Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method

Abstract

This research deals with the dynamic instability analysis of double-walled carbon nanotubes (DWCNTs) conveying pulsating fluid under 2D magnetic fields based on the sinusoidal shear deformation beam theory (SSDBT). In order to present a realistic model, the material properties of DWCNTs are assumed viscoelastic using Kelvin–Voigt model. Considering the strain gradient theory for small scale effects, a new formulation of the SSDBT is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The van der Waals interactions between the adjacent walls of the nanotubes are taken into account. The size dependent motion equations and corresponding boundary conditions are derived based on the Hamilton’s principle. The differential quadrature method in conjunction with Bolotin method is applied for obtaining the dynamic instability region. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of DWCNTs. The results depict that the surface stress effects on the dynamic instability of visco-DWCNTs are very significant. Numerical results of the present study are compared with available exact solutions in the literature. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems in advanced biomechanics applications with magnetic field as a parametric controller.

Appendix 1

$$\begin{aligned} B_{1} = & \frac{{\tau_{s} hP_{{0_{S} }} }}{\pi }{ ,} \quad B_{2} = \frac{8}{15}{\mkern 1mu} \frac{{l_{1}^{2} G^{*} \pi^{2} L}}{{h^{2} }} + \frac{1}{4}\frac{{l_{2}^{2} G^{*} L{\mkern 1mu} \pi^{2} }}{{h^{2} }} + G^{*} O{ ,} \\ B_{{3_{i} }} = & - l_{1}^{2} \left( {\frac{32}{15}{\mkern 1mu} G^{*} T_{0} + \frac{16}{15}{\mkern 1mu} G_{s}^{*} T_{{0_{s} }} } \right) + \frac{8}{15}{\mkern 1mu} l_{1}^{2} \left( {G^{*} A_{i} + G_{s}^{*} S_{i} } \right) + l_{1}^{2} \left( {\frac{32}{15}{\mkern 1mu} G^{*} O + \frac{8}{15}{\mkern 1mu} G_{s}^{*} O_{s} } \right) - l_{2}^{2} \left( {G^{*} T_{0} + 2{\mkern 1mu} G_{s}^{*} T_{{0_{s} }} } \right) \\ & + l_{2}^{2} \left( {\frac{1}{4}G^{*} O + G_{s}^{*} O_{s} } \right) + l_{2}^{2} \left( {G^{*} A_{i} + G_{s}^{*} S_{i} } \right) - 4{\mkern 1mu} l_{0}^{2} \left( {G^{*} T_{0} + G_{s}^{*} T_{{0_{s} }} } \right) + 2{\mkern 1mu} l_{0}^{2} \left( {G^{*} O + G_{s}^{*} O_{s} } \right) \\ & + 2{\mkern 1mu} l_{0}^{2} \left( {G^{*} A_{i} + G_{s}^{*} S_{i} } \right) + EI + E_{s} I_{s} - 2{\mkern 1mu} \frac{h}{\pi }\left( {E^{*} P_{1} + E_{s}^{*} P_{{1_{s} }} } \right) + \frac{{h^{2} }}{{\pi^{2} }}\left( {E^{*} L + E_{s}^{*} L_{s} } \right){ ,} \\ B_{4}\,=\,& \frac{4}{5}l_{1}^{2} \left( {G^{*} I + G_{s}^{*} I_{s} } \right) + \frac{4}{5}{\mkern 1mu} \frac{{l_{1}^{2} h^{2} }}{{\pi^{2} }}\left( {G^{*} L + G_{s}^{*} L_{s} } \right) - \frac{8}{5}\frac{{l_{1}^{2} h}}{\pi }\left( {G^{*} P_{1} + G_{s}^{*} P_{{1_{s} }} } \right) + 2{\mkern 1mu} l_{0}^{2} \left( {G^{*} I + G_{s}^{*} I_{s} } \right) \\ & - 4{\mkern 1mu} \frac{{l_{0}^{2} h}}{\pi }\left( {G^{*} P_{1} + G_{s}^{*} P_{{1_{s} }} } \right) + 2{\mkern 1mu} \frac{{l_{0}^{2} h^{2} }}{{\pi^{2} }}\left( {G^{*} L + G_{s}^{*} L_{s} } \right){ ,} \\ B_{5}\,=\,& \frac{4}{5}{\mkern 1mu} l_{1}^{2} G^{*} L - \frac{4}{5}\frac{{l_{1}^{2} G^{*} \pi {\mkern 1mu} P_{1} }}{h}{ ,} \quad B_{6} = \frac{4}{5}\frac{{l_{1}^{2} G^{*} \pi {\mkern 1mu} P_{0} }}{h}{ ,} \quad B_{7} = 2{\mkern 1mu} \frac{h}{\pi }\left( {E^{*} P_{0} + E_{s}^{*} P_{{0_{s} }} } \right){ ,} \\ B_{8}\,=\,& \frac{8}{5}{\mkern 1mu} \frac{{l_{1}^{2} h}}{\pi }\left( {G^{*} P_{0} + G_{s}^{*} P_{{0_{s} }} } \right) + 4{\mkern 1mu} \frac{{l_{0}^{2} h}}{\pi }\left( {G^{*} P_{0} + G_{s}^{*} P_{{0_{s} }} } \right){ ,} \quad B_{9} = - \frac{16}{15}{\mkern 1mu} \frac{{l_{1}^{2} G^{*} \pi^{2} L}}{{h^{2} }} - \frac{1}{2}{\mkern 1mu} \frac{{l_{2}^{2} G^{*} L{\mkern 1mu} \pi^{2} }}{{h^{2} }} - 2{\mkern 1mu} G^{*} O{ ,} \\ B_{10}\,=\,& - l_{1}^{2} \left( {\frac{64}{15}{\mkern 1mu} G^{*} O + \frac{16}{15}{\mkern 1mu} G_{s}^{*} O_{s} } \right) + l_{1}^{2} \left( {\frac{32}{15}{\mkern 1mu} G^{*} T_{0} + \frac{16}{15}{\mkern 1mu} G_{s}^{*} T_{{0_{s} }} } \right) - l_{2}^{2} \left( {\frac{1}{2}{\mkern 1mu} G^{*} O + 2{\mkern 1mu} G_{s}^{*} O_{s} } \right) \\ & + l_{2}^{2} \left( {G^{*} T_{0} + 2{\mkern 1mu} G_{s}^{*} T_{{0_{s} }} } \right) + 4{\mkern 1mu} l_{0}^{2} \left( {G^{*} T_{0} + G_{s}^{*} T_{{0_{s} }} } \right) - 4{\mkern 1mu} l_{0}^{2} \left( {G^{*} O + G_{s}^{*} O_{s} } \right) - 2{\mkern 1mu} \frac{{h^{2} }}{{\pi^{2} }}\left( {E^{*} L + E_{s}^{*} L_{s} } \right) \\ & + 2{\mkern 1mu} \frac{h}{\pi }\left( {E^{*} P_{1} + E_{s}^{*} P_{{1_{s} }} } \right){ ,} \\ B_{11} = & - \frac{4}{5}{\mkern 1mu} l_{1}^{2} G^{*} L + \frac{4}{5}{\mkern 1mu} \frac{{l_{1}^{2} G^{*} \pi P_{1} }}{h}{ ,} \\ B_{12} = & - \frac{8}{5}\frac{{l_{1}^{2} h^{2} }}{{\pi^{2} }}\left( {G^{*} L + G_{s}^{*} L_{s} } \right) + \frac{8}{5}{\mkern 1mu} \frac{{l_{1}^{2} h}}{\pi }\left( {G^{*} P_{1} + G_{s}^{*} P_{{1_{s} }} } \right) - 4{\mkern 1mu} \frac{{l_{0}^{2} h^{2} }}{{\pi^{2} }}\left( {G^{*} L + G_{s}^{*} L_{s} } \right) \\ & + 4{\mkern 1mu} \frac{{l_{0}^{2} h}}{\pi }\left( {G^{*} P_{1} + G_{s}^{*} P_{{1_{s} }} } \right){ ,} \\ B_{{13_{i} }}\,=\,& \tau_{s} S_{i} { ,}\,B_{{14_{i} }} = E^{*} A_{i} + E_{s}^{*} S_{i} { ,}\,B_{{15_{i} }} = \frac{4}{5}l_{1}^{2} \left( {G^{*} A_{i} + G_{s}^{*} S_{i} } \right) + 2{\mkern 1mu} l_{0}^{2} \left( {G^{*} A_{i} + G_{s}^{*} S_{i} } \right){ ,} \\ B_{16} = & - 2{\mkern 1mu} \frac{h}{\pi }\left( {E^{*} P_{0} + E_{s}^{*} P_{{0_{s} }} } \right){ ,} \quad B_{17} = - \frac{4}{5}\frac{{l_{1}^{2} G^{*} \pi {\mkern 1mu} P_{0} }}{h}{ ,} \\ B_{18} = & - \frac{8}{5}{\mkern 1mu} \frac{{l_{1}^{2} h}}{\pi }\left( {G^{*} P_{0} + G_{s}^{*} P_{{0_{s} }} } \right) - 4{\mkern 1mu} \frac{{l_{0}^{2} h}}{\pi }\left( {G^{*} P_{0} + G_{s}^{*} P_{{0_{s} }} } \right){ ,} \quad B_{19} = \frac{8}{15}{\mkern 1mu} \frac{{l_{1}^{2} G^{*} \pi^{2} L}}{{h^{2} }} + \frac{1}{4}\frac{{l_{2}^{2} G^{*} L{\mkern 1mu} \pi^{2} }}{{h^{2} }} + G^{*} O{ ,} \\ B_{20} = & - \frac{{\tau_{s} hP_{{0_{s} }} }}{\pi }{ ,} \\ B_{21}\,=\,& l_{1}^{2} \left( {\frac{32}{15}{\mkern 1mu} G^{*} O + \frac{8}{15}{\mkern 1mu} G_{s}^{*} O_{s} } \right) + l_{2}^{2} \left( {\frac{1}{4}{\mkern 1mu} G^{*} O + G_{s}^{*} O_{s} } \right) + 2{\mkern 1mu} l_{0}^{2} \left( {G^{*} O + G_{s}^{*} O_{s} } \right) + \frac{{h^{2} }}{{\pi^{2} }}\left( {E^{*} L + E_{s}^{*} L_{s} } \right){ ,} \\ B_{22}\,=\,& \frac{4}{5}{\mkern 1mu} \frac{{l_{1}^{2} h^{2} }}{{\pi^{2} }}\left( {G^{*} L + G_{s}^{*} L_{s} } \right) + 2{\mkern 1mu} \frac{{l_{0}^{2} h^{2} }}{{\pi^{2} }}\left( {G^{*} L + G_{s}^{*} L_{s} } \right){ ,}\,B_{23} = \frac{4}{5}{\mkern 1mu} l_{1}^{2} G^{*} L{ ,}\,B_{24} = - \frac{4}{5}{\mkern 1mu} l_{1}^{2} G^{*} L{ ,} \\ \end{aligned}$$

(55)

where the following integrals are defined

$$\left( {\begin{array}{*{20}c} {(A_{i} ,I,P_{0} ,P_{1} ,T_{0} ,L,O)} \\ {(S_{i} ,I_{s} ,P_{{0_{s} }} ,P_{{1_{s} }} ,T_{{0_{s} }} ,L_{s} ,O_{s} )} \\ \end{array} } \right) = \int_{A,S} {\left( {\begin{array}{*{20}c} \beta & 0 \\ 0 & \beta \\ \end{array} } \right)} \left( {\begin{array}{*{20}c} {dA_{i} } \\ {dS_{i} } \\ \end{array} } \right),$$

(56)

where

$$\beta = \left( {1,z^{2} ,f^{(\sin )} ,f^{{({ \cos })}} ,zf^{{({ \cos })}} ,\left( {f^{(\sin )} } \right)^{2} ,\left( {f^{{({ \cos })}} } \right)^{2} } \right).$$

(57)