Wave dispersion characteristics of fluid-conveying magneto-electro-elastic nanotubes

Abstract

In this paper, the wave propagation analysis of fluid-conveying magneto-electro-elastic (MEE) nanotube incorporating fluid effect is investigated. To take into account the small-scale effects, the nonlocal elasticity theory of Eringen is employed. Nonlocal governing equations of MEE-FG nanotube have been derived utilizing Hamilton’s principle. The results of this study have been verified by checking them with antecedent investigations. An analytical solution of governing equations is used to acquire wave frequencies and phase velocities. The Knudsen number is applied to study the effect of slip boundary wall of nanotube and flow. Effect of Knudsen number, different modes, length parameter, nonlocal parameter, fluid velocity, fluid effect and slip boundary condition on wave propagation characteristics of fluid-conveying MEE nanotube is investigated, and the results are presented in detail.

Appendix

$$ K_{11} = - A_{11} k^{2} - A_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - A_{11} l^{2} k^{4} - \frac{{A_{66} }}{{R^{2} }}n^{2} - A_{66} \frac{{l^{2} }}{{R^{4} }}n^{4} - A_{66} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} $$

(68)

$$ K_{12} = - \frac{{A_{12} }}{R}kn - A_{12} \frac{{l^{2} }}{{R^{3} }}kn^{3} - A_{12} \frac{{l^{2} }}{R}k^{3} n - \frac{{A_{66} }}{R}kn - A_{66} \frac{{l^{2} }}{{R^{3} }}kn^{3} - A_{66} \frac{{l^{2} }}{R}k^{3} n $$

(69)

$$ K_{13} = + \frac{{A_{12} }}{R}ki + A_{12} \frac{{l^{2} }}{{R^{3} }}kn^{2} i + A_{12} \frac{{l^{2} }}{R}k^{3} i $$

(70)

$$ K_{14} = 0 $$

(71)

$$ K_{15} = 0 $$

(72)

$$ K_{21} = \frac{{A_{66} }}{R}kn - A_{66} \frac{{l^{2} }}{{R^{3} }}kn^{3} - A_{66} \frac{{l^{2} }}{R}k^{3} n - \frac{{A_{12} }}{R}kn - A_{12} \frac{{l^{2} }}{{R^{3} }}kn^{3} - A_{12} \frac{{l^{2} }}{R}k^{3} n $$

(73)

$$ \begin{aligned} K_{22} & = - A_{66} k^{2} - A_{66} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - A_{66} l^{2} k^{4} - \frac{{A_{11} }}{{R^{2} }}n^{2} - A_{11} \frac{{l^{2} }}{{R^{4} }}n^{4} - A_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - \frac{{D_{66} }}{{R^{2} }}k^{2} \\ & \quad - D_{66} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{2} - D_{66} \frac{{l^{2} }}{{R^{2} }}k^{4} - \frac{{D_{11} }}{{R^{4} }}n^{2} - D_{11} \frac{{l^{2} }}{{R^{6} }}n^{4} - D_{11} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{2} \\ \end{aligned} $$

(74)

$$ \begin{aligned} K_{23} & = + \frac{{A_{11} }}{{R^{2} }}ni + A_{11} \frac{{l^{2} }}{{R^{4} }}n^{3} i - A_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} ni + 2\frac{{D_{66} }}{{R^{2} }}k^{2} ni + 2D_{66} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{3} i + 2D_{66} \frac{{l^{2} }}{{R^{2} }}k^{4} ni \\ & \quad + \frac{{D_{12} }}{{R^{2} }}k^{2} ni + D_{12} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{3} i + D_{12} \frac{{l^{2} }}{{R^{2} }}k^{4} ni + \frac{{D_{11} }}{{R^{4} }}n^{3} i + D_{11} \frac{{l^{2} }}{{R^{6} }}n^{5} i + D_{11} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{3} \\ \end{aligned} $$

(75)

$$ K_{24} = \frac{{E_{31} }}{{R^{2} }}ni + E_{31} \frac{{l^{2} }}{{R^{4} }}n^{3} i + E_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} ni $$

(76)

$$ K_{25} = \frac{{Q_{31} }}{{R^{2} }}ni + Q_{31} \frac{{l^{2} }}{{R^{4} }}n^{3} i + Q_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} ni $$

(77)

$$ K_{31} = - \frac{{A_{12} }}{R}ki - A_{12} \frac{{l^{2} }}{{R^{3} }}kn^{2} i - A_{12} \frac{{l^{2} }}{R}k^{3} i $$

(78)

$$ \begin{aligned} K_{32} & = + \frac{{D_{12} }}{{R^{2} }}k^{2} ni + D_{12} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{3} i + D_{12} \frac{{l^{2} }}{{R^{2} }}k^{4} ni + \frac{{2D_{66} }}{{R^{2} }}k^{2} ni + 2D_{66} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{3} i \\ & \quad + 2D_{66} \frac{{l^{2} }}{{R^{2} }}k^{4} ni - \frac{{D_{11} }}{{R^{4} }}n^{3} i - D_{11} \frac{{l^{2} }}{{R^{6} }}n^{5} i - D_{11} \frac{{l^{2} }}{{R^{4} }}n^{3} k^{2} i - \frac{{A_{11} }}{{R^{2} }}ni - A_{11} \frac{{l^{2} }}{{R^{4} }}n^{3} i \\ & \quad - A_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} ni \\ \end{aligned} $$

(79)

$$ \begin{aligned} K_{33} & = - D_{11} k^{4} - D_{11} \frac{{l^{2} }}{{R^{2} }}k^{4} n^{2} - D_{11} l^{2} k^{6} - 2\frac{{D_{12} }}{{R^{2} }}k^{2} n^{2} - 2D_{12} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{4} \\ & \quad - 2D_{12} \frac{{l^{2} }}{{R^{2} }}k^{4} n^{2} - \frac{{4D_{66} }}{{R^{2} }}k^{2} n^{2} - 4D_{66} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{4} - 4D_{66} \frac{{l^{2} }}{{R^{2} }}k^{4} n^{2} - \frac{{D_{11} }}{{R^{4} }}n^{4} \\ & \quad - D_{11} \frac{{l^{2} }}{{R^{6} }}n^{6} - D_{11} \frac{{l^{2} }}{{R^{4} }}k^{2} n^{4} - \frac{{A_{11} }}{{R^{2} }}W - A_{11} \frac{{l^{2} }}{{R^{4} }}n^{2} - A_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} + \frac{{N_{\theta 0} }}{{R^{2} }}n^{2} \\ & \quad + N_{x0} k^{2} + (e_{0} a)^{2} \left( {\frac{{N_{\theta 0} }}{{R^{4} }}n^{4} + \frac{{N_{\theta 0} }}{{R^{2} }}k^{2} n^{2} + N_{x0} k^{4} + \frac{{N_{x0} }}{{R^{2} }}k^{2} n^{2} } \right) \\ & \quad + \frac{{\rho_{f} R}}{2}\left( {(V_{\text{f}} {\text{VCF}})^{2} k^{2} } \right) + (e_{0} a)^{2} \frac{{\rho_{f} R}}{2}\left( {(V_{\text{f}} {\text{VCF}})^{2} k^{4} + (V_{\text{f}} {\text{VCF}})^{2} \frac{{k^{2} n^{2} }}{{R^{2} }}} \right) \\ \end{aligned} $$

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$$ K_{34} = - E_{31} k^{2} - E_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - E_{31} l^{2} k^{4} + \frac{{E_{31} }}{{R^{2} }}n^{2} - E_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - E_{31} \frac{{l^{2} }}{{R^{4} }}n^{4} $$

(81)

$$ K_{35} = - Q_{31} k^{2} - Q_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - Q_{31} l^{2} k^{4} - \frac{{Q_{31} }}{{R^{2} }}n^{2} - Q_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - Q_{31} \frac{{l^{2} }}{{R^{4} }}n^{4} $$

(82)

$$ K_{41} = 0 $$

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$$ K_{42} = \frac{{E_{31} }}{{R^{2} }}ni + E_{31} \frac{{l^{2} }}{{R^{4} }}n^{3} i + E_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} ni $$

(84)

$$ K_{43} = E_{31} k^{2} + E_{31} l^{2} k^{4} + E_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} + \frac{{E_{31} }}{{R^{2} }}n^{2} + E_{31} \frac{{l^{2} }}{{R^{4} }}n^{4} + E_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} $$

(85)

$$ \begin{aligned} K_{44} & = - X_{11} k^{2} - X_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - X_{11} l^{2} n^{4} - X_{22} n^{2} - X_{22} \frac{{l^{2} }}{{R^{2} }}n^{4} - X_{22} l^{2} k^{2} n^{2} - X_{33} \\ & \quad - X_{33} k^{2} - X_{33} \frac{{n^{2} }}{{R^{2} }} \\ \end{aligned} $$

(86)

$$ \begin{aligned} K_{45} & = - Y_{11} k^{2} - Y_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - Y_{11} l^{2} k^{4} - Y_{22} n^{2} - Y_{22} \frac{{l^{2} }}{{R^{2} }}n^{4} - Y_{22} l^{2} k^{2} n^{2} - Y_{33} - Y_{33} \frac{{l^{2} }}{{R^{2} }}n^{2} \\ & \quad - Y_{33} l^{2} k^{2} \\ \end{aligned} $$

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$$ K_{51} = 0 $$

(88)

$$ K_{52} = \frac{{Q_{31} }}{{R^{2} }}ni + Q_{31} \frac{{l^{2} }}{{R^{4} }}n^{3} i + Q_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} ni $$

(89)

$$ K_{53} = Q_{31} k^{2} + Q_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} + Q_{31} l^{2} k^{4} + \frac{{Q_{31} }}{{R^{2} }}n^{2} + Q_{31} \frac{{l^{2} }}{{R^{4} }}n^{4} + Q_{31} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} $$

(90)

$$ \begin{aligned} K_{54} & = - Y_{11} k^{2} - Y_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - Y_{11} l^{2} k^{4} - Y_{22} n^{2} - Y_{22} \frac{{l^{2} }}{{R^{2} }}n^{4} - Y_{22} l^{2} k^{2} n^{2} - Y_{33} \\ & \quad - Y_{33} \frac{{l^{2} }}{{R^{2} }}n^{2} - Y_{33} l^{2} k^{2} \\ \end{aligned} $$

(91)

$$ \begin{aligned} K_{55} & = - T_{11} k^{2} - T_{11} \frac{{l^{2} }}{{R^{2} }}k^{2} n^{2} - T_{11} l^{2} k^{4} - T_{22} n^{2} - T_{22} \frac{{l^{2} }}{{R^{2} }}n^{4} - T_{22} l^{2} k^{2} n^{2} - T_{33} \\ & \quad - T_{33} \frac{{l^{2} }}{{R^{2} }}n^{2} - T_{33} l^{2} k^{2} \\ \end{aligned} $$

(92)

$$ M_{11} = M_{22} = - I_{1} - I_{1} (e_{0} a)^{2} \left( {k^{2} + \frac{{n^{2} }}{{R^{2} }}} \right) $$

(93)

$$ M_{33} = - I_{1} - I_{1} (e_{0} a)^{2} \left( {k^{2} + \frac{{n^{2} }}{{R^{2} }}} \right) + \frac{{\rho_{\text{f}} R}}{2} + (e_{0} a)^{2} \frac{{\rho_{\text{f}} R}}{2}\left( {k^{4} + \frac{{k^{2} n^{2} }}{{R^{2} }}} \right) $$

(94)

$$ C_{33} = - 2\frac{{\rho_{\text{f}} R}}{2}(V_{\text{f}} {\text{VCF}})k - 2(e_{0} a)^{2} \frac{{\rho_{\text{f}} R}}{2}\left( {k^{3} + \frac{{kn^{2} }}{{R^{2} }}} \right). $$

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