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Nasibullayev I.Sh. Reducing the systematic error of the average fluid flow rate in axisymmetric computer model of piezoelectric micropump. Multiphase Systems. 16 (2021) 1. 22–33 (in Russian).
2021. Vol. 16. Issue 1, Pp. 22–33
URL: http://mfs.uimech.org/mfs2021.1.004,en
DOI: 10.21662/mfs2021.1.004
Reducing the systematic error of the average fluid flow rate in axisymmetric computer model of piezoelectric micropump
Nasibullayev I.Sh.
Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia

Abstract

Mathematical and computer models of an axisymmetric coupled problem of interaction of a fluid and an elastic body for the solving partial differential equations by the finite element method FreeFem ++ software are proposed. In these models, periodic vibrations of annular piezoelectric actuators placed on an elastic tube of circular cross-section lead to radial deformations of the tube. With a synchronous oscillation of the system of piezoelectric actuators, the average fluid flow rate is zero. An asymmetric oscillation pattern (micropump mode) results in a nonzero average flow rate. Two types of boundary conditions are considered in the region of contact between the piezoelectric drive and the tube — Dirichlet and Neumann. The computer model was tested for unsteady fluid flow induced by a pressure gradient applied along the pipe with a circular cross section. With synchronous oscillation of piezoelectric actuators, a nonzero average fluid flow is a systematic error. This value was compared with the asymmetric oscillation pattern of piezoelectric actuators to determine the contribution of the systematic error to the pump-generated fluid flow rate. Based on the fluid flow velocity profile and the results of profiling the program code, the optimal parameters of the computational meshes for the channel (dense uniform) and tube walls (nonuniform, consistent with the velocity profile and reconstructing at each time step) were determined, which reduce the magnitude of the systematic error. The main way to reduce the error is to increase the density of the computational mesh, which leads to an increase in the required computing resources. An alternative method is proposed for reducing the systematic error due to an additional pressure drop applied to the channel. Depending on the type of boundary conditions and the number of piezoelectric actuators, this method can reduce the systematic error by 1 − 2 orders of magnitude without increasing the simulation time.

Keywords

hydrodynamics,
hydraulic resistance,
piezoelectric actuator,
linear elasticity,
microchannel,
finite element method,
micropump

Article outline

Mathematical and computer models of an axisymmetric coupled problem of interaction of a fluid and an elastic body for solving partial differential equations by the finite element method FreeFem++ software are proposed. In these models, periodic vibrations of annular piezoelectric actuators placed on an elastic tube of circular cross-section lead to radial deformations of the tube. With a synchronous oscillation of the system of piezoelectric actuators, the average fluid flow rate is zero. An asymmetric oscillation pattern (micropump mode) results in a non-zero average flow rate. The equations were written in variational form. Time discretization was carried out by the implicit Euler method. For nonlinear terms in the Navier-Stokes equation, the Picard scheme was used. To increase the stability of the numerical scheme, artificial compressibility was added to the continuity equation. Two types of boundary conditions were considered in the area of contact between the piezoelectric drive and the tube - Dirichlet (”hard“ piezoelectric element) and Neumann (”soft“ piezoelectric element). The equations were solved in the FreeFem++ package by the finite element method. To prevent the appearance of an error associated with the displacement of the nodes of one finite element inside another finite element (the appearance of negative volumes), when remeshing the computational mesh for the channel, the algorithm of the ”safe“ remeshing of the computational mesh, developed earlier by the author, was used.

A specific feature of the problem under consideration is that the deformations of the microchannel are small (0.5% of the radius) and at high oscillation frequencies the fluid flow velocity is high (several mm/s). The computer model was tested for an unsteady fluid flow induced by a pressure gradient applied along the layer in a pipe with a circular cross section (Poiseuille flow). The errors of numerical simulation for the instantaneous value of the fluid flow rate and the characteristic relaxation time of the velocity (the time for establishing a constant flow) did not exceed 1 % relative to the analytical solution according to the Poiseuille formula and the characteristic relaxation time of the velocity. With synchronous oscillation of piezoelectric elements, a nonzero average fluid flow is a systematic error. This value was compared with the asymmetric oscillation pattern of piezoelectric elements to determine the contribution of the systematic error to the pump-generated liquid flow rate.

Based on the fluid flow velocity profile and the results of profiling the program code, the optimal parameters of the computational grids for the channel (dense uniform) and tube walls (non-uniform, consistent with the velocity profile and reconstructing at each time step) were determined, which reduce the magnitude of the systematic error. The computational meshes for the microchannel and the tube are consistent: the boundary elements on the contact line of the grids coincide. The main way to reduce the error is to increase the density of the computational mesh, which leads to an increase in the required computing resources.

An alternative method is proposed for reducing the systematic error due to an additional pressure drop applied to the channel. Depending on the type of boundary conditions and the number of piezoelectric actuators, this method can reduce the systematic error by 1-2 orders of magnitude without increasing the simulation time. The approach presented in the work to reduce the systematic error will improve the accuracy of modeling the operating modes of a piezoelectric micropump.

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