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Wave propagation in double-walled carbon nanotube conveying fluid considering slip boundary condition and shell model based on nonlocal strain gradient theory

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Abstract

In this paper, wave propagation in fluid-conveying double-walled carbon nanotube (DWCNT) was investigated by using the nonlocal strain gradient theory. In so doing, the shear deformable shell theory was used, taking into consideration nonlocal and material length scale parameters. The effect of van der Waals force between the two intended walls and the DWCNT surroundings was modeled as Winkler foundation. The classical governing equations were derived from Hamilton’s principle. Results were validated by comparing them to the results of the references obtained through molecular dynamic method, and a remarkable consistency was found between the results. According to the findings, the effects of nonlocal and material length scale parameters, wave number, fluid velocity and stiffness of elastic foundation are more considerable in the nonlocal strain gradient theory than in classical theory.

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Correspondence to Yaghoub Tadi Beni.

Appendix

Appendix

In equations of motion (37)–(41), the constant coefficients are written as follows:

$$\begin{aligned} a_{1}^{i} & = A_{11} ,\quad a_{2}^{i} = \frac{{A_{33} }}{{R_{i}^{2} }},\quad a_{3}^{i} = - l_{2}^{2} A_{11} ,\quad a_{4}^{i} = - \frac{{l_{2}^{2} A_{33} }}{{R_{i}^{4} }},\quad a_{5}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{2} }}\left( {A_{11} + A_{33} } \right), \\ a_{6}^{i} & = \frac{1}{{R_{i} }}\left( {A_{12} + A_{33} } \right),\quad a_{7}^{i} = - \frac{{l_{2}^{2} }}{{R_{i} }}\left( {A_{12} + A_{33} } \right),\quad a_{8}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{3} }}\left( {A_{12} + A_{33} } \right),\quad a_{9}^{i} = \frac{{A_{12} }}{{R_{i} }}, \, \\ a_{10}^{i} & = - \frac{{l_{2}^{2} A_{12} }}{{R_{i} }},\quad a_{11}^{i} = - \frac{{l_{2}^{2} A_{12} }}{{R_{i}^{3} }}. \\ \end{aligned}$$
(45)
$$\begin{aligned} b_{1}^{i} & = \frac{{A_{12} }}{{R_{i} }} + \frac{{A_{33} }}{{R_{i} }},\quad b_{2}^{i} = - \frac{{l_{2}^{2} }}{{R_{i} }}\left( {A_{12} + A_{33} } \right),\quad b_{3}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{3} }}\left( {A_{12} + A_{33} } \right),\quad b_{4}^{i} = A_{33} + \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i}^{2} }}, \\ b_{5}^{i} & = - l_{2}^{2} A_{33} ,\quad b_{6}^{i} = \frac{{A_{22} }}{{R_{i}^{2} }} + \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i}^{4} }},\quad b_{7}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{2} }}\left( {A_{22} + A_{33} } \right),\quad b_{8}^{i} = - \frac{{l_{2}^{2} A_{22} }}{{R_{i}^{4} }},\quad b_{9}^{i} = - \frac{{A_{55} k_{s} }}{{R_{i}^{2} }}, \\ b_{10}^{i} & = \frac{{A_{22} }}{{R_{i}^{2} }} + \frac{{A_{55} k_{s} }}{{R_{i}^{2} }},\quad b_{11}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{2} }}\left( {A_{22} + A_{55} k_{s} } \right),\quad b_{12}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{4} }}\left( {A_{22} + A_{55} k_{s} } \right),\quad b_{13}^{i} = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i} }}, \\ b_{14}^{i} & = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i}^{3} }},\quad b_{15}^{i} = \frac{{A_{55} k_{s} }}{{R_{i} }}. \\ \end{aligned}$$
(46)
$$\begin{aligned} c_{1}^{i} & = - \frac{{A_{12} }}{{R_{i} }},\quad c_{2}^{i} = \frac{{l_{2}^{2} A_{12} }}{{R_{i} }},\quad c_{3}^{i} = \frac{{l_{2}^{2} A_{12} }}{{R_{i}^{3} }},\quad c_{4}^{i} = - \frac{1}{{R_{i}^{2} }}\left( {A_{22} + A_{55} k_{s} } \right),\quad c_{5}^{i} = \frac{{l_{2}^{2} }}{{R_{i}^{2} }}\left( {A_{22} + A_{55} k_{s} } \right), \\ c_{6}^{i} & = \frac{{l_{2}^{2} }}{{R_{i}^{4} }}\left( {A_{22} + A_{55} k_{s} } \right),\quad c_{7}^{i} = A_{44} k_{s} + \frac{{l_{2}^{2} A_{22} }}{{R_{i}^{2} }},\quad c_{8}^{i} = \frac{{A_{55} k_{s} }}{{R_{i}^{2} }} + \frac{{l_{2}^{2} A_{22} }}{{R_{i}^{4} }},\quad c_{9}^{i} = - l_{2}^{2} k_{s} A_{44} , \\ c_{10}^{i} & = - \frac{{k_{s} l_{2}^{2} }}{{R_{i}^{2} }}\left( {A_{44} + A_{55} } \right),\quad c_{11}^{i} = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R^{4} }},\quad c_{12}^{i} = - \frac{{A_{22} }}{{R_{i}^{2} }},\quad c_{13}^{i} = - l_{2}^{2} k_{s} A_{44} , \\ c_{14}^{i} & = \frac{{l_{2}^{2} B_{12} }}{{R^{3} }} - \frac{{k_{s} l_{2}^{2} A_{44} }}{{R_{i}^{2} }},\quad c_{15}^{i} = \frac{{B_{12} }}{{R_{i} }} - A_{44} k_{s} ,\quad c_{16}^{i} = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i} }},\quad c_{17}^{i} = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i}^{3} }},\quad c_{18}^{i} = \frac{{A_{55} k_{s} }}{{R_{i} }}. \\ \end{aligned}$$
(47)
$$\begin{aligned} d_{1}^{i} & = - A_{44} k_{s} ,\quad d_{2}^{i} = l_{2}^{2} A_{44} k_{s} ,\quad d_{3}^{i} = \frac{{l_{2}^{2} k_{s} A_{44} }}{{R_{i}^{2} }},\quad d_{4}^{i} = - l_{2}^{2} D_{11} ,\quad d_{5}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{2} }}\left( {D_{33} + D_{11} } \right), \\ d_{6}^{i} & = - \frac{{l_{2}^{2} D_{33} }}{{R_{i}^{4} }},\quad d_{7}^{i} = l_{2}^{2} A_{44} k_{s} + D_{11} ,\quad d_{8}^{i} = \frac{1}{{R_{i}^{2} }}\left( {l_{2}^{2} k_{s} A_{44} + D_{33} } \right),\quad d_{9}^{i} = - A_{44} k_{s} , \\ d_{10}^{i} & = - \frac{{l_{2}^{2} }}{{R_{i} }}\left( {D_{33} + D_{12} } \right),\quad d_{11}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{3} }}\left( {D_{33} + D_{12} } \right),\quad d_{12}^{i} = \frac{1}{{R_{i} }}\left( {D_{33} + D_{12} } \right). \\ \end{aligned}$$
(48)
$$\begin{aligned} e_{1}^{i} & = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i} }},\quad e_{2}^{i} = - \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i}^{3} }},\quad e_{3}^{i} = - \frac{{A_{55} k_{s} }}{{R_{i}^{2} }},\quad e_{4}^{i} = \frac{{A_{55} k_{s} }}{{R_{i} }},\quad e_{5}^{i} = \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i} }}, \\ e_{6}^{i} & = \frac{{l_{2}^{2} A_{55} k_{s} }}{{R_{i}^{3} }},\quad e_{7}^{i} = - \frac{{l_{2}^{2} }}{{R_{i} }}\left( {D_{12} + D_{33} } \right),\quad e_{8}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{3} }}\left( {D_{12} + D_{33} } \right),\quad e_{9}^{i} = \frac{1}{{R_{i} }}\left( {D_{12} + D_{33} } \right), \\ e_{10}^{i} & = - l_{2}^{2} D_{33} ,\quad e_{11}^{i} = - \frac{{l_{2}^{2} }}{{R_{i}^{2} }}\left( {D_{22} + D_{33} } \right),\quad e_{12}^{i} = - \frac{{l_{2}^{2} D_{22} }}{{R_{i}^{4} }},\quad e_{13}^{i} = l_{2}^{2} A_{55} k_{s} + D_{33} , \\ e_{14}^{i} & = \frac{1}{{R_{i}^{2} }}\left( {l_{2}^{2} A_{55} k_{s} + D_{22} } \right),\quad e_{15}^{i} = - A_{55} k_{s} . \\ \end{aligned}$$
(49)
$$I_{ji} = \int_{{ - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\rho_{ti} } z^{j} {\text{d}}z,\quad (i = 1,2),\quad (j = 0,2)$$
(50)

In the above equation, the constant coefficient is defined by:

$$\left\{ {A_{ij} , \, D_{ij} } \right\} = \int_{{ - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {C_{ij} } \left\{ {1, \, z^{2} } \right\}{\text{d}}z$$
(51)

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Zeighampour, H., Beni, Y.T. & Karimipour, I. Wave propagation in double-walled carbon nanotube conveying fluid considering slip boundary condition and shell model based on nonlocal strain gradient theory. Microfluid Nanofluid 21, 85 (2017). https://doi.org/10.1007/s10404-017-1918-3

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