Pulsating ow induced parametric instabilities of a smart ...scientiairanica.sharif.edu/article_21201_efb7ce186236d77...dynamics of pipes conveying pulsating uid with a combination

Scientia Iranica B (2020) 27(2), 730{744

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Pulsating ow induced parametric instabilities of asmart embedded micro-shell based on nonlocalpiezoelasticity theory

V. Atabakhshian and A. Shooshtari�

Department of Mechanical Engineering, Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran.

Received 21 January 2018; received in revised form 16 September 2018; accepted 29 December 2018

KEYWORDSDynamical instability;Pulsating ow;Smart materials;Cylindrical shell;Nonlocal theory.

Abstract. In this study, the dynamical instabilities of a smart embedded micro-shellconveying pulsating uid ow are investigated based on nonlocal piezoelasticity theory andnonlinear cylindrical shell model. The micro-shell is surrounded by an elastic foundation,which is suitable for both Winkler spring and Pasternak shear modules. The internal uid ow is considered to be purely harmonic, irrotational, isentropic, Newtonian, andincompressible and is mathematically modeled using linear potential ow theory, time meanNavier Stokes equations, and Knudsen number. To bring the micro-scale problem closerto the reality, the pulsating viscous e�ects and the slip boundary condition are also takeninto account. Employing the modi�ed Lagrange equations of motion for open systems,the nonlinear coupled governing equations are achieved and, consequently, the instabilityboundaries are obtained using Bolotin's method. In the section of numerical results, acomprehensive discussion about the dynamical instabilities of the system is presented (suchas divergence, utter, and parametric resonance). It was found that the application of thepositive electric potential �eld would improve the stability of the system as an actuator oras a vibration amplitude controller in the micro electromechanical systems.© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

Undoubtedly, nanotechnology is and will be the basisof the future industrial revolution, and the discovery ofnanotubes is one of the major improvements that canspeed up this revolution. Generally, nanotechnologyinvolves the construction and application of biological,chemical, and physical systems, ranging from microto nanometer. Nowadays, uid-conveying nanotubesare actively researched and studied due to their exten-sive possible applications in biological systems (e.g.,biological separation, biosensing, molecular imaging),

*. Corresponding author. Tel.: +98 81 38272410E-mail address: [email protected] (A. Shooshtari)

doi: 10.24200/sci.2018.50293.1627

medicine (e.g., drug delivery), chemistry (e.g., fuelstoring cells, chemical analyses), physics (e.g., opticalstructures), and other research areas [1{3].

The dynamic and instability analyses of uid-conveying pipes are one of the important challengesin the �eld of uid conveying systems, which basi-cally started by the Trans-Arabian pipeline project in1950 [4]. Certainly, the widespread applications ofpipes conveying or containing uids in various indus-tries such as oil and gas, water and sewage, militaryand aerospace, and medicine and chemistry are enoughto attract researchers' attention in this regard. Someof the valuable pioneering works are as follows.

Paidoussis [5] examined uid and body interac-tions in slender structures that convey axial ows. Hemodeled and analyzed the dynamics and instabilities of uid-conveying systems. Amabili [6] studied nonlinear

V. Atabakhshian and A. Shooshtari/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 730{744 731

dynamics and instabilities of circular cylindrical shellsconveying uid and classi�ed uid instabilities forthem. He utilized various nonlinear shell models,modal expansion analyses, and energy approaches topresent a comprehensive analysis of the problem. Dy-namical behavior of the uid-conveying pipes basedon various types of beam theories was investigated byReddy and Wang [7]. They obtained equation of mo-tion via energy approaches, in which the �nite elementmethod was employed to solve the problem. Pellicanoand Amabili [8] studied dynamics and instabilities ofempty and uid-�lled circular cylindrical shells underexternal loadings. They used modal analysis approachto simulate the problem. Sadeghi and Karimi-Dona [9]employed Finite Element Method (FEM) and the statespace approach besides MATLAB program to modelthe dynamical behaviors of uid-conveying pipes witha moving sprung mass on it. Gu et al. [10] used theintegral transform technique to obtain the dynamicalresponse of uid-conveying pipes.

In the micro- and nano-scale problems and inthe �eld of biomechanics, the utter phenomenon ofveins conveying blood ow was investigated by Kammand Pedley [11]. Paidoussis [12] proposed a modelfor the Fluid-Structure Interaction (FSI) between theblood ow and veins. Moreover, in the �eld of nanomechanics, Yan et al. [13] studied the stability andcritical ow velocity of multi-walled carbon nanotubes(CNTs) conveying uid. Nonlinear vibration analysisof double-walled CNTs conveying uid was carried outby Kuang et al. [14]. Ghorbanpour Arani et al. [15]studied the ow-induced vibration and instability ofa smart polymeric micro tube subjected to the axialelectric �eld. They found that the imposition of electricpotential �eld had a signi�cant e�ect on the stability ofthe smart micro tube. In another work, GhorbanpourArani et al. [16] studied the nonlinear dynamicalresponse of embedded uid-conveyed micro-tubes rein-forced by Boron nitride nanotubes (BNNTs). In otherworks, Ghorbanpour Arani et al. [17,18] studied thedynamic stability of double-walled BNNTs conveyingviscose uid. Atabakhshian et al. [19] studied nonlineardynamic and instability of an elastically coupled CNT-PPB system with internal ow. They proposed anelastically coupled nano-beam system composed ofCNT conveying uid and Piezoelectric Polymeric Beam(PPB) polarized in the axial direction. They showedthat this system would improve the stability of CNTconveying uid by applying positive electric potentialto the coupled PPB.

In all of the above investigations, a constant owvelocity has been considered inside a structure, whichis not suitable for many empirical problems like thosewith pumps or turbines in the pipelines. Amongst thetime-dependent ow regimes inside a circular cylindri-cal shell, pulsating ow with purely harmonic nature

has attracted much attention amongst researchers dueto its important applications in industries. Practically,special pumps generate pulsating ow (e.g., beatingpumps in biomechanics) and, arbitrarily, a wide rangeof applications are classi�ed as purely harmonic irre-spective of small velocity perturbations. Moreover, ageneral time-dependent ow regime may be decoupledinto harmonic components as a Fourier series and,in such cases, the analysis may be considered for asingle Fourier component [20]. In this regard, nonlineardynamics of pipes conveying pulsating uid with acombination of parametric and internal resonances wasstudied by Panda and Kar [21]. Azrar et al. [22] studiedthe dynamic and parametric instabilities of single-walled CNTs conveying pulsating and viscous uid.Liang and Su [23] utilized the averaging method forthe stability analysis of single-walled carbon nanotubesconveying pulsating and viscous uid. Pipes conveyingpulsating uid and exposed to the external vortex wereanalyzed by Da et al. [24]. They employed multipletime-scale method to obtain the parametric resonancesof the system. Nonlinear dynamics of simply supportedpipes with motion constraints and internal pulsating uid was investigated by Wang [25]. Recently, micropulsating uid ow in heat pipes with alternate witheswas examined by Yang et al. [26]. Tubaldi et al. [27]modeled nonlinear dynamics of shells subjected topulsating uid ow. They utilized a special pulsatiletime-dependent function for blood ow velocity basedon the heart beating period. They also recentlyexamined in another work [28] the e�ects of pulse-wavepropagation on nonlinear dynamics of a shell conveyingpulsatile ow.

The simulation of the small-scale structures isusually done using two main categories: the MolecularDynamics (MD) method and the classical continuummechanics. It is clear that the MD method is acostly and time-consuming method whose applica-tion is limited to smaller nanostructures with a fewmolecules. Contrary to the MD method, the nonlocalelasticity theory was later developed [29] for larger scalenano/micro-structures based on classical continuummechanics such as classical Euler Bernoulli beam,Timoshenko beam, and shell theories [30{34].

Based on the above survey, the lack of properresearch on instability prediction and instability smartcontrol of uid-conveying microtubes can be clearlyfelt. Hence, this study investigates the ow-inducedvibrations and instabilities of a smart embedded micro-shell conveying pulsating uid, as schematically shownin Figure 1. Parametric excitation due to harmonicpulsation in the ow velocity, Knudsen number e�ects,pulsation viscose e�ects, small-scale parameter, Win-kler and Pasternak modules, and combined electro-mechanical loadings are taken into account. Theunsteady uid motion and the e�ect of viscosity for

732 V. Atabakhshian and A. Shooshtari/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 730{744

Figure 1. Schematic of a smart embedded micro-shellconveying pulsating uid.

turbulent ow are modeled using potential ow theoryand unsteady time-averaged Navier-Stokes equations,respectively. The governing equations of motion are de-rived using modal expansion analyses and the modi�edLagrangian approach for open systems. In the resultssection, divergence, utter, and parametric resonanceinstabilities are examined in detail. It is concludedthat imposing axial electric �eld may be used as acontrolling factor to improve the stability of the system.The results obtained in this study can be particu-larly utilized to design and manufacture nano/microelectromechanical systems in advanced biomechanicsapplications with electrical �elds as in hydraulic sensorsand actuators.

2. Constitutive equations

2.1. Piezoelasticity theoryBased on piezoelasticity theory, stresses f�g and strainsf"g, as well as electric displacement fDg and electric�eld fEg vectors, in piezoelectric materials, may becombined as follows [35]:�

�D

�=�C �eeT 2

��"E

��p

�� ; (1)

where the coe�cients of thermal expansion, pyroelec-tric, and temperature change are shown by f�g, fPg,and ��, respectively. Eq. (1) can be expanded inEq. (2) as shown in Box I; the component of theelectric �eld can be written in terms of scalar functions

of electric potential � as follows [36]:

E = �r�; (3)

in which r is the gradient operator.

2.2. Nonlocal piezoelasticity theoryBased on the theory of nonlocal elasticity for piezo-electric materials, the stress and electric displacement�elds at a reference point (x) of a body are respectivelyfunctions of the strain and electrical �elds at eachpoint of the body [37{39]. The nonlocal constitutiveequations for the piezoelectric material are expressedas follows [40]:

�nlij (x)=ZV

� (jx� x0j ; �) �lij (x0) dV (x0) ; 8x 2 V;(4a)

Dnli (x)=

ZV

� (jx� x0j ; �) Dli (x0) dV (x0) 8x 2 V;

(4b)

where �nlij and Dnli are the components of nonlocal

stresses and nonlocal electric displacements, respec-tively. The kernel function � (jx� x0j ; �) is the nonlo-cal modulus, and jx�x0j is the Euclidean distance. � =e0a=l is de�ned as the scale coe�cient that incorporatesthe small-scale factor, where e0 is a constant appro-priate for each material and obtained experimentally,and a and l are the internal and external charac-teristic lengths (e.g., crack length and wavelength),respectively. In addition, �lij and Dl

i imply the localstresses and local (i.e., classical) electric displacementsobtained from Eq. (2), respectively. According tothe axial polarization of the smart micro tube, thenonlocal constitutive equations may be written asfollows [41,42]:�

1� (e0a)2r2� �nlij = �lij ; (5a)

�1� (e0a)2r2� Dnl

ij = Dlij ; (5b)

8>>>>>>>>>>>><>>>>>>>>>>>>:

�1�2�3�23�31�12D1D2D3

9>>>>>>>>>>>>=>>>>>>>>>>>>;=

8>>>>>>>>>>>><>>>>>>>>>>>>:

C11 C12 C13 0 0 0 0 0 �e31C12 C22 C23 0 0 0 0 0 �e32C13 C23 C33 0 0 0 0 0 �e330 0 0 C44 0 0 0 �e24 00 0 0 0 C55 0 �e15 0 00 0 0 0 0 C66 0 0 00 0 0 0 e15 0 211 0 00 0 0 e24 0 0 0 222 0e31 e32 e33 0 0 0 0 0 233

9>>>>>>>>>>>>=>>>>>>>>>>>>;

8>>>>>>>>>>>><>>>>>>>>>>>>:

"1"2"3 23 31 12E1E2E3

9>>>>>>>>>>>>=>>>>>>>>>>>>;�

8>>>>>>>>>>>><>>>>>>>>>>>>:

�1�2�3�4�5�6p1p2p3

9>>>>>>>>>>>>=>>>>>>>>>>>>;�� (2)

Box I


where r2 represents the Laplace operator. The aboveimplicit relations couple the nonlocal values of stressand electric displacement to their local values. Toachieve explicit relations, an iterative-based methodcan be utilized as follows:�

�nlijk+1

= (e0a)2r2 ��lijk +��lij; k � 0�

Dnlijk+1

= (e0a)2r2 �Dlijk

+�Dlij; k � 0�

�nlij0

=��lij;�

Dnlij0

=�Dlij: (6)

In the above iteration loop, the �rst step is startedby the local results and k is the iteration number.Clearly, increasing the iteration number improves theaccuracy of the nonlocal results. In addition, the orderof the small-scale parameter shows that just a fewsteps of iteration give enough accuracy, as shown inthe numerical results section. From now on, nonlocalcomponents of stresses and electric displacement areused without superscripts.

2.3. Kinematic relationsAccording to nonlinear shallow shell models, the dis-placement components of an arbitrary point of thebody are expressed as follows:

~U (x; �; z; t) = u (x; �; t)� z @w (x; �; t)@x

;

~V (x; �; z; t) = v (x; �; t)� z 1R@w (x; �; t)

@�;

~W (x; �; z; t) = w (x; �; t) ; (7)

where u, v and w are the components of the shellmid-plane displacement along x, �, and z coordinates,respectively, and t is the time. Based on Donnell'snonlinear shell theory, the strain-displacement relationsyield the following:8>>>>>><>>>>>>:

"x = @u@x + 1

2

�@w@x

�2 + z(�@2w@x2 )

"� = 1R@�@� + w

R + 12R2

�@w@�

�2 + z(� 1R2

@2w@�2 )

x� = 1R@u@� + @�

@x + 1R@w@�

@w@x + z(� 2

R@2w@x@� )

(8)

2.4. Mode expansion of displacements andelectric potential

By considering simply supported boundary conditionsat both ends of the cylindrical shell, the following

mode expansion for components of displacement maybe considered [43]:

u =MXm=1

NXn=0

sin�

(2m� 1)�xL

��U cmn (t) cos (n�)

+ Usmn (t) sin (n�)�;

v =MXm=1

NXn=0

sin�

(2m� 1)�xL

��V cmn (t) cos (n�)

+ V smn (t) sin (n�)�; (9)

w =MXm=1

NXn=0

sin�

(2m� 1)�xL

��W cmn (t) cos (n�)

+W smn (t) sin (n�)

�;

where m and n denote the longitudinal half wavenumber and circumferential wave number, respec-tively, M and N are the maximum values of wavenumbers. U cmn(t), Usmn(t), V cmn(t), V smn(t), W c

mn(t),and W s

nm(t) are time-dependent Degrees Of Free-dom (DOF), and fqdg = [U cmn(t); Usmn(t); V cmn(t);V smn(t);W c

mn(t);W smn(t)] is de�ned as the vector of

DOF. The vector dimension of fqdg denotes the numberof DOFs, which may be calculated as:

NDOF = 3� (2M �N +M) :

For electric potential, �, the following mode expansioncan be proposed [44]:

� = �0

�1� x

L

�+

MXm=1

�sm sin�

(2m� 1)�xL

�; (10)

where �sm is the time-independent amplitude compo-nent of electric potential, fq�g = f�smg is de�ned asthe vector of electric potential amplitudes, and �0 in-dicates the magnitude of the imposed electric potential.Because of the unidirectional, longitudinally polarizedelectric �eld, the electric potential is expanded as afunction of the longitudinal coordinate x. In thenumerical results section, the accuracy of the resultswill be examined with respect to di�erent values of Mand N .

In addition, a vector of unknowns can be intro-duced as follows:

fqg = ffqdg ; fq�ggT : (11)


2.5. Energy functionsBased on the theory of piezoelectricity, the totalpotential and kinetic energies of the micro tube withlength L, cross-section A, volume 8, and density �s areobtained as follows [45]:

US =12

Z L

0

ZA

[� D]�

"�E�dAdx; (12)

TS =12�Sy8~V :~V d8; (13)

where � and D are the nonlocal components ofstress and electric displacement and obtained throughEqs. (5). In addition, ~V corresponds to the velocityvector of the micro tube, which leads to the followingrelation:

TS =12�SZ L

0

ZA

"�@u@t� z

�@2w@x@t

��2

+�@w@t

�2#

dAdx: (14)

The external works done by the surrounded elasticfoundation are expressed as follows:

Welastic foundation=12

Z L

0

��KWw+GPr2w�wdx;

(15)

where KW and GP are Winkler's spring foundationmodulus and Pasternak's shear foundation modulus,respectively. Moreover, the energies related to the uid ow are evaluated separately in the following section.

3. Pulsating uid ow modeling

This study considers visco pulsating micro uid ow in-side the micro shell, which is one of the most empiricalmodels. The micro uid ow is assumed to be turbulentfully developed, Newtonian, irrotational, isentropic,and incompressible. The unsteady uid motion isobtained based on potential ow theory and, to bringthe micro scale problem closer to reality, the e�ectsof pulsating viscosity and slip boundary condition areconsidered based on unsteady time-averaged Navies-Stokes equations and Knudsen number theory [6,10].

Let us assume that no cavitation occurred atthe uid and shell interfaces. An unsteady potentialfunction ' may be found, which satis�es the Laplaceequation and following boundary condition as [46]:

r2' =@2'@x2 +

@2'@r2 +

1r@'@r

+1r2@2'@�2 = 0; (16)

@'@r

�� r=R =�@w@t

+ Uf@w@x

�; (17)

where Uf is the pulsating ow velocity. By employing

the separation of variables method, the general solutionof Eq. (16) may be obtained as follows:

' (x; r; �; t)=MXm=1

NXn=0

�m (x) �m;n(r) cos (n�)fm;n (t):(18)

By substituting Eq. (18) into (16) besides applyingthe regularity condition at r = 0, the unknownfunctions will be obtained. By utilizing the boundarycondition (17), Eq. (18) will be evaluated as follows:

� (x; r; �; t) =MXm=1

NXn=0

Lm�

In (m�r=L)I 0n (m�R=L)�

@wm;n@t

+ Uf@wm;n@x

�; (19)

where In and I 0n are respectively the modi�ed Besselfunction of the �rst kind of order and its derivative.Consequently, the perturbation pressure at the innerwall of the shell is found as follows [6]:

p = ��FMXm=1

NXn=0

Lm�

In (m�r=L)I 0n (m�R=L)�

@@t

+ Uf@@x

�2

wm;n; (20)

in which �F is the density of uid ow.

3.1. Potential energy of uid owConsider the uid ow inside the micro tube withoccupied volume � and surrounded surface �. Thetotal energy related to the uid- ow ETF is given by:

ETF =12�fy

�~Vf : ~Vf d�: (21)

According to Green's theorem, Eq. (21) is evaluated asfollows:

ETF =12�fy

�r:rd� =

12�fx

�

�@@�

�d�;

(22)

where is the potential function of the uid ow, and� is the normal vector of boundary surfaces, which isconsidered positive outward. Integration of Eq. (22)gives:

ETF =12�fZ 2�

0

Z L

0

�@@r

�r=R

dxRd�

+12�fZ 2�

0

Z R

0

�@@x

�x=L

rdrd�

� 12�fZ 2�

0

Z R

0

�@@x

�x=0

rdrd�; (23)

in which the boundary conditions are:


@@r

=@'@r;

@@x

= Uf +@'@x

;

jx=L = UfL+ 'jx=L;

jx=0 = 'jx=L: (24)

By using Eq. (24), Eq. (23) may be obtained as follows:

ETF =EpF +12�fZ 2�

0

Z L

0

�Ufx

@'@r

�r=R

dxRd�

+12�fZ 2�

0

Z R

0

�UfL

@'@x

�x=L

rdrd�

+12�fU2

f�R2L; (25)

where EpF denotes the perturbation velocity potentialof uid and is given by:

EpF =12�fZ 2�

0

Z L

0

�'@'@r

�r=R

dxRd�: (26)

Moreover, other terms in Eq. (25) have no in uence onthe total energy of uid ow. By substituting Eq. (16)into Eq. (17) and employing Eq. (7), we yield thefollowing:

EpF =12�fZ 2�

0

Z L

0(')r=R

�@w@t

+ Uf@w@x

�dxRd�:

(27)

Using Eq. (9), Eq. (27) may be obtained as follows:

EpF =12�fZ 2�

0

Z L

0

MXm=1

MXl=1

NXn=0

NXk=0

Lm�

In (m�R=L)I 0n (m�R=L)

��

_wm;n+Uf@wm;n@x

��_wl;k+Uf

@wl;k@x

�dxRd�:

(28)

By substituting Eq. (28) into Eq. (25), the total energyof uid is derived and divided as follows:

ETF = TF � VF + EG; (29)

where TF , EG, and VF are kinetic, gyroscopic, andpotential energies of the uid ow, respectively, andare expressed as follows [6]:

TF =12�f

MXm=1

NXn=0

Z 2�

0

Z L

0

Lm�


_w2m;n

dxRd�; (30)

VF =� 12�f

MXm=1

NXn=0

Z 2�

0

Z L

0

Lm�


U2f

�@wm;n@x

�2

dxRd�; (31)

EG=12�f

MXm=1

MXl=1

NXn=0

NXk=0

Z 2�

0

Z L

0

Lm�


Uf�_wm;n

@wl;k@x

+ _wl;k@wm;n@x

�dxRd�: (32)

The unsteady ow velocity Uf (t) for pulsating uid ow is expressed by the following time-dependentfunction as follows:Uf (t) = U0 (1 + � cos (t)) ; (33)

where U0 is the mean value of ow velocity, � is theamplitude of the harmonic uctuation, and is itsfrequency. In addition, it is clear that the expressionsof TF , EG, and VF will be time dependent in this caseand, therefore, this leads to the addition of a new termin the Lagrange equations of motion in Section 4.

3.2. Viscous e�ects for pulsatile owIn this section, the pulsating uid viscosity e�ect issimulated as an external force acting on the inner wallof the shell. Considering a fully developed turbulentand incompressible axial ow regime with uctuatingvelocities ux, u�, ur in the axial, circumferential,and radial directions, respectively, the time-averagedNavier-Stokes equations are given by:

1�f@P@x

= �1rddr

(r�ux�ur) +�rddr

�rdUfdr

�;

1�f@P@r

= �1rddr�r�u2r�

+�u2�r;

0 =ddr

(r�ur�u�) + 2�ur�u�r

; (34)

where � is the kinematic viscosity, and the overbarindicates the time-averaged components. In the fol-lowing, by using the Computational Fluid Dynamics(CFD), the pressure distribution will be obtained asfollows [47]:

P (x; r) = �2�fRU2� x� �f �u2

r + �fZ r

R

�u2� � �u2

rr

dr

+P (0; R); (35)

where the stress velocity, U� , is expressed as follows:

U� =�� dUf

dr

�1=2

r=R=��w�f

�1=2

=�

18fU2

f

�1=2

;(36)

where f and �w are the friction factor and the uid


frictional force per unit area of the shell, respectively.The distribution of pressure on the shell interfaces andthe drop pressure in the axial direction of the shell aregiven by:

P (x;R) = �2�fRU2� x+ P (0; R)

= ��f�f

4RU2f +

dUfdt

�x+ P (0; R); (37)

�Pw=P (0; R)�P (L;R)=�f�f

4RU2f +

dUfdt

�L:(38)

By ignoring the e�ects of ow acceleration, the con-stant friction traction force in the axial direction isgiven by:

�w = f�fU2f�

8: (39)

Based on the classical uid mechanic, the friction factor(f) of the micro tube is a function of the Reynoldsnumber (Re) and the mean roughness (�) of the innerwall of the shell. The friction factor is estimatedthrough empirical Colebrook equation as follows:

1pf

+ 2 log��=2R3:7

+2:51

Repf

�= 0; (40)

in which Re = 2RUf=�.Lastly, the total work associated with the uid

viscosity is given by:

WFluid Viscosity =Z 2�

0

Z L

0(�PW + �Wu)dxRd�:

(41)

3.3. E�ect of slip boundary conditionsOften, the FSI problems are assumed with no-slipboundary conditions in the uid and shell interfaces.However, this assumption will not be valid for themicro-scale problems due to the nonlocality and Knud-sen number e�ects [48]. In other words, based on Kntheory, the value of Kn in the micro-scale problems willexceed 0.01 and, therefore, an average uid VelocityCorrection Factor (VCF) should be applied to all ofthe following equations as follows [49]:

V CF =Vavg; slip

Vavg; no�slip=�

4�

2��

��Kn

1+Kn

�+1�;(42)

where �v is the tangential momentum accommodationcoe�cient and is measured to be 0.7 for most practicalpurposes [50].

4. Method of solution

This study utilized the modi�ed Lagrange equationsof motion for open systems with a control volume

consisting of the micro shell and the uid ow passingthrough it [27,51]:

ddt

�@ (TS + TF + EG)

@ _qk

�� @ (TS + TF + EG)

@qk

+@ (US + VF )

@qk= Qk; k = 1; ::::; 3N; (43)

in which qk is the element of vector fqg previouslyintroduced in Section 2.5, the terms @TS

@qk and @TF@qk

are zero, and Qk is the generalized force obtainedby di�erentiation of the virtual work done throughexternal forces:

Qk =@W@qk

: (44)

By employing mode expansions (9) and (10), Eq. (43)is evaluated as follows:

[M ] f�qg+ [C(t)] f _qg+ [K(t)] fqg = fQg ; (45)

where [M ] and [C(t)] are the mass and dampingmatrices, respectively, and [K(t)] is the sti�ness matrixcomposed of linear and nonlinear terms as follows:

[K(t)] = [KL(t) +KNL(t)] : (46)

Because of static coupling between mechanical andelectric displacement, Eq. (45) may be rearrangedbased on Eq. (11) as follows:�

Mdd 00 0

��qd�q�

�+�Cdd 00 0

��_qd_q�

�+�Kdd Kd�K�d K��

��qdq�

�=�QdQ�

�: (47)

According to the lack of external electrical load, thesecond sets of Eq. (47) lead to the following:

fq�g = � hK�1��K�d

i fqdg : (48)

Through Eq. (48), q� in the �rst sets of Eq. (47) iseliminated and yields a set of modi�ed equations ofmotion as:

[Mdd] f�qdg+ [Cdd(t)] f _qdg+ [KM (t)] fqdg = fQMg ;(49)

where:KM = Kdd �Kd�K�1

�� K�d;

QM = Qd �Kd�K�1��Q�: (50)

Of note, the nonlinear modi�ed sti�ness matrix[KM (t)] and the damping matrix [Cdd(t)] in Eq. (49)are harmonically time dependent by considering thepulsatile ow regime according to Eq. (34) as follows:

KM (t) = f1(sin t; cos t);

Cdd(t) = f2(sin t; cos t): (51)


4.1. Dynamic instability formulationFor static uid velocity condition (� = 0), the followinggeneral solution can be expressed for Eq. (49) asfollows:

fqdg = fq̂dg exp (! t) ; (52)

in which fq̂dg is the amplitude of the displacementsvector, and ! is a complex circular frequency composedof real and imaginary parts that represent damping andnatural frequencies, respectively.

By introducing the secondary vector fSg =!fqdg, Eq. (49) is converted to the following state spaceproblem:

[] fXg = ! fXg ; (53)

where:

[] =�

0 I�M�1

dd KM �M�1dd Cdd

�;

fXg = [fqdg ; fSg]T : (54)

Eq. (53) is an eigenvalue problem with nonlinearsti�ness terms that can be solved according to thefollowing iterative procedure [52]:

Step 1: Calculating linear eigenvalues and eigen-vectors by ignoring all of the nonlinear terms in thesti�ness matrix;Step 2: Scaling up the eigenvectors and estimatingnonlinear terms in the sti�ness matrix;Step 3: Considering nonlinear and linear compo-nents of the sti�ness matrix together and evaluatingeigenvectors and eigenvalues of the updated eigen-value problem;Step 4: Comparing new and previous eigenvectorsuntil reaching the convergence conditions; otherwise,Steps 2 to 4 must be repeated.

In this study, the above procedure is performeduntil the maximum relative error between the twosuccessive iterations becomes less than 0.1%.

4.2. Parametric instability formulationIn order to study the parametric instability of thesystem, it is usual to study the e�ects of amplitude (�)and frequency () of the pulsating ow on the systemstability at various static uid velocity U0 levels. Theboundaries of the regions of parametric resonance maybe obtained usually via the Bolotin's [53] or Floquetmethods. Based on Bolotin's method, the generalizedcoordinate is assumed to be periodic and is expressedin the following form:

qd(t) =�NX

k=0

�ak sin

�kt

2

�+ bk cos

�kt

2

��: (55)

By substituting Eq. (55) into Eq. (49) and setting thecoe�cients of harmonics to zero, the following algebraicequation is obtained:

[C] fY g = f0g ; (56)

where fY g = fb0; a1; b1; ::::; a �N ; b �NgT is the am-plitude vector. The characteristic equation of thisproblem is expressed as follows:

f(C) = det(C) = 0: (57)

This equation is a nonlinear algebraic equation on, which is solved numerically by the mathematicalsoftware to detect the system's parametric resonanceinstability boundaries.

5. Numerical results

In this section, the dynamical instabilities of a smartmicro tube surrounded by an elastic foundation andunder internal pulsating micro ow and axial electric�eld are examined, as shown in Figure 1. To thisend, divergence and utter instabilities, which occur byincreasing the mean ow velocity (U0), and parametricresonance instability in the case of increasing dimen-sionless pulsation amplitude (�) are studied. The microtube is considered to be made of piezoelectric materialsas PZT4 and its structural properties are presented inTable 1. A comprehensive parametric study has beenmade based on the e�ects of uid ow components(mean ow velocity and the frequency and pulsatingmagnitudes), small-scale parameter, elastic foundationmodules, Knudsen number, uid viscosity, geometricnonlinearity, and applied electric voltage.

5.1. Accuracy of the resultsThe accuracy of the results is examined in Table 2 fordi�erent magnitudes of M and N and for both linearand nonlinear approaches. From this table, it can be

Table 1. Properties of PZT4.

Properties of PZT4 Ref. [45]

C11 (GPa) 139C22 (GPa) 115C12 (GPa) 77.8e11 (C/m2) 15.1e12 (C/m2) {5.2�11 (C2/Nm2) 6:46� 10�9

p11 (C/m2K) �2:5� 10�5�1 (1/K) 7:41� 10�6

�2 (1/K) 2:11� 10�6

� (kg/m3) 7500L=R 20


Table 2. Accuracy of the results for various mode expansions.N(U�0 = 0:0001; � = 0) N(U�0 = 0:001; � = 0)

M 1 2 3 1 2 3

1NDOF = 9Im(!�)NL=0:00934Im(!�)L=0:00925

NDOF = 15Im(!�)NL=0:00942Im(!�)L=0:00925

NDOF = 21Im(!�)NL=0:00947Im(!�)L=0:00925

NDOF = 9Im(!�)NL=0:00498Im(!�)L=0:00460

NDOF = 15Im(!�)NL=0:00523Im(!�)L=0:00460

NDOF = 21Im(!�)NL=0:00537Im(!�)L=0:00460

2NDOF = 18Im(!�)NL=0:00943Im(!�)L=0:00925

NDOF = 30Im(!�)NL=0:00946Im(!�)L=0:00925

NDOF = 42Im(!�)NL=0:00950Im(!�)L=0:00925

NDOF = 18Im(!�)NL=0:00531Im(!�)L=0:00460

NDOF = 30Im(!�)NL=0:00542Im(!�)L=0:00460

NDOF = 42Im(!�)NL=0:00548Im(!�)L=0:00460

3NDOF = 27Im(!�)NL=0:00950Im(!�)L=0:00925

NDOF = 45Im(!�)NL=0:00950Im(!�)L=0:00925

NDOF = 63Im(!�)NL=0:00950Im(!�)L=0:00925

NDOF = 27Im(!�)NL=0:00550Im(!�)L=0:00460

NDOF = 45Im(!�)NL=0:00552Im(!�)L=0:00460

NDOF = 63Im(!�)NL=0:00552Im(!�)L=0:00460

concluded that the accuracy of the results is su�cientlyenough for linear analyses if (M;N) = (1; 1) (i.e., DOF= 9) and for nonlinear analyses if (M;N) = (3; 1) (i.e.,DOF = 27).

5.2. Validation of the resultsIn order to validate the present results, at �rst, thepiezoelectric properties of the structure, small-scaleparameters, and elastic medium are neglected, andthe uid ow velocity is assumed constant. The shellspeci�cations used in this simulation include:

R = 0:041275 m; L = 0:1206 m;

h = 0:127 mm; E = 70� 109 Pa;

�=2700 kg=m3; �F =1000 kg=m3; �=0:33:

However, the results are compared with the �ndings ofAmabili and Graziera [54], as shown in Figure 2, forvibration and stability of the shell conveying uid. Itcan be seen that the obtained results are close to thosereported by Amabili and Graziera [54], demonstratingthe validity of this work.

For another comparison, the piezoelectric proper-ties of the structure and elastic medium are neglected.In other words, the vibration of a cylindrical shellconsidering size e�ects is studied considering:

h = 0:34 mm; E = 1:6T Pa;

D = 0:678 nm; � = 0:19:

Table 3 shows the frequency of the structure fordi�erent length-to-diameter ratios. It is found that the

Figure 2. Comparison of the dimensionless naturalfrequency and dimensionless ow velocity for thesimpli�ed analyses of the present work and Amabili andGraziera [54].

results of this paper are in good agreement with thedata of Mohammadi et al. [55].

Now, let us introduce the dimensionless parame-ters used in this study as follows:

U�0 = U0

q�f=C11; � = L

q�=C11;

!� = !Lq�=C11; � =

e0aL; �� =

�AfpEI mf

;

K�w = Kw=C11; G�P = GP =C11Rh; �W =wL;

�X =xL: (60)

The results will be presented in the dimensionless form.

Table 3. Comparison of the natural frequencies (THz) obtained by the nonlocal elasticity theory.

L=D Mohammadi et al. [55] Present work Error (%)4.86 0.776 0.768 1.037.47 0.346 0.341 1.4413.89 0.161 0.158 1.8617.47 0.114 0.111 2.63


5.3. Modal analyses and critical ow velocitiesfor static uid ow (� = 0)

E�ects of increasing U�0 on the natural and dampingfrequencies of the smart micro tube for the �rst threevibrational modes are studied in Figures 3(a) and3(b). As can be seen from these �gures, by increasingU�0 , the imaginary part of natural frequencies willdecrease until reaching the value zero at U�0 = 0:0012,0.0023, and 0.0048 (for the �rst, second, and thirdmodes, respectively), which are introduced as thecritical ow velocities. It is clear from Figure 3(b)that, at these points, the real part of frequenciesbecomes nonzero and divergence instability will occur.For more clarity, let us consider the increasing owvelocity for the �rst mode (fundamental mode). Inthis mode, at �rst, the imaginary part of frequencydecreases by increasing ow velocity, which implies lessstable conditions. The critical ow velocity will beU�0 = 0:0012, and the divergence instability occursvia a pitchfork bifurcation point, as shown in Fig-ure 3(b). In addition, by increasing ow velocities toa greater extent, it becomes clear that in the range

Figure 3(a). Dimensionless natural frequencies versusdimensionless ow velocity for the 1st to 3rd modes.

Figure 3(b). Dimensionless damping frequencies versusdimensionless uid velocity for the 1st to 3rd modes.

of 0:0012 < U�0 < 0:0048, the �rst and second modeswill merge, which is another kind of instability knownas utter instability. Flutter instability occurs moreoften at higher ow velocities. Figure 4. depicts thee�ects of the small-scale parameter on the fundamentalfrequency and the critical ow velocity. It is clear thatthe small-scale parameter should be considered in theorder of structural characteristic length of the system.Therefore, in this study, a maximum value of about120 nm on a small scale is considered. According to this�gure, by increasing ow velocity, the frequency andthe critical ow velocity will decrease at an increasingrate. The variation of uid viscosity magnitude isdemonstrated in Figure 5. It can be observed thatthe in uence of viscosity becomes more remarkablewhen the uid velocity increases. Additionally, thefundamental frequency and the critical ow velocityof the system will increase when the viscosity of uidincreases, which means that the stability of the systemhas improved. E�ects of externally applied voltage onthe fundamental frequency are shown in Figure 6. It

Figure 4. Dimensionless ow velocity versusdimensionless natural frequency for various small-scaleparameters for � = 0.

Figure 5. Dimensionless ow velocity versusdimensionless natural frequency for various magnitude of uid viscosity for � = 0.


is concluded that, at a constant ow velocity, applyingpositive (direct) electric potential increases the funda-mental frequency and the critical ow velocity, whichmeans that the stability of the system has increased. Itis also found that imposing negative (indirect) electricpotential decreases the fundamental frequency of thesystem. This statement means that the applicationof external electric �eld is an e�ective controllingparameter for the smart uid-conveying systems. Tocompare the magnitudes of linear and nonlinear naturalfrequencies, Figure 7 shows the variations of frequencyratio versus mean ow velocity for di�erent small-scaleparameters. As can be seen from this �gure, whenthe mean ow velocity is lower than U�0 < 0:0004, thedi�erence between nonlinear and linear frequencies islower than 3% for static uid ow (� = 0) and is lowerthan 5% for dynamic uid ow (� = 0:02). It can beconcluded that the e�ects of geometric nonlinearity willincrease in case of an increase in the velocity of uid ow, and this is more remarkable when the uid ow isdynamic. E�ects of the small-scale parameter are also

Figure 6. Dimensionless ow velocity versusdimensionless natural frequency for various appliedvoltages for � = 0.

Figure 7. Frequency ratio versus dimensionless owvelocity for di�erent values of the small-scale parameter.

considered in this �gure. It is shown that consideringthe small-scale parameter increases the frequency ratioof the system for both dynamic and static uid ows.

5.4. Parametric instability of dynamic uid ow (� 6= 0)

By enhancing the dimensionless pulsation amplitudeof uid ow (�), the parametric resonance instabilityoccurs, and its boundary curves are achieved throughEq. (57). According to previous literatures, the resultsof this analysis will be commonly examined in thefrequency, , amplitude, �, plane at various �xed levelsof all other in uencing parameters such as mean owvelocity, viscosity, small-scale parameter, etc. Figures 8to 12 show the instability boundary curves for thesystem. From all of these �gures, the regions outsideand inside the curves are related to the stable andunstable conditions, respectively. According to these�gures, by increasing the mean ow velocity, theinstability boundaries move to the down and left sidesand will expand beyond the previous ones. This meansthat a smaller pulsating amplitude and a wider rangeof pulsating frequencies lead to instability.

E�ects of the small-scale parameter are particu-larly studied in Figure 8. As can be seen from this�gure, the e�ects of nonlocality are more signi�cantat higher ow velocities. It is shown that, giventhe nonlocality a�ects, the resonance regions movefrom higher frequencies to lower ones (towards leftand slightly down). This means that the naturalfrequencies of the system have decreased. In addition,at a constant (�), the resonance region will be narrowerthan previous one, which means also a more stablecondition in this case. Figure 9 shows the instabilityboundaries of the viscous- uid conveying beam. Itis observed that the e�ects of uid viscosity stronglydepend on the mean ow velocity. As the mean owvelocity increases, the in uences of the uid viscositybecome more signi�cant. Additionally, with an increase

Figure 8. Instability region of a smart micro shell forU�0 = 0:0001 (right side) and U�0 = 0:001 (left side) andvarious dimensionless small-scale parameters.


Figure 9. Instability boundaries of a smart micro shellfor U�0 = 0:0001 (right side) and U�0 = 0:001 (left side) andvarious dimensionless uid viscosities.

Figure 10. E�ects of applied electric �eld on instabilityboundaries of a smart micro shell for U�0 = 0:0001 (rightside) and U�0 = 0:001 (left side).

in uid viscosity, the area of the instability regionreduces and the instability boundaries slightly shift tothe right. This is because of the increasing naturalfrequencies of the system, as observed in Figure 6. Toevaluate the e�ects of the piezoelectric properties of thesmart shell, the instability response of the system isobtained by applying direct and indirect external volt-ages. Figure 10 illustrates the instability boundaries ofthe system under di�erent magnitudes of the appliedvoltage. It is found that by applying direct voltage, theinstability curves move to the right, which means thatthe resonance frequencies of the system have increased.In other words, direct voltage increases the stabilityof the system and indirect voltage does the opposite.Figure 11 illustrates the e�ects of both Winkler andPasternak elastic foundations on the instability bound-aries of the system for U�0 = 0:001. It is shown that theunstable areas of the system shift to the left when theelastic foundation exists, because the sti�ness of theelastic foundation absorbs the vibrating energy of thesystem and makes a sti�er and more stable structure.

Figure 11. E�ects of elastic medium on instabilityboundaries of a smart micro shell for U�0 = 0:001.

Figure 12. E�ects of Kn on instability boundaries of asmart micro shell for U�0 = 0:001.

Figure 11 depicts a more stable condition from thePasternak foundation (considering both normal andshear e�ects) than the Winkler foundation with onlynormal loads. For a micro uid ow, the value of Kn isin the range of 10�2 < Kn < 10�1 and, therefore, theslippery regime occurs. It can be seen from Figure 12that the enhancement of Kn value moves the instabilityarea to the right and down, which means a less stablecondition. In fact, increasing Kn means increasing themean free path of uid and results in the lower sti�nessof the system.

6. Conclusion

Instability prediction of a uid-conveying pipe (as ahistorical challenge for the engineers) on a micro scalewith an internal pulsating ow regime was investigatedin this study. To obtain the stability active control of


the system (sensing and actuating), the piezoelectricmaterial was employed for the body, and e�ects ofapplying external electric �eld on the stability of thesystem were investigated. Nonlocal piezoelasticitytheory, nonlinear cylindrical shell model, and energyapproaches were employed to obtain the nonlinearelectro statically coupled equations of motion. Byapplying mode expansion analyses, the equations ofmotion were discretized and solved via the state spaceproblem. Consequently, the boundaries of the para-metric resonance instability were achieved via Bolitin'smethod. In the numerical results, e�ects of variousparameters such as mean ow velocity, small-scaleparameter, applied voltage, uid viscosity, Knudsennumber, and elastic foundation constants on dynamicalinstabilities of the system were studied. It was shownthat the imposed positive electric potential �eld alongthe micro shell increased natural frequency, critical ow velocity, and the stability of the structure andvice versa. In addition, it was observed that the uidviscosity and small-scale parameter had major e�ectson the critical ow velocity and the stability of thesystem. This work was presented to extend or completethe application of hydraulic sensors or actuators inadvanced nano/micro electro-mechanical systems.

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Biographies

Vahid Atabakhshian received his BSc degree fromthe Islamic Azad University in Kashan, Kashan, Iran

in 2009. He then received his MS degree from theUniversity of Kashan, Kashan, Iran in 2013. He iscurrently a PhD Student at Bu-Ali Sina Universityin Hamedan, Hamedan, Iran. His current researchinterests are ow-induced vibration, smart materials,nanomechanics, nonlinear dynamic, buckling, and com-posites.

Alireza Shooshtari is an Associate Professor ofMechanical Engineering at Bu-Ali Sina University,Hamedan, Iran. He received his BS degree in Me-chanical Engineering from Tehran University, Tehran,Iran and his MS and PhD degrees in Applied Me-chanics from Bu-Ali Sina University, Hamedan, Iranand Tarbiat Modarres University, Tehran, Iran, respec-tively. He has been an academic sta� in MechanicalEngineering Department, Bu-Ali Sina University since2006. His research interests are in the �eld of dynamicsand vibration, especially in nonlinear vibration and dy-namics of smart structures, continuous systems, modalanalysis, vibration of nanostructures, and compositesstructures.

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