On Accuracy Order of Fourier Coefficients Computation for Periodic Signal Processing Models

The article is devoted to construction piecewise constant functions for modelling periodic signal. The aim of the paper is to suggest a way to avoid discontinuity at points where waveform values are obtained. One solution is to introduce shifted step function whose middle points within its partial intervals coincide with points of observation. This means that large oscillations of Fourier partial sums move to new jump discontinuities where waveform values are not obtained. Furthermore, any step function chosen to model periodic continuous waveform determines a way to calculate Fourier coefficients. In this case, the technique is certainly a weighted rectangular quadrature rule. Here, the weight is either unit or trigonometric. Another effect of the solution consists in following. The shifted function leads to application midpoint quadrature rules for computing Fourier coefficients. As a result the formula for zero coefficient transforms into trapezoid rule. In the same time, the formulas for other coefficients remain of rectangular type

On the Reduction of the Driving Force in Shear-driven Flows

In shear-driven flows, an external driving force is needed to maintain the relative movement of horizontal plates. This thesis presents a systematic analysis on using spatially periodic heating and grooved surfaces to control this force. It is found that the use of periodic heating creates a buoyancy-driven effect that always reduces this force. The use of proper heating may even lead to the complete elimination of this force. It is further found that the use of isothermal grooved surfaces always enhances flow resistance, resulting in an increase of this force. When grooves and heating are applied together, their interaction induces a horizontal pressure force that can either increase or decrease the driving force, depending on the relative positions of the groove and heating patterns. Mechanisms leading to such changes of the driving force are discussed