Size-dependent bifurcations of microtubes conveying fluid flow embedded in a nonlinear elastic medium

The size-dependent bifurcation behaviour of a fluid-conveying microtube taking into account the effect of internal energy loss is studied in this paper. It is assumed that the viscoelastic microscale tube is externally excited by a transverse harmonic force. In addition, the viscoelastic microscale system is surrounded by a nonlinear spring bed. To take into account the influence of the internal energy loss on the size-dependent bifurcation behaviour, the Kelvin-Voigt scheme of viscoelasticity is employed. The modified couple stress theory (MCST), as a size-dependent theory, and the Hamilton principle, as an energy/work law, are utilised for deriving the governing coupled equations for the bifurcation response of the viscoelastic fluid-conveying microtube. The displacement along the transverse direction as well as the axial displacement are incorporated into the size-dependent continuum model, leading to an accurate coupled continuum-based model. The Galerkin weighted-residual scheme, as a decomposition approach, is then applied to the derived nonlinear differential equations. Clamped-clamped boundary conditions are taken into consideration for extracting numerical results. The sizedependent bifurcation behaviour of the fluid-conveying viscoelastic microtube is finally predicted by a numerical timeintegration technique. Introduction Fluid-structure interactions at microscale levels have recently been at the centre of attention of many scientific communities owing to the fact that some useful microscale devices such as microfluidics and microscale biosensors work based on these interactions [1, 2]. It is important to improve the knowledge level on the mechanical behaviour such as the bifurcation of fluid-conveying microscale structures under various external loads since in order to design a microscale system with fluidstructure interactions, the proper prediction of its timedependent deformation is required. Some modified elasticity models have been introduced to capture the size influence on the dynamic response of fluidconveying small-scale structures. The most popular ones are the nonlocal theory [3-7], the couple stress model [8-10] and the nonlocal strain gradient elasticity [11, 12]. In this analysis, the modified couple stress theory (MCST) is applied since it is able to capture the size influence at microscale levels. Although the dynamic response of fluid-conveying macroscale structures have been widely studied in the literature [13, 14], the number of research investigations on the mechanics of small-scale structures conveying flowing fluid is limited. Some of the most relevant investigations are concisely explained in the following. Wang [15] explored the linear scale-dependent oscillation of tubes containing fluid flow at microscale levels; he applied the MCST to capture the size influence. Kural and Özkaya [16] also studied the effect of an elastic foundation on the vibration of microscale beams conveying fluid; they utilised a perturbation technique to present an approximate solution. In addition, Dehrouyeh-Semnani et al. [17] examined the vibrational behaviour of micropipes containing fluid flow via an Euler–Bernoulli model together with the MCST; they took into consideration von Kármán's geometrical nonlinearity in the formulation. In another paper, Hosseini and Bahaadini [18] developed a scale-dependent theoretical model for analysing the stability of cantilever pipes containing fluid flow at microscale levels via a modified strain gradient model. Furthermore, a parametric size-dependent investigation was presented in the literature on the flow-induced time-dependent deformation of pipes with clamped-free ends [19]. Tang et al. [20] also investigated the large amplitude vibrational behaviour of curved microscale tube containing fluid flow via the MST. More lately, the effects of thermal stresses on the forced vibrational response of carbon nanotubes have been explored by Askari and Esmailzadeh [21]. Moreover, a nonlinear analysis has been reported on the flow-induced vibrational response of piezoelectric nanoscale tubes incorporating the influence of large deformations [22]. In this analysis, a scale-dependent continuum model is proposed to examine the bifurcation behaviour of a fluidconveying microtube resting on a nonlinear elastic medium in the super critical regime. The influences of the geometrical nonlinearity and the internal energy loss as well as the size influences are incorporated in the continuum modelling and the solution method. It is assumed that the microtube is subject to transverse harmonic loading. Using the Kelvin-Voigt scheme, MCST and a Hamilton approach, the differential equations of motion are derived. Both two types of displacement components (i.e. the axial and transvers ones) are simultaneously considered in the scale-dependent nonlinear modelling. Galerkin’s procedure was implemented in order to obtain a set of ordinary differential equations from the coupled partial differential equations of the microscale system. Finally, to solve the obtained ordinary differential equations, a continuation scheme is applied. Numerical results are presented for clamped-clamped microtubes in the supercritical regime. Scale-dependent formulation Figure 1 illustrates the schematic diagram of a microtube conveying fluid flow resting on a nonlinear elastic medium. The small-scale tube is subject to an excitation force given by      cos F x t where F denotes the forcing amplitude, and  is the excitation frequency. The length of the microtube and the velocity of the fluid flow are denoted by L and Uf, respectively. The plug flow assumption is considered as the boundary layer thickness is negligible. Moreover, Di and D are respectively the inner and outer diameters of the microscale tube. Based on von Kármán's geometrical nonlinearity and the EulerBernoulli model of beams, one can write                            2 2 ˆ ˆ ( , ) ( , ) ( , ) , , 1 1 , xx w x t u x t x t x z t z x x x (1) in which  xx , ŵ , û and  represent the strain, transverse deflection, axial displacement and tube rotation, respectively. Taking into account the internal loss using Kelvin-Voigt viscoelasticity, the stress-strain relation is expressed as                              , . xx e xx e v xx v E