Abstract
We derive equations of motion that describe the dynamics of a fluid confined within an elastic nanotube subject to periodic bending deflections. We use the principle of least action applied to a continuous open system at constant temperature. We solve the equations analytically in two limiting situations: when the tube oscillations are so small that they do not affect the fluid motion, but this one affects the tube dynamics; and when the flow magnitude is so small that it has no influence on the tube dynamics, but this one affects fluid motion. In the first case, we find out that the characteristic bending frequency spectrum of the tube depends not only on the magnitude of flow velocity, as previously stated in the literature, but also on the fluid velocity profile. This could constitute the basis of a strategy for indirect determination of the slip length in carbon nanotubes conveying flow via measurement of the buckling speed. In the second case, we find that tube vibrations can modify the dynamics of the fluid. Particularly, for a fluid subject to a constant pressure gradient, the tube motion induces an oscillatory motion in the fluid with twice the frequency of the tube. Moreover, the amplitude of the oscillatory fluid motion persists at high frequencies. This could constitute a strategy to generate high-frequency flows at nanoscales. Our results open up a panorama to control flow across nanotubes via tube vibrations, which could be complementary to chemical functionalization of nanostructures.
Highlights
Understanding the flow dynamics across nanometric channels plays an important role in the development of biomedical and chemical technology, from biosensors for 916 A16-1U
Two frames of reference arise in the study of a fluid confined within an oscillating tube: a static frame, (x, y, z), which is an inertial frame of reference and is used to describe the tube motion; and a dynamic frame, which is a non-inertial frame and is used to describe the fluid motion, which consists of cylindrical coordinates (r, θ, z ) or equivalently, (x, y, z ) such that the z -axis is located at the centre of the tube as it moves
It is possible to solve these equations, analytically, in regimes in which one of the dynamic variables is not strongly dependent on the other one. These regimes correspond to different physical considerations and they are: (i) a regime in which the tube deformation is very small, and the fluid dynamics is not affected by the tube oscillation; and (ii) a regime in which the fluid flow magnitude is very small, and the tube dynamics is not affected by fluid motion
Summary
Understanding the flow dynamics across nanometric channels plays an important role in the development of biomedical and chemical technology, from biosensors for. We consider that a model with these features can be derived by means of simple physical and geometrical constraints, via a formulation based on the principle of least action (Djukic & Vujanovic 1971; Lebon & Lambermont 1973; Leech 1977; Bedford & Drumheller 1983; Salmon 1983; Bedford 1985; Sieniutycz & Berry 1989; Shepherd 1990; Benaroya & Wei 2000) This approach has been useful to study complex geometries, such as the flow dynamics inside compressible nanobubbles (Teshukov & Gavrilyuk 2002) and systems subject to very complicated physical interactions, such as magnetorheological fluids (Sun, Zhou & Zhang 2003). We predict an oscillating velocity for the fluid within the nanotube, with twice the frequency of the latter one, that persists at high frequencies, even for a fluid driven by a constant pressure drop
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