U[nm]U[pn]]}
(80)
p=0,1,2,3
Since at 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
U[0]=Uo=χandU[1]=0
(81)
then
U[2]=χ2M{Vχ2+K}U[3]=16{C(χ3VKχ)M2}U[4]=112M{32Vχ2(χ3VKχ)M+12C2(χ3VKχ)M212K(χ3VKχ)M}U[5]=120M{Vχ2C(Vχ3Kχ)2MC3M2(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+KC(Vχ3Kχ)6M2}U[6]=130M{(Vχ24M(3Vχ2(Vχ2Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+3χ(Vχ3Kχ)24M2)(C4M(Vχ2C(Vχ3Kχ)2M2C3M(3Vχ2(Vχ3Kχ)2MC2(Vχ3Kχ)2MK(Vχ3Kχ)2M)+KC(Vχ3Kχ)26M2))K12M(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2MK(Vχ3Kχ)2M)}
(82)
Using t*=τ, the series solution for the temporal part from DTM is expressed as
U(τ)=k=0Tkτkusingthefirstseventerms
(83)
which
U(τ)=U[0]+U[1]τ+U[2]τ2+U[3]τ3+U[4]τ4+U[5]τ5+U[6]τ6+
(84)
The temporal part becomes
U(τ)=χ(χ2M{Vχ2+K})τ216(C(χ3VKχ)M2)τ3+112M(32Vχ2(χ3VKχ)M+12C(χ3VKχ)M212K(χ3VKχ)M)τ4+120M{Vχ2C(Vχ3Kχ)2MC3M2(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+KC(Vχ3Kχ)6M2}τ5+130M{(Vχ24M(3Vχ2(Vχ2Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+3χ(Vχ3Kχ)24M2)(C4M(Vχ2C(Vχ3Kχ)2M2C3M(3Vχ2(Vχ3Kχ)2MC2(Vχ3Kχ)2MK(Vχ3Kχ)2M)+KC(Vχ3Kχ)26M2))K12M(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2MK(Vχ3Kχ)2M)}τ6
(85)
For pinned–pinned (simply supported), the spatial part using Table 2 is
Ψ(x*)=sin(nπLx*)
(86)
Table 2

Trial functions for the four boundary conditions of the SWCNT

Trial function Ψ(x*)
BCsHyperbolic–trigonometric functionEquivalent polynomiala4
P–Psin(nπLx)(X − 2X3 + X4)a43.20
C–Ccoshβnxcosβnx(sinhβnL+sinβnLcoshβnLcosβnL)(sinhβnxsinβnx)(X2 − 2X3 + X4)a425.20
C–Pcoshβnxcosβnx(coshβnLcosβnLsinhβnLsinβnL)(sinhβnxsinβnx)(32X252X3+X4)a411.625
C–Fcoshβnxcosβnx(coshβnL+cosβnLsinhβnL+sinβnL)(sinhβnxsinβnx)(6X2 − 4X3 + X4)a40.6625
Trial function Ψ(x*)
BCsHyperbolic–trigonometric functionEquivalent polynomiala4
P–Psin(nπLx)(X − 2X3 + X4)a43.20
C–Ccoshβnxcosβnx(sinhβnL+sinβnLcoshβnLcosβnL)(sinhβnxsinβnx)(X2 − 2X3 + X4)a425.20
C–Pcoshβnxcosβnx(coshβnLcosβnLsinhβnLsinβnL)(sinhβnxsinβnx)(32X252X3+X4)a411.625
C–Fcoshβnxcosβnx(coshβnL+cosβnLsinhβnL+sinβnL)(sinhβnxsinβnx)(6X2 − 4X3 + X4)a40.6625
But the deflection of SWCNT is
w*(x*,t*)=Ψ(x*)U(t*)
(87)
Therefore, the deflection when considering pinned–pinned supported nanobeam becomes
w(x*,τ)={χ(χ2M{Vχ2+K})τ216(C(χ3VKχ)M2)τ3+112M(32Vχ2(χ3VKχ)M+12C(χ3VKχ)M212K(χ3VKχ)M)τ4+120M{Vχ2C(Vχ3Kχ)2MC3M2(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+KC(Vχ3Kχ)6M2}τ5+130M{(Vχ24M(3Vχ2(Vχ2Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+3χ(Vχ3Kχ)24M2)(C4M(Vχ2C(Vχ3Kχ)2M2C3M(3Vχ2(Vχ3Kχ)2MC2(Vχ3Kχ)2MK(Vχ3Kχ)2M)+KC(Vχ3Kχ)26M2))K12M(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2MK(Vχ3Kχ)2M)}τ6}{sinnπLx*}
(88)

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

For clamped–pinned supported nanobeam:
w(x*,τ)={χ(χ2M{Vχ2+K})τ216(C(χ3VKχ)M2)τ3+112M(32Vχ2(χ3VKχ)M+12C(χ3VKχ)M212K(χ3VKχ)M)τ4+120M{Vχ2C(Vχ3Kχ)2MC3M2(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+KC(Vχ3Kχ)6M2}τ5+130M{(Vχ24M(3Vχ2(Vχ2Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+3χ(Vχ3Kχ)24M2)(C4M(Vχ2C(Vχ3Kχ)2M2C3M(3Vχ2(Vχ3Kχ)2MC2(Vχ3Kχ)2MK(Vχ3Kχ)2M)+KC(Vχ3Kχ)26M2))K12M(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2MK(Vχ3Kχ)2M)}τ6}{coshβnx*cosβnx*(coshβnLcosβnLsinhβnLsinβnL)(sinhβnx*sinβnx*)}
(89)
For clamped–clamped supported nanobeam:
w(x*,τ)={χ(χ2M{Vχ2+K})τ216(C(χ3VKχ)M2)τ3+112M(32Vχ2(χ3VKχ)M+12C(χ3VKχ)M212K(χ3VKχ)M)τ4+120M{Vχ2C(Vχ3Kχ)2MC3M2(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+KC(Vχ3Kχ)6M2}τ5+130M{(Vχ24M(3Vχ2(Vχ2Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+3χ(Vχ3Kχ)24M2)(C4M(Vχ2C(Vχ3Kχ)2M2C3M(3Vχ2(Vχ3Kχ)2MC2(Vχ3Kχ)2MK(Vχ3Kχ)2M)+KC(Vχ3Kχ)26M2))K12M(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2MK(Vχ3Kχ)2M)}τ6}{coshβnx*cosβnx*(sinhβnL+sinβnLcoshβnLcosβnL)(sinhβnx*sinβnx*)}
(90)
For clamped–free (cantilever) supported nanobeam:
w(x*,τ)={χ(χ2M{Vχ2+K})τ216(C(χ3VKχ)M2)τ3+112M(32Vχ2(χ3VKχ)M+12C(χ3VKχ)M212K(χ3VKχ)M)τ4+120M{Vχ2C(Vχ3Kχ)2MC3M2(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+KC(Vχ3Kχ)6M2}τ5+130M{(Vχ24M(3Vχ2(Vχ2Kχ)2M+C2(Vχ3Kχ)2M2K(Vχ3Kχ)2M)+3χ(Vχ3Kχ)24M2)(C4M(Vχ2C(Vχ3Kχ)2M2C3M(3Vχ2(Vχ3Kχ)2MC2(Vχ3Kχ)2MK(Vχ3Kχ)2M)+KC(Vχ3Kχ)26M2))K12M(3Vχ2(Vχ3Kχ)2M+C2(Vχ3Kχ)2MK(Vχ3Kχ)2M)}τ6}{coshβnx*cosβnx*(coshβnL+cosβnLsinhβnL+sinβnL)(sinhβnx*sinβnx*)}
(91)

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.

Fig. 2
(a) Effects of boundary condition on mode shape for pinned–pinned supports, (b) effects of boundary condition on mode shape for clamped–clamped supports, and (c) effects of boundary condition on mode shape for clamped–pinned supports
Fig. 2
(a) Effects of boundary condition on mode shape for pinned–pinned supports, (b) effects of boundary condition on mode shape for clamped–clamped supports, and (c) effects of boundary condition on mode shape for clamped–pinned supports
Close modal

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.

Fig. 3
(a) Verification of polynomial function for pinned–pinned support, (b) verification of polynomial function for clamped–clamped supports, (c) verification of polynomial function for clamped–pinned support, and (d) verification of polynomial function for clamped–free supports
Fig. 3
(a) Verification of polynomial function for pinned–pinned support, (b) verification of polynomial function for clamped–clamped supports, (c) verification of polynomial function for clamped–pinned support, and (d) verification of polynomial function for clamped–free supports
Close modal

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.

Fig. 4
Impact of end conditions on nonlinear dimensionless amplitude-dimensionless frequency response of the nanotube
Fig. 4
Impact of end conditions on nonlinear dimensionless amplitude-dimensionless frequency response of the nanotube
Close modal

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 Uc 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 Us 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.

Fig. 5
(a) Impact of branch angle on nanotube’s stability for linear prebifurcation, (b) impact of branch angle on nanotube’s stability for nonlinear prebifurcation, (c) impact of branch angle on nanotube’s stability for linear post-bifurcation, (d) impact of branch angle on nanotube’s stability for nonlinear post-bifurcation, (e) impact of branch angle on nanotube’s stability for linear pre- and post-bifurcation, and (f) impact of branch angle on nanotube’s stability for nonlinear pre- and post-bifurcation
Fig. 5
(a) Impact of branch angle on nanotube’s stability for linear prebifurcation, (b) impact of branch angle on nanotube’s stability for nonlinear prebifurcation, (c) impact of branch angle on nanotube’s stability for linear post-bifurcation, (d) impact of branch angle on nanotube’s stability for nonlinear post-bifurcation, (e) impact of branch angle on nanotube’s stability for linear pre- and post-bifurcation, and (f) impact of branch angle on nanotube’s stability for nonlinear pre- and post-bifurcation
Close modal

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

Figures 6 and 7 depict the effect of βf 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 βf increases at this region. Figure 7 illustrates a means of obtaining damping values. It is shown that the critical value of βf is 0.40 because an incessant increase in βf above this limit results in an effect analogous to when values lower than βf = 0.4 were used.

Fig. 6
Impact of mass ratio term on frequency for varying dimensionless flow velocity
Fig. 6
Impact of mass ratio term on frequency for varying dimensionless flow velocity
Close modal
Fig. 7
Impact of mass ratio term on dimensionless damping frequency for varying dimensionless flow velocity
Fig. 7
Impact of mass ratio term on dimensionless damping frequency for varying dimensionless flow velocity
Close modal

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.

Fig. 8
(a) Effect of stiffness coefficients on nanotube’s stability For linear Winkler foundation, (b) effect of stiffness coefficients on nanotube’s stability for nonlinear Winkler foundation, (c) effect of stiffness coefficients on nanotube’s stability for linear Pasternak foundation, and (d) effect of stiffness coefficients on nanotube’s stability for nonlinear Pasternak foundation
Fig. 8
(a) Effect of stiffness coefficients on nanotube’s stability For linear Winkler foundation, (b) effect of stiffness coefficients on nanotube’s stability for nonlinear Winkler foundation, (c) effect of stiffness coefficients on nanotube’s stability for linear Pasternak foundation, and (d) effect of stiffness coefficients on nanotube’s stability for nonlinear Pasternak foundation
Close modal

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.

Fig. 9
(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90
Fig. 9
(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90
Close modal
Fig. 10
(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90
Fig. 10
(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90(a) Effect of velocity on stability curve when downstream angle, ϕ = 0, (b) effect of velocity on stability curve when downstream angle, ϕ = 15, (c) effect of velocity on stability curve when downstream angle, ϕ = 30, (d) effect of velocity on stability curve when downstream angle, ϕ = 45, (e) effect of velocity on stability curve when downstream angle, ϕ = 60, (f) effect of velocity on stability curve when downstream angle, ϕ = 75, (g) effect of velocity on stability curve when downstream angle limit, ϕ = 79, and (h) effect of velocity on stability curve when downstream angle, ϕ = 90
Close modal
Fig. 11
(a) Effect of different end shapes on stability curve without magnetic effect and (b) effect of different end shapes on stability curve with magnetic effect
Fig. 11
(a) Effect of different end shapes on stability curve without magnetic effect and (b) effect of different end shapes on stability curve with magnetic effect
Close modal

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.

Fig. 12
Impact of tension on nanotube’s stability
Fig. 12
Impact of tension on nanotube’s stability
Close modal

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.

Fig. 13
(a) Effect of nano-term on the frequency–velocity curve without magnetic effect and (b) effect of nano-term on the frequency–velocity curve with magnetic effect
Fig. 13
(a) Effect of nano-term on the frequency–velocity curve without magnetic effect and (b) effect of nano-term on the frequency–velocity curve with magnetic effect
Close modal
Fig. 14
Effect of nano-term on the dimensionless frequency–dimensionless velocity curve for pre- and post-buckling
Fig. 14
Effect of nano-term on the dimensionless frequency–dimensionless velocity curve for pre- and post-buckling
Close modal

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.

Fig. 15
(a) Thermal effects on nanotube’s stability for linear Winkler foundation and without magnetic effect, (b) thermal effects on nanotube’s stability for nonlinear Winkler foundation and without magnetic effect, (c) thermal effects on nanotube’s stability for linear Pasternak foundation and without magnetic effect, and (d) thermal effects on nanotube’s stability for nonlinear Pasternak foundation and without magnetic effect
Fig. 15
(a) Thermal effects on nanotube’s stability for linear Winkler foundation and without magnetic effect, (b) thermal effects on nanotube’s stability for nonlinear Winkler foundation and without magnetic effect, (c) thermal effects on nanotube’s stability for linear Pasternak foundation and without magnetic effect, and (d) thermal effects on nanotube’s stability for nonlinear Pasternak foundation and without magnetic effect
Close modal
Fig. 16
(a) Thermal effects on nanotube’s stability for linear Winkler foundation and with magnetic effect, (b) thermal effects on nanotube’s stability for nonlinear Winkler foundation and with magnetic effect, (c) thermal effects on nanotube’s stability for linear Pasternak foundation and with magnetic effect, and (d) thermal effects on nanotube’s stability for nonlinear Pasternak foundation and with magnetic effect
Fig. 16
(a) Thermal effects on nanotube’s stability for linear Winkler foundation and with magnetic effect, (b) thermal effects on nanotube’s stability for nonlinear Winkler foundation and with magnetic effect, (c) thermal effects on nanotube’s stability for linear Pasternak foundation and with magnetic effect, and (d) thermal effects on nanotube’s stability for nonlinear Pasternak foundation and with magnetic effect
Close modal

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.

Fig. 17
(a) Impact of shear modulus on nanotube’s stability without magnetic effect and (b) impact of shear modulus on nanotube’s stability with magnetic effect
Fig. 17
(a) Impact of shear modulus on nanotube’s stability without magnetic effect and (b) impact of shear modulus on nanotube’s stability with magnetic effect
Close modal

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 Uc 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.

Fig. 18
Influence of Knudsen on the dimensionless frequency of the SWCNT
Fig. 18
Influence of Knudsen on the dimensionless frequency of the SWCNT
Close modal

4.14 Frequency Against Fluid Velocity for Different Modes.

Figures 19(a)19(f) depict the effect of βf 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 βf increases at this region.

Fig. 19
(a) Frequency against fluid velocity for two modes for pinned–pinned supported nanotube, (b) impact of mass ratio on nanotube’s stability for pinned–pinned supported nanotube, (c) frequency against fluid velocity for two modes for clamped–pinned supported nanotube, (d) impact of mass ratio on nanotube’s stability for clamped–pinned supported nanotube, (e) frequency against fluid velocity for two modes for clamped–fixed supported nanotube, and (f) impact of mass ratio on nanotube’s stability for clamped–fixed supported nanotube
Fig. 19
(a) Frequency against fluid velocity for two modes for pinned–pinned supported nanotube, (b) impact of mass ratio on nanotube’s stability for pinned–pinned supported nanotube, (c) frequency against fluid velocity for two modes for clamped–pinned supported nanotube, (d) impact of mass ratio on nanotube’s stability for clamped–pinned supported nanotube, (e) frequency against fluid velocity for two modes for clamped–fixed supported nanotube, and (f) impact of mass ratio on nanotube’s stability for clamped–fixed supported nanotube
Close modal

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.

Fig. 20
Assessment of linear and nonlinear dynamic responses of nanotube
Fig. 20
Assessment of linear and nonlinear dynamic responses of nanotube
Close modal

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.

Fig. 21
(a) Dynamic response of nanotube for mode 1 with B = 0 for the linear model, (b) dynamic response of nanotube for mode 1 with B = 50 for the linear model, (c) dynamic response of nanotube for mode 1 with B = 100 for the linear model, and (d) dynamic response of nanotube for mode 1 with B = 120 for the linear model
Fig. 21
(a) Dynamic response of nanotube for mode 1 with B = 0 for the linear model, (b) dynamic response of nanotube for mode 1 with B = 50 for the linear model, (c) dynamic response of nanotube for mode 1 with B = 100 for the linear model, and (d) dynamic response of nanotube for mode 1 with B = 120 for the linear model
Close modal
Fig. 22
(a) Dynamic response of nanotube for mode 2 with B = 0 for the nonlinear model, (b) dynamic response of nanotube for mode 2 with B = 50 for the nonlinear model, (c) dynamic response of nanotube for mode 2 with B = 100 for the nonlinear model, and (d) dynamic response of nanotube for mode 2 with B = 120 for the nonlinear model
Fig. 22
(a) Dynamic response of nanotube for mode 2 with B = 0 for the nonlinear model, (b) dynamic response of nanotube for mode 2 with B = 50 for the nonlinear model, (c) dynamic response of nanotube for mode 2 with B = 100 for the nonlinear model, and (d) dynamic response of nanotube for mode 2 with B = 120 for the nonlinear model
Close modal

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.

Fig. 23
(a) Effect of change in temperature on deflection when B = 0, (b) effect of change in temperature on deflection when B = 50, (c) effect of change in temperature on deflection when B = 100, and (d) effect of change in temperature on deflection when B = 120
Fig. 23
(a) Effect of change in temperature on deflection when B = 0, (b) effect of change in temperature on deflection when B = 50, (c) effect of change in temperature on deflection when B = 100, and (d) effect of change in temperature on deflection when B = 120
Close modal

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.

Fig. 24
(a) Effect of foundation parameter on deflection with low Δθ and (b) effect of foundation parameter on deflection with high Δθ
Fig. 24
(a) Effect of foundation parameter on deflection with low Δθ and (b) effect of foundation parameter on deflection with high Δθ
Close modal

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 (eoa) = 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.

Fig. 25
Model validation

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˙ =

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

mj =

mass of the junction

I0 =

Bessel term

Mf =

mass of fluid

Ri =

radius of CNT

ICNT =

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

(eoa) =

nonlocal parameter

ρf =

fluid density,

ρjf =

density of fluid in junction

ρtf =

density of fluid in tube

σv =

tangential momentum

=

gradient

Appendix

The parameters used are defined as
α1=(eoa)*2Bo2qsin2αα2=μeff(eoa)*2U*α3=(eoa)*2σα4=μeff(eoa)*2βα5={1(eoa)*2Mf*U*2(eoa)*2U*2cosϕ+(eoa)*2G*+(eoa)*2Bt2cos2α+Bt2qsin2α}α6=μeffU*2(eoa)*2U*2Mf*2α7=2(eoa)*2U*[β+βjf]α8={Mf*2U*2Bt2cos2αU*2cosϕG*+P*T*Kp*Δθ*(eoa)*2U*2Mf*2+(eoa)*2kw*}α9={(eoa)*2c*+μeffβ+4(eoa)*2U*βjf+(eoa)*2Bf2cos2α}α10=σ(eoa)*2{1+βT2}α11=2U*[β+βjf2(eoa)*2βjf]α12=2(eoa)*2βT2α13=c*+Bf2cos2αα14=1+βT2(eoa)*2βT2α15=kw*andα16=k3*

References

1.
Iijima
,
S.
,
1991
, “
Helical Micro Tubes of Graphitic Carbon
,”
Nature
,
354
(
e1991
), pp.
56
58
. DOI: 10.1038/354056a0
2.
Terrones
,
M.
,
Banhart
,
F.
,
Grobert
,
N.
,
Charlier
,
J.
,
Terrones
,
C.
, and
Ajayan
,
H. P. M.
,
2002
, “
Molecular Junctions by Joining Single-Walled Carbon Nanotubes
,”
Phys. Rev. Lett.
,
89
(
7
), p.
07550
.
3.
Nagy
,
P.
,
Ehlich
,
R.
,
Biro
,
L. P.
, and
Gjyulai
,
J.
,
2000
, “
Y-Branching of Single Walled Carbon Nanotubes
,”
Appl. Phys. A
,
70
(
4
), pp.
481
483
.
4.
Chernozatonskii
,
L. A.
,
1992
, “
Carbon Nanotubes Connectors and Planar Jungle Gyms
,”
Appl. Phys. A
,
172
, pp.
173
176
.
5.
Biro
,
L. P.
,
Horvath
,
Z. E.
,
Mark
,
G. I.
,
Osvath
,
Z.
,
Koos
,
A. A.
,
Benito
,
A. M.
,
Maser
,
W.
, and
Lambin
,
P. H.
,
2004
, “
Carbon Nanotube Y Junctions: Growth and Properties
,”
Diamond Relat. Mater.
,
13
(
2
), pp.
241
249
.
6.
Andriotis
,
A. N.
,
Menon
,
M.
,
Srivastava
,
D.
, and
Chernozatonskii
,
L.
,
2001
, “
Rectification Properties of Carbon Nanotube “Y-Junction”
,”
Phys. Rev. Lett.
,
87
(
6
), p.
66802
.
7.
Castrucci
,
P.
,
Scarselli
,
M.
,
De Crescenzi
,
M.
,
El Khakani
,
M. A.
,
Rosei
,
F.
,
Braidy
,
N.
, and
Yi
,
J. H.
,
2004
, “
Effect of Coiling on the Electronic Properties Along Single-Wall Carbon Nanotubes
,”
Appl. Phys. Lett.
,
85
(
17
), pp.
3857
3859
.
8.
Tsai
,
P. C.
,
Jeng
,
Y. R.
, and
Fang
,
T. H.
,
2006
, “
Coalescence, Melting, and Mechanical Characteristics of Carbon Nanotube Junctions
,”
Phys. Rev. B
,
74
(
4
), p.
045406
.
9.
Liu
,
Q.
,
Liu
,
W.
,
Cui
,
Z. M.
,
Song
,
W. G.
, and
Wan
,
L. J.
,
2007
, “
Synthesis and Characterization of 3D Double Branched K Junction Carbon Nanotubes and Nanorods
,”
Carbon
,
45
(
2
), pp.
268
273
.
10.
Zhao
,
B.
,
Hu
,
H.
,
Mandal
,
S. K.
, and
Haddon
,
R. C.
,
2005
, “
A Bone Mimic Based on the Self-Assembly of Hydroxyapatite on Chemically Functionalized Single-Walled Carbon Nanotubes
,”
Chem. Mater.
,
17
(
12
), pp.
3235
3241
.
11.
Kam
,
N. W. S.
, and
Dai
,
H.
,
2005
, “
Carbon Nanotubes as Intracellular Protein Transporters: Generality and Biological Functionality
,”
J. Am. Chem. Soc.
,
127
(
16
), pp.
6021
6026
.
12.
Yoon
,
J. W.
, and
Hwang
,
H. J.
,
2011
, “
Molecular Dynamics Modeling and Simulations of a Single-Walled Carbon-Nanotube-Resonator Encapsulating a Finite Nanoparticle
,”
Comput. Mater. Sci.
,
50
, p.
27414
.
13.
Wu
,
C. D.
,
Fang
,
T. H.
, and
Chan
,
C. Y.
,
2011
, “
A Molecular Dynamics Simulation of the Mechanical Characteristics of a C60-Filled Carbon Nanotube Under Nano Indentation Using Various Carbon Nanotube Tip
,”
Carbon
,
49
(
6
), pp.
2053
2061
.
14.
Ge
,
Y.
,
Yang
,
F.
,
Liang
,
Q.
, and
Dong
,
J.
,
2011
, “
Coherent Phonons in Excited State Carbon Nanotubes: A Simulation by Tight-Binding Molecular Dynamics
,”
Physica E
,
43
(
9
), pp.
1585
1591
.
15.
Ohta
,
Y.
,
Okamoto
,
Y.
,
Irle
,
S.
, and
Morokuma
,
K.
,
2009
, “
Density-Functional Tight-Binding Molecular Dynamics Simulations of SWCNT Growth by Surface Carbon Diffusion on an Iron Cluster
,”
Carbon
,
47
(
5
), pp.
1270
1275
.
16.
Ghorbanpour
,
A. A.
,
Roudbari
,
M. A.
, and
Amir
,
S.
,
2012
, “
Nonlocal Vibration of SWBNNT Embedded in Bundle of CNTs Under a Moving Nanoparticle
,”
Physica B
,
407
(
17
), pp.
3646
3653
.
17.
Ghorbanpour
,
A. A.
,
Shajari
,
A. R.
,
Atabakhshian
,
V.
,
Amir
,
S.
, and
Loghman
,
A.
,
2013
, “
Nonlinear Dynamical Response of Embedded Fluidconveyed Micro-Tube Reinforced by BNNTs
,”
Composites Part B
,
44
(
1
), pp.
424
432
.
18.
Khodami
,
M. Z.
,
Ghorbanpour
,
A. A.
,
Kolahchi
,
R.
,
Amir
,
S.
, and
Bagheri
,
M. R.
,
2013
, “
Nonlocal Vibration and Instability of Embedded DWBNNT Conveying Viscose Fluid
,”
Composites Part B
,
45
(
1
), pp.
423
432
.
19.
Chang
,
T. P.
, and
Liu
,
M. F.
,
2011
, “
Flow-Induced Instability of Double-Walled Carbon Nanotubes Based on Nonlocal Elasticity Theory
,”
Physica E
,
43
(
8
), pp.
1419
1426
.
20.
Wang
,
L.
,
2009
, “
Vibration and Instability Analysis of Tubular Nano- and Micro-Beams Conveying Fluid Using Nonlocal Elastic Theory
,”
Physica E
,
41
(
10
), pp.
1835
1840
.
21.
Ghorbanpour
,
A. A.
,
Shajari
,
A. R.
,
Amir
,
S.
, and
Loghman
,
A.
,
2012
, “
Electrothermo-mechanical Nonlinear Nonlocal Vibration and Instability of Embedded Micro-Tube Reinforced by BNNT, Conveying Fluid
,”
Physica E
,
45
, pp.
109
121
.
22.
Lee
,
H. L.
, and
Chang
,
W. J.
,
2008
, “
Free Transverse Vibration of the Fluidconveying Single-Walled Carbon Nanotube Using Nonlocal Elastic Theory
,”
J. Appl. Phys.
,
103
(
2
), p.
024302
.
23.
Chang
,
T. P.
,
2012
, “
Thermal–Mechanical Vibration and Instability of a Fluid-Conveying Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based on Nonlocal Elasticity Theory
,”
Appl. Math. Model.
,
36
(
5
), pp.
1964
1973
.
24.
Yan
,
Y.
,
Wang
,
W.
, and
Zhang
,
L.
,
2012
, “
Free Vibration of the Fluid-Filled Single Walled Carbon Nanotube Based on a Double Shell-Potential Flow Model
,”
Appl. Math. Model.
,
36
(
12
), pp.
6146
6153
.
25.
Kiani
,
K.
,
2013
, “
Vibration Behavior of Simply Supported Inclined Single Walled Carbon Nanotubes Conveying Viscous Fluids Flow Using Nonlocal Rayleigh Beam Model
,”
Appl. Math. Model.
,
37
(
4
), pp.
1836
1850
.
26.
Kaviani
,
F.
, and
Mirdamadi
,
H. R.
,
2012
, “
Influence of Knudsen Number on Fluid Viscosity for Analysis of Divergence in Fluid Conveying Nanotubes
,”
Comput. Mater. Sci.
,
61
, pp.
270
277
.
27.
Lei
,
X. W.
,
Natsuki
,
T.
,
Shi
,
J. X.
, and
Ni
,
Q. Q.
,
2012
, “
Surface Effects on the Vibrational Frequency of Double-Walled Carbon Nanotubes Using the Nonlocal Timoshenko Beam Model
,”
Composites, Part B
,
43
(
1
), pp.
64
69
.
28.
Wang
,
L.
,
2010
, “
Vibration Analysis of Fluid-Conveying Nanotubes With Consideration of Surface Effects
,”
Physica E
,
43
(
1
), pp.
437
439
.
29.
Lin
,
R. M.
,
2012
, “
Nanoscale Vibration Characterization of Multi-layered Graphene Sheets Embedded in an Elastic Medium
,”
Comput. Mater. Sci.
,
53
(
1
), pp.
44
52
.
30.
Pradhan
,
S. C.
, and
Phadikar
,
J. K.
,
2009
, “
Small Scale Effect on Vibration of Embedded Multilayered Graphene Sheets Based on Nonlocal Continuum Models
,”
Phys. Lett. A
,
373
(
11
), pp.
1062
1069
.
31.
Ghorbanpour
,
A. A.
,
Zarei
,
M. S.
,
Amir
,
S.
, and
Khoddami
,
M. Z.
,
2013
, “
Nonlinear Nonlocal Vibration of Embedded DWCNT Conveying Fluid Using Shell Model
,”
Physica B
,
410
, pp.
188
196
.
32.
Ghorbanpour
,
A. A.
, and
Amir
,
S.
,
2013
, “
Electro-thermal Vibration of Visco-Elastically Coupled BNNTsystems Conveying Fluid Embedded on Elastic Foundation Via Strain Gradient Theory
,”
Physica B
,
419
, pp.
1
6
.
33.
Eichler
,
A.
,
Moser
,
J.
,
Chaste
,
J.
,
Zdrojek
,
M.
,
Wilson-Rae
,
I.
, and
Bachtold
,
A.
,
2011
, “
Nonlinear Damping in Mechanical Resonators Made From Carbon Nanotubes and Grapheme
,”
Nat. Nanotechnol.
,
6
(
6
), pp.
339
342
.
34.
Yinusa
,
A. A.
, and
Sobamowo
,
M. G.
,
2019
, “
Analysis of Dynamic Behaviour of a Tensioned Carbon Nanotube in Thermal and Pressurized Environment
,”
Karbala Int. J. Mod. Sci.
,
5
(
1
), pp.
2
11
.
35.
Ouyang
,
J.
,
2019
, “
Applications of Carbon Nanotubes and Graphene for Third-Generation Solar Cells and Fuel Cells
,”
Nano Mater. Sci.
,
1
(
2
), pp.
101
115
.
36.
Guo
,
H.
,
Ruicong
,
L.
, and
Shulin
,
B.
,
2019
, “
Recent Advances on 3D Printing Graphene-Based Composites
,”
Nano Mater. Sci.
,
1
(
2
), pp.
77
90
.
37.
Shuang
,
W.
,
Zengtao
,
C.
, and
Cunfa
,
G.
,
2019
, “
Analytic Solution for a Circular Nano-Inhomogeneity in a Finite Matrix
,”
Nano Mater. Sci.
,
1
(
2
), pp.
116
120
.
38.
Xie
,
X.
,
Chaoyue
,
C.
,
Gang
,
J.
,
Run
,
X.
, and
Hanlin
,
L.
,
2019
, “
A Novel Approach for Fabricating a CNT/AlSi Composite With the Self-Aligned Nacre-Like Architecture by Cold Spraying
,”
Nano Mater. Sci.
,
1
(
2
), pp.
137
141
.
39.
Smith
,
A.
,
Anna
,
M. L.
,
Songshan
,
Z.
,
Bin
,
L.
, and
Luyi
,
S.
,
2019
, “
Synthesis, Properties, and Applications of Graphene Oxide/Reduced Graphene Oxide and Their Nanocomposites
,”
Nano Mater. Sci.
,
1
(
1
), pp.
31
47
.
40.
Dodds
,
H. L.
, and
Runyan
,
H.
,
1965
, “
Effects of High Velocity Fluid Flow in the Bending Vibrations and Static Divergence of a Simply Supported Pipe
,”
National Aeronautical and Space Adminitration Report
, NASA TN, D-2870.
41.
Sobamowo
,
M. G.
,
2017
, “
Nonlinear Thermal and Flow Induced Vibration Analysis of Fluid-Conveying Carbon Nanotube Resting on Winkler and Pasternak Foundations
,”
Ther. Sci. Eng. Prog.
,
4
, pp.
133
149
.
42.
Sobamowo
,
M. G.
, and
Yinusa
,
A. A.
,
2018
, “
Thermo-fluidic Parameters Effects on Nonlinear Vibration of Fluid Conveying Nanotube Resting on Elastic Foundations Using Homotopy Perturbation Method
,”
J. Therm. Eng.
,
4
(
4, Spec. Issue 8
), pp.
2211
2233
.
43.
Ashgharifard
,
S. P.
, and
Haeri
,
Y. M. R.
,
2013
, “
Nonlinear Free Vibrations of Functionally Graded Nanobeams With Surface Effects
,”
Composites, Part B
,
45
(
1
), pp.
581
586
.