Nonlinear vibration analysis of protein microtubules in cytosol conveying fluid based on nonlocal elasticity theory using differential quadrature method

In this article, nonlinear vibration of protein microtubules in cytosol with internal flow is studied. Based on the Euler–Bernoulli beam theory with von Kármán nonlinearity type and using Hamilton’s principle, the equations of motion for fluid-conveying microtubules are derived. The size effect is taken into account using Eringen’s nonlocal elasticity theory; moreover, the effect of an elastic surrounding filament network and the surface traction of cytosol are studied. The governing differential equations for vibration response of microtubules are solved using the differential quadrature method. The nonlinear frequency response of microtubules, considering the effect of microtubule properties, size effect, the surrounding elastic media, and the fluid motion are reported in this article. It has been found that the effect of nonlocal parameter on the vibration behavior and instability of the embedded microtubule conveying fluid are significant. In this regard, we need to point out that the critical flow velocity for a range of nonlocality parameter from 0 to 2 nm varies between 41 and 47 m/s, which should be avoided due to instability of the microtubule system. Therefore, they should be taken into account in the design of nano/micro-devices for measuring density of a fluid, such as drugs flowing through such microtubules, with great applications in biomechanics.