A fractional differential equation for a MEMS viscometer used in the oil industry

Authors:
A. D. Fitt;A. R. H. Goodwin;K. A. Ronaldson;W. A. Wakeham
Affiliations:
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK;Schlumberger, 125 Industrial Blvd., Sugar Land, TX 77478, USA;School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK;School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

Citing 2
Cited 3

On Mittag-Leffler-type functions in fractional evolution processes

Journal of Computational and Applied Mathematics - Special issue on higher transcendental functions and their applications
Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

Technical communique: A note on fractional-order derivatives of periodic functions

Automatica (Journal of IFAC)
Quadratic spline solution for boundary value problem of fractional order

Numerical Algorithms
Application of the collocation method for solving nonlinear fractional integro-differential equations

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	7.30

Visualization

Abstract

A mathematical model is developed for a micro-electro-mechanical system (MEMS) instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration. It is shown that, in one mode of operation, the displacement of the device satisfies a fractional differential equation (FDE). The theory of FDEs is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme. It is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised. It is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring, where the resulting FDE is nonlinear.