WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS Sayyid H. Hashemi Kachapi

Nonclassical vibration analysis of piezoelectric nanosensor conveying viscous fluid

SAYYID H. HASHEMI KACHAPI Department of Mechanical Engineering Babol Noshirvani University of Technology Shariati Street, Babol, Mazandaran 47148-71167 [email protected], IRAN

Abstract: - In this paper, the natural frequency, critical fluid velocity and stability analysis of piezoelectric biomedical nanosensor (PBMNS) based on cylindrical nanoshell conveying viscous fluid is investigated using the electro-elastic Gurtin–Murdoch surface/interface theory. This system subjected to nonlinear electrostatic field and viscoelastic medium including visco-pasternak and damping coefficients. Hamilton’s principle and the assumed mode method combined with Euler – Lagrange is used for the governing equations and boundary conditions. It is shown that fluid velocity due to motion of biomarkers has major unpredictable effects on natural frequency and critical fluid velocity of the system and one should precisely consider their effects. Key-Words: - Piezoelectric biomedical nanosensor, Natural frequency, Instability, Critical fluid velocity, Nanoshell, Gurtin–Murdoch surface/interface theory, Electrostatic force, Visco-Pasternak medium.

1 Introduction 11]. Also, a nanosensor is proposed for detection of Nanotechnology is a multidisciplinary branch of cancer cells located in a particular region of a blood science which encompasses numerous diverse fields vessel [12] and for detection of cancer biomarkers in of science and technology, pharmaceutical, serum at ultralow concentrations [13]. Recently, a agricultural, environmental, advanced materials, research project will lead to development of a new chemical science, physics, electronics, information category of nanometer-sized chemical and technology, and specially biomedical fields such as biological sensors that are compatible with the imaging agents, drug delivery vehicle, diagnostic intracellular environment and will enable new tools, etc. to save human life along with other areas hypotheses to be tested in the role of metabolic by application of engineering skills in surgical coupling in pancreatic alpha cells and a range of diagnosis, monitoring, treatment, and therapy etc. In other cellular systems involved in the regulation of recent years, the application of nanotechnology normal human health [14]. With the development of shows further advancement in several specific areas material science, the piezoelectric nanosensor and in biomedical such as drug targeting, bio- nano actuator play an important role for medicine diagnostics, bioimaging, and genetic manipulation applications and have been fabricated as nano- [1]. For example, in head and neck cancer (tumors) beams, nano-plates, nano-membranes and nano- and breast cancer cells, image-guided laser ablation shells [15-17]. One of the most important scientific and photothermal therapy with laser light and concepts in the design and fabrication of this plasmon naturals are the promising minimally nanosensor, and due to the high sensitivity that can invasive techniques currently being investigated as be found in medical applications, is the analysis of an alternative to conventional surgical interventions. dynamic and vibrations considering of nano- For increase the conformality of therapy delivery, mechanical theories. Because of the large ratio of the use of tumortargeted nanoparticles, especially surface area to volume in nano-scaled structure, the metallic nanoshell, which are preferentially targeted behaviour of surfaces and interfaces lead to a to the tumor and, thereby enhance the safety and significant factor in controlling the vibration efficacy of the overall procedure, are proposed [2- analysis of piezoelectric Nano biomedical sensor 5]. For detection of a wide range of biomarkers such based on nano-shell. In this case some non-classical as cancer biomarkers many of materials such as carbon nanotubes, magnetic nanoparticles, gold Acknowledgement nanoparticles and nanowires and other materials I highly appreciate the time and efforts of all the have been presented and developed [6]. Recently reviewers devoted to review my manuscript and I Gold nanoshells (AuNSs) and nanotube-based thank the Associate Editor and the reviewers for composite sensors have been intensively many helpful and constructive comments. investigated and applied for medical application [7-

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continuum theories such as the electro-elastic subjected to nonlinear electrostatic field, Gurtin–Murdoch surface elasticity theory have been viscoelastic medium including visco-pasternak and introduced to develop the size-dependent continuum damping coefficients. Hamilton’s principle and also models [18-19]. In the past two decades, the assumed mode method combined with Euler – investigating the surface effects on the mechanical Lagrange is used for the governing equations, behavior of nanostructures has become one of the boundary conditions and for changing the partial attractive research areas in nanomechanics, as differential equations into ordinary differential evidenced by the large number of publications on equations. Also stability analysis and dimensionless this issue [20-25]. Recently, Zhu and Fang et al. natural frequency (Ω) versus fluid velocity () studied free and forced vibration analysis of nano- dimensionless of a piezoelectric biomedical sized single and double shell structures with nanosensor are accurately studied with respect to the piezoelectric layer based on GM surface/interface different geometrical and material parameters. theory [20-21]. Ghorbanpour Arani et al. studied on vibration analysis of double walled visco-carbon nanotubes under magnetic fields [22]. In another 2 Mathematical formulation research by Ghorbanpour Arani et al., nonlinear A piezoelectric biomedical nanosensor shown in vibration of nano sheet with small scale and surface Figure 1 based on cylindrical nanoshell embedded effects are investigated by using of nonlocal and with a visco-Pasternak medium and electrostatic surface piezoelasticity theories [23]. Also, vibration force with incoming bloodstream as viscous fluid. analysis of viscoelastic DWCNT unified with ZnO layers and subjected to magnetic and electric fields are studied by Fereidoon et al. [24]. Surface stress effect on the vibration of nanoscale pipes based on a size-dependent Timoshenko beam model is investigated by Ansari et al. [25]. Ye et al. presented a unified solution method for the free vibration analysis of composite shallow shells with general elastic boundary conditions [26]. Fazollari developed an analytical formulation for free Fig. 1. Piezoelectric biomedical nanosensor vibration analysis of doubly curved laminated (PBMNS) conveying viscous bloodstream composite shallow shells by combining the dynamic stiffness method and a higher order shear The length of nano shell is , the geometrical deformation theory [27]. Mirza and Alizadeh parameters of the cylindrical shell are mid-surface investigated the effects of detached base length on radius thickness of cylindrical shell 2 the natural frequencies and modal shapes of , cylindrical shells [28]. Loy et al. presented the free thickness of piezoelectric material layer . With vibration analysis of cylindrical shells using an the origin of coordinate system located on the improved version of the differential quadrature middle surface of nano-shell, the coordinates of a method [29]. An analytical procedure to study the typical point in the axial, circumferential and radius free vibration characteristics of thin circular directions are described by , , and , respectively. Also, , and are stiffness coefficient of cylindrical shells was presented by Naeem and Sharma, in which Ritz polynomial functions are Winkler foundation, shear layer of Pasternak used for solution of the problem [30]. Zeighampour foundation and the damping factor of the visco et al. investigated wave propagation in viscoelastic medium for the transverse motion, respectively. single walled carbon nanotubes by accounting for Young modulus, Poisson ratio and the mass density the simultaneous effects of the nonlocal constant of cylindrical nano-shell represent , and , and the material length scale parameter and visco- respectively. In the present nano-shell, it is assumed Pasternak foundations [31]. that the mention material properties vary through In the present study, the natural frequency, critical the thickness of nano-shell according to the power- fluid velocity and stability analysis of piezoelectric law function. They are written as biomedical nanosensor (PBMNS) based on 2 )1( cylindrical nanoshell conveying viscous fluid is 2 investigated using the electro-elastic Gurtin– 2 Murdoch surface/interface theory and considering (2) 2 von-karman-Donnell's shell model. This system

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2 (3) , 2 1 where is the power-law exponent. The subscripts , and represent the properties of the nano-shell at )7( 1 the upper and lower layers, respectively. Young modulus, Poisson ratio, piezoelectric and 21 dielectric constants and the mass density of piezoelectric layer are , , , , and Since the piezoelectric layers are very thin, and . Due to the nano-sized property, the ratio of are assumed to be zero ( 0), and surface to the volume becomes large, and the only the radial component of electric field is surface energy around the shell expresses significant considered. Consequetly, can be written as [34] effect on the vibration of nano-structure. According 0 to the electro-elastic surface/interface theory, the 0 , )8( surface/interface region adhered to the neigh boring ⁄ solids is several atomic sizes, and has its own electromechanical properties. The surface at the where is the voltage applied to piezoelectric outer piezoelectric layer is denoted by , and the layers. In addition, the voltages at the piezoelectric inner surface is denoted by , as shown in Fig. 1. surface and are

The material properties of surface are Lamé’s and , respectively. Based on these assumptions constants , , residual stress and mentioned above, the radial component of electric displacement can be presented as piezoelectric constants ,. Those of the inner surface are Lamé’s constants , , and )9( residual stress . Due to the character of nano-shell, the state of generalized plane stress of shells is assumed, and 3 Non- classical Shell theory the normal stress in the radial direction is zero. In Within the framework of classical shell the cylindrical nano-shell, the constitutive relation theory, the displacement fields of the nano-shell can can be expressed as [32, 33]; be written as , 0 , , , , (10) 0 , )4( , 00 , , , , (11) , In the outside piezoelectric shell, the constitutive , , , , (12) relation can be expressed as [17] where , and stand for the middle surface displacements in the , and directions, 0 respectively. The nonlinear deflection and 0 curvatures are defined by von-karman-Donnell's 00 theory as [32, 33] 00 )5( , 00 00 0 or , 1 In which the subscripts and represent the (13) 2 cylindrical nano-shell and piezoelectric layers, 1 1 respectively. is the vector of electric field for 2 piezoelectric layers. and are the matrixes 1 1 of elastic constants, and they can be denoted as , 1 )6( , 1 21

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nonclassical continuum model, this assumption does not satisfy the surface conditions. Thus, it is 1 supposed that varies linearly through the thickness and satisfies the balance conditions on the surfaces [35, 36], i.e. 2 16) 1 ( 1 in which ، and are the middle surface strains, and , and are the curvature 2 1 components of the nano-shell. Since the dimension of the shell is at nanometer 1 scale, the surface effect needs to be considered. On 1 based of the Gurtin–Murdoch surface elasticity theory, the constitute relations for surfaces can be 2 1 written as [18-20] (14) For simplification, the material properties of 2 surfaces and interfaces are selected as , , , , , , , , (17) ,,,, By means of Eqs. (15) and (16), can be 2 ,, σ zz in which is the Kronecker delta function. rewritten as Furthermore, the components of stress at the surfaces can be expressed as 2 2 (15) 1 (18) 2 , 2 2 , 2 2 1 , According to Eq. (18), the normal stresses σ and 2 xx 2 σ Eqs. (4) and (5) can be rewrite ten as θθ , , , (19a) 2 1 2 , 1 , 1 (19b) 2 1 , (19c) 2 , , (19d) , 1 , , (19e) 1 1 , 1 2 , , (19f) , 1,2 4 Governing equations In this section, the governing equations of Based on the classical continuum models, is motion of the piezoelectric cylindrical nanoshell are neglected due to its small value as compared to obtained by applying the assumed mode method. other normal stress components. But, in the present

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The total strain and kinetic energies considering the distance to nanoshell, nanoshell radius and the air surface stress effect are expressed as: permittivity [39]. 1 The equation for the fluid (in this paper, 2 bloodstream) motion may be described by the well- known Navier–Stokes equation as [40] 1 2 , (25) 1 where V≡,, is the flow velocity vector in 2 ( 20) cylindrical coordinate system with components in 1 , , and directions. This vector can be expressed 2 as follows cos, (26a) 1 . 2 , (26b) sin, (26c) 1 (21) Where and is the constant velocity of 2 the fluid (blood). Also , , and are the pressure, where the effective fluid viscosity, and the mass density of the fluid, respectively, and represents the body forces. Furthermore, and are gradient (22) | and Laplasian operators, respectively and is 2 | perfect derivative which can be defined as below: Which , and are the mass density of , (27) nanoshell, piezoelectric layer and surfaces, respectively. Substituting Eqs. (26) and (27) into (25) we obtain In Eq. (20), the stresses and moment resultants are the pressure of fluid as: defined in appendix A. The work done by the surrounded viscoelastic 2 medium including the visco-Pasternak medium and viscous damping and the electrostatic force can be (28) written as [37-39] . (23) With multiple both side of above equation at the cross-sectional area of the internal fluid ( ), radial force is expressed as: 2 Υ cos (24) (29) 2 . cosh 1 For electrostatic force, , and Υ 8.85 10, respectively, are electrode

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Considering the slip boundary condition in the There are two ways to obtain the equations of analysis for exact investigation of the behaviour of motion for a mechanical system: Hamilton's the nanoflow, the average velocity correction factor principle and Euler–Lagrange equation. Hamilton's is introduced to establish the relation between principle is the main procedure to obtain the slip average velocity and no-slip average equations of motion and boundary conditions. The velocity , i.e.; equations of motion and corresponding boundary conditions of the piezoelectric nano shell can be , (30) derived from Hamilton’s principle and by taking the variations of displacements , and and where coefficient can be written as [40]: , 1 integrating by parts, and by equating the coefficients 2 of , and to zero, the governing equations 1 , (31) 4 of motion are derived as: 1 1 where the slip of flow from inner nanotube through : , (35) Knudsen number () is considered and for practical purposes, the tangential momentum 1 : , (36) accommodation coefficient 0.7. Other parameters are: 2 1 2 : , 1 64 (32) , 2 1 1 3 1 Where 4 and 0.4. And also for slip boundary condition, the value of parameter is set to be 1. Hence, the external work of the fluid can be expressed as: 2 (37) 2 Υ cos , (33 2 cosh 1 ) and boundary conditions are obtained as follows: 1 0 0, (38a) 1 0 0, (38b) 1 0 (38c) 1 1 The equations of motion and corresponding boundary conditions of the piezoelectric shell can be 0, derived from Hamilton’s principle (34)

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1 ΩL 0 0, ̅ , ̅ , (38d) 1 1 2 0 0, , (38e)

But Euler–Lagrange method is an easier way to 2 obtain the equations of motion. The resulting ̅ , , equations can be solved by the assumed mode method. In the assumed modes method, the Υ , , , admissible functions must satisfy the geometric boundary conditions and there is no need to satisfy Respectively, dimensionless strain and kinetic the natural boundary conditions. Hence, solving the energies are obtained as follows: equations with this method reduces significantly the 1 complexity of the problem. Therefore, in this work, 2 the assumed mode method is used to obtain the ̅ ̅ equations of motion using Euler–Lagrange method. Following dimensionless parameters are used: ,̅ , , , , ̅ ̅ ̅ , , ̅ ̅ ̅ , , ∗ ̅∗ ̅ ̅ , , ∗ ∗ , , ∗ ∗ ∗ ∗ , , ∗ ∗ ∗ ,∗ , ( 40) ∗ ∗ ∗ ,∗̅ , ̅ ∗ ∗ (39) ∗ ̅ ∗ ̅ ̅ , , ∗ ∗ ∗ ,∗ , ∗ ∗ ∗ ∗ , , ∗ ∗ ,̅ , , 1 , , ,

, , 2 Ω,Ω , 2 Ω ̅ , ,

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cos ̅ 2 cosh 1 ,

̅ Using electrostatic force in according to Eq. (42c), 1 complicate solution of the obtained equation of (41) 2 motion. Hence it is necessary to express it to the polynomial form. For this purpose, Taylor where coefficients of 1. .51 are introduced expansion or curve fitting methods can be used. For in Appendix B. curve fitting method, the nonlinear electrostatic Also, dimensionless work done by viscoelastic force is approximated with polynomial with desired medium, potential due to external electric voltage, order. This is a standard problem in optimization visco-pasternak effect and harmonic force, theory and can be implemented in different respectively, obtained as follows: mathematical software such as Matlab, Maple and Mathematica. For example lsqcurvefit function in Matlab which solve nonlinear curve-fitting (data- fitting) problems in least-squares sense can be used here. Hence with expressing the electrostatic force (42a) as a polynomial form, the dimensionless work done by electrostatic force can be express as follows: ̅ cos (43) ̅ ̅ ̅ ̅ ⋯̅ which ̅ ̅ are constant. 2̅ ̅ 5 Solution procedure (42b) In this section, first, discretizing equations of motion is expressed by applying the assumed mode ̅ method and then the complexification-averaging method is applied for studying of the steady state ̅ response of the system. ̅ 5.1 Discretizing equations of motion ̅ In this section, by applying the assumed mode method, the in-plane, transverse and shear deformations can be expressed as general coordinates and mode shape functions that satisfy (42c) the geometric boundary conditions, as follows [33]: , , , , (44) , ,

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cos coefficients of mass, stiffness, linear and nonlinear ,, term matrixes and applied loads Eqs. (46) - (48) are ,, sin presented in Appendix C. sin ,, Natural frequencies and mode shapes can be ,, cos obtained from solving following eigenvalue ,, cos equation: sin ,, (49) , 0,

,

, 6 Results and Discussions , In this section, first, comparing the present numerical results of macroscopic cylindrical shell ,, with previously published in the literature is where , and are modal functions presented for arbitrary boundary conditions and which satisfy the required geometric boundary convergence study of a Piezoelectric biomedical conditions. , and are unknown nanosensor (PBMNS) based on nano-shell with functions of time and are related to dynamical simply supported boundary condition is investigated response. for Gurtin–Murdoch surface/interface theory. Then Substituting Eqs. (44) into Eqs. (40)- (43) and estimation of the natural frequency, critical fluid applying the Lagrange equtions: velocity and stability analysis of piezoelectric biomedical nanosensor conveying viscous (45) bloodstream is investigated using the electro-elastic Gurtin–Murdoch surface/interface theory and , ,̅, considering von-karman-Donnell's shell model. In Where . order to simplify the presentation, CC, SS, CS and Results in the following reduced-order model of the CF represent clamped edges, simply supported system: edges, clamped-simply supported edges and clamped-free edges, respectively and also for simplification of surface effect is represented SE. ̅ (46) , The material properties for nonhomogeneous nano- shell and piezoelectric layer are shown in Table 1-2, ̅ respectively [42]. ̅ (47) , Table 1. Properties of stainless steel and nickel [42] Stainless steel Nichel ̅ ̅ 208 0.381 8166 205 0.31 8900 Table 2. Properties of PZT-4 [42] (48) ⁄ ⁄ 10 95 0.3 5.2 5.2 560 7500 2 , ̅ ̅ The material and geometrical parameters used in all ̅ ̅ following results are shown in Table 3. where , and are mass, damping and linear stiffness matrixes. , , and Table 3. The material and geometrical parameters are second-order nonlinear stiffness [20, 37, 43-44] ⁄ ⁄ ⁄ ⁄ ⁄ matrixes and is third-order nonlinear stiffness matrix. Also, , and are the 110 10 0.02 0.02 4.488 2.774 ⁄ ⁄ ⁄ linear stiffness, second and third order nonlinear ⁄ stiffness matrixes for electrostatic force expansion, ⁄ respectively. Also, , and are applied 0.6 3.17 1 110 4.488 2.774 0.6 ⁄ . ⁄ ⁄ ⁄ loads by piezoelectric voltage and surface stress. All ⁄ ⁄

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3 10 3 10 5.61 1 1 8.99950 2.07127 10 10 Complete convergence for the dimensionless natural ⁄ ⁄ . frequencies Ω and dimensionless damping 3 1 2 1 310 1060 frequencies Ω of SS piezoelectric biomedical nanosensor considering with the Gurtin–Murdoch Of course, the geometrical parameters can be surface/interface theory and the material and varying according to the type of problem. In this geometrical parameters of Tables (1-3) is shown in paper, the results are presented in dimensionless Tables 5 and 6, respectively. form and thus the results are not limited to a particular type of matter. The data presented in the Table 5. Convergence of dimensionless natural form of sample data to approximate the numbers frequencies Ω of the SS piezo piezoelectric elastic used in the actual range. biomedical nanosensor 1 2 3 5 2 0.8107340 0.8107351 0.8107351 0.8107351 6.1 Convergence and comparison studies 4357 0782 0781 6796 The method proposed in this paper is validated by 3 1.8151243 1.8151572 1.8151571 1.8151580 comparing the present numerical results with 0896 0073 8273 8628 previously published in the literature. If we neglect 4 2.8327282 2.8328471 2.8328467 2.8328504 the piezoelectric, visco-Pasternak and surface 9570 2539 7062 1730 effects, the present model can be reduced to the macroscopic cylindrical shell model. The Table 6. Convergence of dimensionless damp frequencies Ω of the SS piezoelectric biomedical dimensionless natural frequencies nanosensor Ω1 ⁄ of present work are compared 1 2 3 5 2 0.8062114 0.8062124 0.8062124 0.8062124 with macroscopic cylindrical shell which previously 8043 2314 2313 7871 given by Loy et al. [29] that is shown in Table 4 for 3 1.8029349 1.8029658 1.8029658 1.8029666 the three classical boundary conditions. The 1873 5533 4049 9801 parameters used in this example are: 1, ⁄ 4 2.8103357 2.8104503 2.8104500 2.8104535 20, ⁄ 0.01, and 0.3. It can be observed 1036 3736 1216 3764 from Table 5 that the present results agree very well with the reference solutions, which indicates that the It is observed that the number of polynomial method presented in this paper is suitable and of terms, , is increased, the value of the frequency high accuracy for free vibration analysis of parameter (Ω,Ω), converges rapidly. With cylindrical shells with classical boundary considering of the two succesasing values of , it is conditions. The slight differences in the results may shows that as increases, the percentage difference be attributed to the different shell theories and between the successive frequency approximations solution approaches adopted in the literature and in decreases. Thus the error as shown above is less 1 this paper. per cent, which is well within the limits of engineering tolerance. The minimum frequency in Table 4. Comparison of dimensionless natural this case is associated with the circumferential wave frequencies Ω1 ⁄ for SS, SC number 2. This assertion is valid for the entire and CC boundary conditions for a homogeneous range of shell. cylindrical shells with 1, ⁄ 20, ⁄ 0.01, and υ0.3. SS CS CC 6.2 Parametric study Presen Loy Presen Loy Presen Loy The convergence and comparison study of the t [29] t [29] t [29] present work was verified in the previous 1 0.016 0.016 0.023 0.023 0.032 0.032 subsection. In this subsection, dimensionless 101 101 299 974 074 885 frequencies Ω versus dimensionless fluid 2 0.009 0.009 0.010 0.011 0.013 0.013 (bloodstream) velocity for stability analysis and 235 382 963 225 202 932 estimation of critical fluid velocity of a SS 3 0.021 0.022 0.020 0.022 0.019 0.022 piezoelectric biomedical nanosensor (PBMNS) are 753 105 953 310 713 672 presented in Figures 2-10 with respect to the 4 0.039 0.042 0.041 0.042 0.041 0.042 307 095 300 139 386 208 different geometrical and material parameters such

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as the ratio of the length of the nanosystem to the velocity for SS piezoelectric biomedical radius ⁄ , the ratio of nanoshell thickness to nanosensor is illustrated in Figure 3. It is evident radius ⁄, the ratio of the piezoelectric thickness that natural frequencies and critical fluid velocity of to the radius ⁄, the effects of the viscoelastic the PBMNS increase with increasing ⁄ . As can medium ( , , ̅ ), the piezoelectric voltage be seen, the critical fluid velocity corresponds to the and also surface energy effects. As shown in these lower value of ⁄ 5 sooner than the rest values figures, the frequency is decreased as the flow of this ratio are reached to be zero. This physically velocity is increased. implies that first the PBMNS in ⁄ 5 losses its Figure 2 demonstrate in the dimensionless stability due to the divergence via a pitchfork frequency (Ω) versus fluid velocity () for the first bifurcation. In addition, the length-to-small radius five vibration modes of the PBMNS. It can be seen ratio of cylindrical shell has an important effect on natural frequency. The reason is that a higher ⁄ that Ω reduces with increased . For zero natural frequency, PBMNS becomes unstable and the ratio leads to increase in the nanoshell stiffness, and corresponding fluid velocity is called the critical cause to higher natural frequencies of nanoshells. flow velocity. As can be seen, the critical fluid velocity correspond to the 1 and 2 modes is reached at the vicinity of ≅ 6.894 . This physically implies that the DWBNNT losses its stability due to the divergence via a pitchfork bifurcation while the 3, 4 and 5 modes are still stable. Thereafter, for the fluid velocity within the range 6.894 7.23, the first amd second modes are zero, which the system becomes unstable. As the flow velocity reaches about ≅7.23 , natural frequencies in the 1 mode return to positive and the system tends to regains stability in the first mode, while, the 2 mode still remain instable. Also, the PBMNS becomes unstable at 3 Fig. 3. The dimensionless frequencies Ω versus fluid mode when ≅ 7.248. This phenomenon may be velocity of SS piezoelectric biomedical observed in different modes for higher velocities in nanosensor for different values of ⁄ ratio this figure. The same behavior can also be observed for other vibration modes of following figures. Figure 4 illustrates the effect of thickness shell to small radius ratio ⁄ on dimensionless natural frequencies Ω versus dimensionless fluid velocity for SS piezoelectric biomedical nanosensor.

Fig. 2. Dimensionless frequencies Ω versus fluid velocity for the first five vibration modes of SS piezoelectric biomedical nanosensor

Fig. 4. The dimensionless frequencies Ω versus fluid The effect of length-to-small radius ratio (⁄ ) on velocity of SS piezoelectric biomedical dimensionless natural Ω versus dimensionless fluid nanosensor for different values of ⁄ ratio

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It can be seen that with the increasing of the nanoshell stiffness ratio (⁄), the natural frequency Ω and the critical fluid velocity decrease. Also, the critical fluid velocity corresponds to the higher value of ⁄ 0.05 sooner than the rest values of this ratio are reached to be zero. This physically implies that first the PBMNS in ⁄ 0.05 losses its stability due to the divergence via a pitchfork bifurcation. The effect of piezoelectric thickness to small radius ratio (⁄) on dimensionless natural frequencies Ω versus dimensionless fluid velocity is presented in Figure 5. It can be seen that unlike Fig. 6. The dimensionless frequencies Ω versus fluid previous results for ⁄ ratio, in this case with the velocity of SS piezoelectric biomedical increasing of the piezoelectric thickness to small nanosensor for different values of radius ratio, the natural frequency Ω and the critical fluid velocity increase. Also, the critical fluid This is perhaps because increasing Winkler and velocity corresponds to the lower value of ⁄ Pasternak coefficients increases the shell stiffness. 0.005 sooner than the rest values of this ratio are For zero natural frequency, PBMNS becomes reached to be zero. This physically implies that first unstable and this physically implies that the the PBMNS in ⁄ 0.005 losses its stability PBMNS losses its stability due to the divergence via due to the divergence via a pitchfork bifurcation. a pitchfork bifurcation.

Fig. 5. The dimensionless frequencies Ω versus fluid velocity of SS piezoelectric biomedical Fig. 7. The dimensionless frequencies Ω versus fluid velocity of SS piezoelectric biomedical nanosensor for different values of ⁄ ratio nanosensor for different values of Figures 6 and 7 respectively illustrate dimensionless stiffness coefficient of Winkler foundation and Figure 8 presents the effect of dimensionless ̅ shear layer of Pasternak foundation on damping coefficient on dimensionless damp frequencies Ω versus dimensionless fluid velocity dimensionless natural frequencies Ω versus for SS piezoelectric biomedical nanosensor. It dimensionless fluid velocity for SS piezoelectric biomedical nanosensor. In both case, it can be seen can be seen that with the increasing of damper coefficient, the natural frequency Ω and the critical that with the increasing of and ratios, the natural frequency Ω and the critical fluid velocity fluid velocity decrease. Also, the critical fluid ̅ increase. velocity corresponds to the higher value of sooner than the rest values of this ratio are reached

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to be zero and system losses its stability due to the divergence via a pitchfork bifurcation.

Fig. 9. The dimensionless frequencies Ω versus fluid velocity of SS piezoelectric biomedical Fig. 8. The dimensionless frequencies Ω versus fluid velocity of SS piezoelectric biomedical nanosensor for different cases of Visco-medium nanosensor for different values of ̅ The effect of piezoelectric voltage on Figure 9 presents the effect of several of dimensionless natural frequencies Ω versus surrounding medium on dimensionless natural dimensionless fluid velocity for SS piezoelectric biomedical nanosensor is illustrated in Figure 10. frequencies Ω with respect to the dimensionless fluid velocity . As can be seen, without medium, Winkler ( ), Pasternak ( ), visco ( ̅ ), visco- Winkler ( ̅ ), visco- Pasternak ( ̅ ), Winkler-Pasternak ( ) and Visco-medium ( ̅ ), are eight assumptions medium in this work. It can be founded, that the frequency and the critical velocity have minimum and maximum values when the system respectively has with and without damping coefficient. In this results indicate that the system tends to be very sensitive to whether or not there is a damping coefficient in all cases. Also, it is worth mentioning that critical velocity in Pasternak foundation is more than critical velocity predicted by Winkler foundation. This is due to the fact that the Winkler foundation describes the effect of the normal stress of the elastic medium, whereas Fig. 10. The dimensionless frequencies Ω versus Pasternak elastic foundation describes the effect of fluid velocity of SS piezoelectric biomedical the tangential and normal stresses of the elastic nanosensor for different values of medium.

As shown in this Figure, with the increasing of , the natural frequency Ω and the critical fluid velocity increase. This is perhaps because increasing piezoelectric voltage increases the nanoshell stiffness. For zero natural frequency, PBMNS becomes unstable and this physically implies that the PBMNS losses its stability due to the divergence via a pitchfork bifurcation. And in end, dimensionless natural frequencies Ω versus dimensionless fluid velocity for SS piezoelectric biomedical nanosensor with and

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without surface energy effects are shown in Figures that the methods are suitable and of high 11. As show in results, the most important accuracy for free vibration analysis of parameters on natural frequencies and fluid velocity cylindrical nanoshell. is surface density (, ) that with regardless of  In convergence study, as the number of this parameter, the critical fluid velocity sooner than polynominal terms, , is increased, the value of the rest parameters are reached to be zero and the frequency parameter, Ω, converges rapidly system losses its stability due to the divergence via a and also the convergence mode number is pitchfork bifurcation but in this case system has 2 and 2 and this means that two maximum natural frequency due to higher stiffness longitudinal mode and in according to Eq. (45), of nanoshell in this case. Also when all surface five peripheral modes, and totally fifteen effect is considered, the critical fluid velocity later number of mode shapes can represent the than the rest parameters are reached to be zero. response of system with sufficient degrees.  In the 1 and 2 modes, at the vicinity of the critical flow velocity 6.894, natural frequency equal to zero, where divergence instability occurs due to pitchfork bifurcation while at this critical flow velocity, the 3, 4 and 5 modes still remain stable.  As the flow velocity reaches about 7.23 , natural frequencies in the 1 mode return to positive and the system tends to regains stability in the first mode, while, the 2 mode still remain instable. The same behavior can also be observed for other vibration modes.  For all following results, zero natural frequency of PBMNS becomes unstable and this

physically implies that the PBMNS losses its Fig. 11. Surface energy effects on the dimensionless stability due to the divergence via a pitchfork frequencies Ω versus fluid velocity of SS bifurcation. piezoelectric biomedical nanosensor  Natural frequencies and critical fluid velocity increase with increasing of ⁄ , ⁄, , and and the critical fluid velocity corresponds 7 Conclusion to the lower value sooner than the rest values of In current study, the natural frequency, this ratio are reached to be zero. This is perhaps critical fluid velocity and stability analysis of because increasing Winkler and Pasternak piezoelectric biomedical nanosensor (PBMNS) coefficients increases the shell stiffness. based on cylindrical nanoshell conveying viscous  with the increasing of damper coefficient and fluid is investigated using the electro-elastic Gurtin– ⁄ ratio the natural frequency Ω and the Murdoch surface/interface theory, Hamilton’s critical fluid velocity decrease. principle and also the assumed mode method combined with Euler – Lagrange. This system  the frequency and the critical velocity have subjected to nonlinear electrostatic field, minimum and maximum values when the viscoelastic medium including visco-pasternak and system respectively has with and without damping coefficients. The convergence, accuracy damping coefficient. In this results indicate that and reliability of the current formulation are the system tends to be very sensitive to whether validated by comparisons with existing results. Also or not there is a damping coefficient in all cases. stability analysis and dimensionless natural  the most important parameters on natural frequency versus dimensionless fluid velocity of a frequencies and fluid velocity is surface density piezoelectric biomedical nanosensor are accurately (, ) that with regardless of this studied with respect to the different geometrical and parameter, the critical fluid velocity sooner than material parameters. the rest parameters are reached to be zero but in Some conclusions are obtained from this study: this case system has maximum natural  With comparing the previously published in the frequency due to higher stiffness of nanoshell in literature, the present results agree very well this case. with the reference solutions, which indicates

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 when all surface effect is considered, the critical ∗ , fluid velocity later than the rest parameters are reached to be zero. , ( A.5) Conflict of interest 1 The authors report no conflict of interest. 1 2 1 ( A.6) ∗ 1 2 1 Funding Acknowledgement ∗ , ‘This research received no specific grant from any funding agency in the public, commercial, or not- 1 2 for-profit sectors’. 1 2 1 Appendix A ( ( A.7) The stresses and moment resultants: 1 2 1 1 2 ,, ∗ 1 ∗ , , ( A.8) , , ( A.1) in which , , ∗ , 1 ∗ , , , 2 ∗ , (A.9) ∗ ∗ ∗ ∗ ∗ ∗ 1 , , ,, ∗ ∗ ∗ ∗ ∗ ∗ 2 , , and ,, , , 1, , , , , (A.10) ,, ,, ( A.2) 1, , , 1 , , 2 , , 1 (A.11) , , , , 2 , , (A.12) 1 2 , , ( A.3) ∗ ∗ 2 2 , 1 ∗ ∗ , ∗ ∗ , (A.13) ∗ , 2 2 1 2 1 ∗ ∗ 2 2 2 , ( A.4) (A.14) ∗ ∗ 1 , ∗

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∗ 2 2 , 1 , 2 2 ∗ ∗ 2 ∗ 2 , ∗ ∗ 2 , , (A.15) 1 ∗ 2 2 Note that, because of geometric symmetry, the , expressions is zero, i.e. 0. 2 ∗ Appendix B 2 1 , ̅ , ̅ , 1 2 2 ∗ (A.16) ̅ ̅, ̅, ̅ ̅, 2 1 , ̅ 1 2 ̅ ̅ , 2 2 2 ̅ ̅ ̅ ∗ , , 2 ̅ ̅ , 2 ̅ ̅ , 2 2 , 1 ̅ ̅, 2 2 ∗ (A.17) 2̅ ̅ ̅ , 1 2 ̅ , ̅ , 2 , 1 ̅ 2 ̅ ̅ , ∗ 1 ̅ ̅ ̅ , 4 2 , ̅ ̅ ̅ , 1 4 2 ̅ ̅ ̅ ∗ (A.18) 4 , 4 ̅ ̅ , 2 2 , ̅ 1 ̅ ̅ , 2 4 ∗ , , (A.19) 1 2,

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4 , , 1 χ 2 2 , αχχ α , , 2 , 1 αχ α 2 , 2 1 2 ̅ 1 ̅ , 1 αχ α 2 2 α χ α χ 2 ̅ 1 ̅ , αχ 2 1 4 αχ 2 , α 1 2 ̅ 1 ̅ , α χ , 2 1 , , 4 , , 1 2 , 2 ̅ ̅ , 1 α χ α , ̅ ̅ , 2 ̅ ̅ , α α 1 ̅ ̅ , 2 α α ̅ ̅ , 1 2 1 2 α α ̅ ̅ 1, α 1 α ̅ ̅ 1, 2 α 1 ̅∗ , ̅ ∗ , 1 2 2 α 1 2 ∗̅ ∗̅ , , 2 2 1 2 ∗̅ ∗̅ ∗̅ 1 , , , 2 4 4 α α 2 ̅ Appendix C 1 ̅ ̅ 2 ̅ ̅ ,

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1 α χ α 2 αχ α χ 4α 4α 1 1 2α α α 2 2α 2 α α 2α α α α 1 α α α 2α 2 α 2α 2α 1 α α , , 2 2α 2α 1 , ̅ , ̅ 2 ̅ , ̅ , ̅ , ̅ ̅ , References: [1] Chatterjee, K., Sarkar, S., Rao K. J. and Paria, α S., Core/shell nanoparticles in biomedical α applications, Advances in Colloid and Interface 2α χ Science, Vol.209, 2014, pp.8–39. [2] Melancon, M. P., Lu, W., Zhong, M., Zhou, 2α χ 1 M., Liang, G., Elliott, A. M., Hazle, J.D., 2 α Myers, J. N., Li, C. and Stafford R. J., Targeted multifunctional gold-based nanoshells for α 2α magnetic natural-guided laser ablation of head and neck cancer, Biomaterials. Vol. 32 No.30, 2α 1 2011, pp. 7600–7608. 2 α [3] Riley, R. S., O’Sullivan, R. K., Potocny, A.M., Rosenthal, J. and Day, E. S., Evaluating α Nanoshells and a Potent Biladiene α Photosensitizer for Dual Photothermal and 2α Photodynamic Therapy of Triple Negative α Breast Cancer Cells, Nanomaterials, 2α doi:10.3390/nano8090658, 2018. [4] Sobral-Filho, R. G., Brito-Silva, A. M., α Isabelle, M., Jirasek, A., Lum, J. J. and Brolo 2α A. G., Plasmonic labeling of subcellular α compartments in cancer cells: multiplexing 1 2α with fine-tuned gold and silver nanoshells, 2 α Chemical Science, Vol. 8, 2017, pp. 3038– 2α 3046. [5] Ayala-Orozco, C., Urban, C., Knight, M. W., α Urban, A. S., Neumann, O., Bishnoi, S. W., 2α Mukherjee, S., Goodman, A. M., Charron, H., α Mitchell, T., Shea, M., Roy, R., Nanda, S., Schiff, R., Halas, N. J. and Joshi, A., Au α Nanomatryoshkas as Efficient Near-Infrared α Photothermal Transducers for Cancer

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Treatment: Benchmarking against Nanoshells, [17] Jalili, N., Piezoelectric-Based Vibration ACS Nano. Vol. 8, No. 6, 2014, pp. 6372-81. Control: From Macro to Micro/Nano Scale [6] Choi, Y.E., Kwak, J.W. and Park, J. W., Systems, Springer, New York, 2010. Nanotechnology for Early Cancer Detection, [18] Gurtin, M.E. and Murdoch, A.I., A continuum Vol. 10, 2010, pp.428-455. theory of elastic material surface, Archive for [7] Singhana, B., Slattery, P., Chen, A., Wallace, Rational Mechanics and Analysis, Vol.57 No.4, M. and Melancon, M. P., Light-Activatable 1975, pp. 291–323. Gold Nanoshells for Drug Delivery [19] Gurtin, M.E. and Murdoch, A.I., Surface stress Applications, AAPS PharmSciTech, Vol. 15 in solids, International Journal of Solids and No.3, 2014, pp. 741–752. Structures, Vol.14 No.6, 1978, pp. 431–40. [8] Riley, R.S., Day, E.S., Gold nanoparticle- [20] Zhu, C.S., Fang, X.Q. and Liu, J.X., Surface mediated photothermal therapy: Applications energy effect on buckling behavior of the and opportunities for multimodal cancer functionally graded nano-shell covered with treatment, Wiley Interdisciplinary Reviews piezoelectric nano-layers under torque, Nanomedicine and Nanobiotechnology, International Journal of Mechanical Sciences, DOI:10.1002/wnan.1449, 2017. Vol.133, 2017, pp. 662–673. [9] Eloy, J.O., Petrilli, R., Lopez, R.F.V. and Lee, [21] Fang, X.Q., Zhu, C.S. and Liu, J.X., Surface R.J., Stimuli-responsive nanoparticles for energy effect on free vibration of nano-sized siRNA delivery, Current Pharmaceutical piezoelectric double-shell structures, Physica Design, Vol.21, 2015, pp. 4131–4144. B: Condensed Matter, Vol. 529 No.15, 2018, [10] Li, C., Thostenson, E. T. and Chou, T.W., pp.41-56. Sensors and actuators based on carbon [22] Ghorbanpour Arani, A., Amir, S., Dashti, P. nanotubes and their composites: A review, and Yousefi M., Flow-induced vibration of Composites Science and Technology, Vol. 68, double bonded visco-CNTs under magnetic 2008, pp.1227–1249. fields considering surface effect, [11] Mousavi, S. M., Hashemi, S. A., Zarei, M., Computational Materials Science, Vol.86, Amani, A. M. and Babapoor, A., Nanosensors 2014, pp.144–154. for Chemical and Biological and Medical [23] Ghorbanpour Arani, A., Kolahchi, R. and Applications, Medicinal Chemistry, Vol. 8 No. Hashemian, M., Nonlocal surface 8, 2018, pp.205-217. piezoelasticity theory for dynamic stability of [12] Mosayebi, R., Ahmadzadeh, A., Wicke, W., double-walled boron nitride nanotube Jamali, V., Schober, R. and Nasiri-Kenari, M., conveying viscose fluid based on different Early Cancer Detection in Blood Vessels Using theories, Proceedings of the Institution of Mobile Nanosensors, arXiv,1805.08777, 2018. Mechanical Engineers, Part C: Journal of [13] Kosaka, P. M., Pini, V., Ruz, J. J., da Silva, R. Mechanical Engineering, Vol.228 No.17, 2014, A., González, M. U., Ramos, D., Calleja, M. pp.3258-3280. and Tamayo, J., Detection of cancer biomarkers [24] Fereidoon, A., Andalib, E. and Mirafzal, A., in serum using a hybrid mechanical and Nonlinear Vibration of Viscoelastic Embedded- optoplasmonic nanosensors, Nature DWCNTs Integrated with Piezoelectric Layers- Nanotechnology, DOI: Conveying Viscous Fluid Considering Surface 10.1038/NNANO.2014.250, 2014. Effects, Physica E: Low-Dimensional Systems [14] Aspinwall, C. A., Nanoshell sensors for cellular and Nanostructures, Vol.81, 2016, pp. 205– analysis, National Institute of Health (NIH), 218. Project number: 5R01GM116946-042019, [25] Ansari, R., Gholami, R., Norouzzadeh, A. and 2019. M. A. Darabi, Surface Stress Effect on the [15] Duan, W. H., Wang, Q. and Quek, S.T., Vibration and Instability of Nanoscale Pipes Applications of Piezoelectric Materials in Conveying Fluid Based on a Size-Dependent Structural Health Monitoring and Repair: Timoshenko Beam Model, Acta Mechanica Selected Research Examples, Materials, Vol.3, Sinica, Vol.31, 2015, pp. 708-719. 2010, pp.5169-5194. [26] Ye, T., Jin, G., Chen, Y., Ma, X. and Su, Z., [16] Rupitsch, S. J., Piezoelectric Sensors and Free vibration analysis of laminated composite Actuators: Fundamentals and Applications, shallow shells with general elastic boundaries, Springer, New York, 2019. Composite Structures, Vol.106, 2013, pp. 470– 490.

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[27] Fazzolari, F.A., A refined dynamic stiffness in bundle of CNTs using nonlocal element for free vibration analysis of cross-ply piezoelasticity cylindrical shell theory, laminated composite cylindrical and spherical Composites: Part B, Vol. 43, 2012, pp.195– shallow shells, Composite Structures, Part B, 203. Vol.62, 2014, pp.143–158. [39] Farokhi, H., Païdoussis, M.P. and Misra, A., A [28] Mirza, S. and Alizadeh, Y., Free vibration of new nonlinear model for analyzing the partially supported cylindrical shells, Shock behaviour of carbon nanotube-based resonators, and Vibration, Vol.2 No. 4, 1995, pp.297–306. Journal of Sound and Vibration, Vol. 378, [29] Loy, C.T., Lam, K.Y. and Shu, C., Analysis of 2016, pp. 56-75. cylindrical shells using generalized differential [40] Wang, L. and Ni, Q., A reappraisal of the quadrature, Shock and Vibration, Vol.4 No.3, computational modelling of carbon nanotubes 1997, pp.193–198. conveying viscous fluid, Mechanics Research [30] Naeem, M.N. and Sharma, C.B., Prediction of Communications, Vol.36, 2009, pp.833–837. natural frequencies for thin circular cylindrical [41] Reddy, J.N. and Wang, C.M., Dynamics of shells, Proceedings of the Institution of fluid-conveying beams: governing equations Mechanical Engineers Part C: Journal of and finite element models, Centre for Offshore Mechanical Engineering Science, Vol.214 Research and Engineering National University No.10, 2000, pp.1313–1328. of Singapore, Core report No. 2004-03, 2004. [31] Zeighampour, H., Tadi Beni, Y. and [42] Jafari, A.A., Khalili, S.M.R. and Tavakolian, Botshekanan, D. M., Wave propagation in M., Nonlinear vibration of functionally graded viscoelastic thin cylindrical nanoshell resting cylindrical shells embedded with a on a visco-Pasternak foundation based on piezoelectric layer, Thin Wall Structures, nonlocal strain gradient theory, Thin-Walled Vol.79, 2014, pp.8–15. Structures, Vol.122, 2018, pp. 378–386. [43] Mohammadimehr, M. and Rostami R., Bending [32] Donnell, L. H., Beam, Plates and Shells, and vibration analyses of a rotating sandwich McGraw-Hill, New York, 1976. cylindrical shell considering nanocomposite [33] Amabili, M., Nonlinear Vibrations and core and piezoelectric layers subjected to Stability of Shells and Plates, Cambridge thermal and magnetic fields, Applied University Press, New York, 2008. Mathematics and Mechanics, Vol.39 No. 2, [34] Sabzikar Boroujerdy, M. and Eslami, M.R., 2018, pp.1-22. Axisymmetric snap-through behavior of Piezo- [44] Lide, D. R., CRC Handbook of Chemistry and FGM shallow clamped spherical shells under Physics (86th Edition). Boca Raton (FL): CRC thermo-electro-mechanical loading, Press (an imprint of Taylor and Francis International Journal of Pressure Vessels and Group): Boca Raton, F.L., 2005, 2544 pp. Piping, Vol.120-121, 2014, pp. 19-26. [35] Kirchhoff, G., Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, Journal fur reine und angewandte Mathematik, Vol.40, 1850, pp.51-88. [36] Lu, P., He, L.H., Lee, H.P. and Lu, C., Thin plate theory including surface effects, International journal Solids and Structure, Vol.43 No.16, 2006, pp. 4631–47. [37] Ghorbanpour Arani, A., Kolahchi, R. and Hashemian M., Nonlocal surface piezoelasticity theory for dynamic stability of double-walled boron nitride nanotube conveying viscose fluid based on different theories, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, DOI: 10.1177/0954406214527270, 2014. [38] Ghorbanpour Arani, A., Amir, S., Shajari, A.R. and Mozdianfard, M.R., Electro-thermo- mechanical buckling of DWBNNTs embedded

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