document: After all micro elements reach a relaxed steady-state, measurements are obtained using a cumulative averaging technique to reduce noise. Each micro element is divided into spatially-oriented bins in the y-direction in order to resolve the velocity and shear-stress profiles. Velocity in each bin is measured using the Cumulative Averaging Method (CAM) [24], while the stress tensor field is measured using the Irving–Kirkwood relationship [25]. A least-squares polynomial fit to the data is performed, which helps reduce noise further. The fit produces a continuous function that avoids stability issues arising from supplying highly fluctuating data to the macro solver. A least-squares fit is applied to an Nth order polynomial for the velocity profile in the core region, and an Mth order polynomial for the velocity profile in the constrained region:(16)〈ui,core〉=∑k=1Nbk,iyi′(N−k),for 0⩽yi′⩽hcore, and(17)〈ui,cs〉=∑k=1Mck,iyi″(M−k),for 0⩽yi″⩽hcs, where bk,i and ck,i are the coefficients of the polynomials used in the core micro region and constrained region respectively. An estimate of the new slip velocity uB for input to the macro solution (6) is taken directly from the compressed wall micro-element solution (16), at yi′=0.
extractive_keyphrases: cam continuous function cumulative averaging method cumulative averaging technique irving–kirkwood relationship macro solution macro solver mth order polynomial new slip velocity nth order polynomial polynomials reduce noise stability issues stress tensor field velocity
abstractive_keyphrases: compressed wall micro-element solution divided into spatially-oriented bins least-squares fit least-squares polynomial fit resolve the velocity and shear-stress profiles shear-stress profiles supplying highly fluctuating data to the macro solver