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Critical rotational speed, critical velocity of fluid flow and free vibration analysis of a spinning SWCNT conveying viscous fluid

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Abstract

In this article, the influences of rotational speed and velocity of viscous fluid flow on free vibration behavior of spinning single-walled carbon nanotubes (SWCNTs) are investigated using the modified couple stress theory (MCST). Taking attention to the first-order shear deformation theory, the modeled rotating SWCNT and its equations of motion are derived using Hamilton’s principle. The formulations include Coriolis, centrifugal and initial hoop tension effects due to rotation of the SWCNT. This system is conveying viscous fluid, and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. The accuracy of the presented model is validated with some cases in the literatures. Novelty of this study is considering the effects of spinning, conveying viscous flow and MCST in addition to considering the various boundary conditions of the SWCNT. Generalized differential quadrature method is used to approximately discretize the model and to approximate the equations of motion. Then, influence of material length scale parameter, velocity of viscous fluid flow, angular velocity, length, length-to-radius ratio, radius-to-thickness ratio and boundary conditions on critical speed, critical velocity and natural frequency of the rotating SWCNT conveying viscous fluid flow are investigated.

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Authors and Affiliations

  1. Department of Mechanics, Faculty of Engineering, Imam Khomeini International University, Postal Code 3414916818, Qazvin, Iran

    Hamed SafarPour & Majid Ghadiri

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  1. Hamed SafarPour
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  2. Majid Ghadiri
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Correspondence to Majid Ghadiri.

Appendix

Appendix

$$\begin{aligned} & N_{xx} = A_{11} \frac{\partial u}{\partial x} + B_{11} \frac{{\partial \psi_{x} }}{\partial x} + A_{12} \left( {\frac{\partial v}{R\partial \theta } + \frac{w}{R}} \right) + B_{12} \frac{{\partial \psi_{\theta } }}{R\partial \theta }, \\ & Q_{xz} = k_{s} A_{55} \left( {\psi_{x} + \frac{\partial w}{\partial x}} \right), \\ & N_{\theta \theta } = A_{11} \left( {\frac{w}{R} + \frac{1}{R}\frac{\partial v}{\partial \theta }} \right) + B_{11} \frac{1}{R}\frac{{\partial \psi_{\theta } }}{\partial \theta } + A_{12} \frac{\partial u}{\partial x} + B_{12} \frac{{\partial \psi_{x} }}{\partial x} \\ & Q_{z\theta } = k_{s} A_{55} \left( {\psi_{\theta } + \frac{1}{R}\frac{\partial w}{\partial \theta } - \frac{v}{R}} \right), \\ & N_{x\theta } = A_{55} \left( {\frac{1}{R}\frac{\partial u}{\partial \theta } + \frac{\partial v}{\partial x}} \right) + B_{55} \left( {\frac{1}{R}\frac{{\partial \psi_{x} }}{\partial \theta } + \frac{{\partial \psi_{\theta } }}{\partial x}} \right), \\ & M_{xx} = B_{11} \frac{\partial u}{\partial x} + D_{11} \frac{{\partial \psi_{x} }}{\partial x} + B_{12} \left( {\frac{\partial v}{R\partial \theta } + \frac{w}{R}} \right) + D_{12} \frac{{\partial \psi_{\theta } }}{R\partial \theta }, \\ & M_{\theta \theta } = B_{11} \left( {\frac{w}{R} + \frac{1}{R}\frac{\partial v}{\partial \theta }} \right) + D_{11} \frac{1}{R}\frac{{\partial \psi_{\theta } }}{\partial \theta } + B_{12} \frac{\partial u}{\partial x} + D_{12} \frac{{\partial \psi_{x} }}{\partial x}, \\ & M_{x\theta } = B_{55} \left( {\frac{1}{R}\frac{\partial u}{\partial \theta } + \frac{\partial v}{\partial x}} \right) + D_{55} \left( {\frac{1}{R}\frac{{\partial \psi_{x} }}{\partial \theta } + \frac{{\partial \psi_{\theta } }}{\partial x}} \right), \\ & Y_{xx} = - A_{55} l^{2} \left( {\frac{{\partial \psi_{\theta } }}{\partial x} + \frac{1}{R}\frac{\partial v}{\partial x} - \frac{1}{R}\frac{{\partial^{2} w}}{\partial x\partial \theta }} \right), \\ & Y_{\theta \theta } = - A_{55} l^{2} \left[ {\frac{1}{R}\left( {\frac{1}{R}\frac{\partial u}{\partial \theta } - \frac{\partial v}{\partial x}} \right) + \frac{1}{R}\frac{{\partial^{2} w}}{\partial x\partial \theta } - \frac{1}{R}\frac{{\partial \psi_{x} }}{\partial \theta }} \right] - B_{55} l^{2} \left( { - \frac{{\partial \psi_{\theta } }}{\partial x}} \right), \\ & Y_{zz} = - A_{55} l^{2} \left[ { - \frac{1}{{R^{2} }}\frac{\partial u}{\partial \theta } + \frac{1}{R}\frac{{\partial \psi_{x} }}{\partial \theta } - \frac{{\partial \psi_{\theta } }}{\partial x}} \right], \\ & Y_{x\theta } = - \frac{{A_{55} l^{2} }}{2}\left( {\frac{1}{{R^{2} }}\frac{\partial v}{\partial \theta } + \frac{{\partial^{2} w}}{{\partial x^{2} }} - \frac{1}{{R^{2} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{1}{R}\frac{{\partial \psi_{\theta } }}{\partial \theta } - \frac{{\partial \psi_{x} }}{\partial x}} \right), \\ & Y_{xz} = - \frac{{A_{55} l^{2} }}{2}\left[ { - \frac{{\partial^{2} v}}{{\partial x^{2} }} - \frac{v}{{R^{2} }} + \frac{1}{R}\frac{{\partial^{2} u}}{\partial x\partial \theta } + \frac{{\psi_{\theta } }}{R} + \frac{1}{{R^{2} }}\frac{\partial w}{\partial \theta }} \right] - \frac{{B_{55} l^{2} }}{2}\left( {\frac{1}{R}\frac{{\partial^{2} \psi_{x} }}{\partial x\partial \theta } - \frac{{\partial^{2} \psi_{\theta } }}{{\partial x^{2} }}} \right), \\ & Y_{\theta z} = - \frac{{A_{55} l^{2} }}{2}\left[ {\frac{1}{{R^{2} }}\frac{{\partial^{2} u}}{{\partial \theta^{2} }} - \frac{1}{R}\frac{{\partial^{2} v}}{\partial x\partial \theta } + \frac{{\psi_{x} }}{R} - \frac{1}{R}\frac{\partial w}{\partial x}} \right] - \frac{{B_{55} l^{2} }}{2}\left( {\frac{1}{{R^{2} }}\frac{{\partial^{2} \psi_{x} }}{{\partial \theta^{2} }} - \frac{1}{R}\frac{{\partial^{2} \psi_{\theta } }}{\partial x\partial \theta }} \right), \\ & T_{xz} = - \frac{{B_{55} l^{2} }}{2}\left[ { - \frac{{\partial^{2} v}}{{\partial x^{2} }} - \frac{v}{{R^{2} }} + \frac{1}{R}\frac{{\partial^{2} u}}{\partial x\partial \theta } + \frac{{\psi_{\theta } }}{R} + \frac{1}{{R^{2} }}\frac{\partial w}{\partial \theta }} \right] - \frac{{D_{55} l^{2} }}{2}\left( {\frac{1}{R}\frac{{\partial^{2} \psi_{x} }}{\partial x\partial \theta } - \frac{{\partial^{2} \psi_{\theta } }}{{\partial x^{2} }}} \right), \\ & T_{\theta z} = - \frac{{B_{55} l^{2} }}{2}\left[ {\frac{1}{{R^{2} }}\frac{{\partial^{2} u}}{{\partial \theta^{2} }} - \frac{1}{R}\frac{{\partial^{2} v}}{\partial x\partial \theta } + \frac{{\psi_{x} }}{R} - \frac{1}{R}\frac{\partial w}{\partial x}} \right] - \frac{{D_{55} l^{2} }}{2}\left( {\frac{1}{{R^{2} }}\frac{{\partial^{2} \psi_{x} }}{{\partial \theta^{2} }} - \frac{1}{R}\frac{{\partial^{2} \psi_{\theta } }}{\partial x\partial \theta }} \right). \\ & T_{\theta \theta } = - B_{55} l^{2} \left[ {\frac{1}{R}\left( {\frac{1}{R}\frac{\partial u}{\partial \theta } - \frac{\partial v}{\partial x}} \right) + \frac{1}{R}\frac{{\partial^{2} w}}{\partial x\partial \theta } - \frac{1}{R}\frac{{\partial \psi_{x} }}{\partial \theta }} \right] - D_{55} l^{2} \left( { - \frac{{\partial \psi_{\theta } }}{\partial x}} \right), \\ & \varsigma_{1} = \rho_{f} h_{f} \quad \varsigma_{2} = \frac{{h_{f} }}{{R^{2} }}\mu n^{2} + \frac{{2h_{f} }}{R}\mu ,\quad \varsigma_{3} = \mu h_{f} ,\quad \varsigma_{4} = \varsigma_{1} ,\quad \varsigma_{5} = 2\varsigma_{1} ,\quad \varsigma_{6} = \varsigma_{2} ,\quad \varsigma_{7} = \varsigma_{3} \\ \end{aligned}$$

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SafarPour, H., Ghadiri, M. Critical rotational speed, critical velocity of fluid flow and free vibration analysis of a spinning SWCNT conveying viscous fluid. Microfluid Nanofluid 21, 22 (2017). https://doi.org/10.1007/s10404-017-1858-y

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