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The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.

The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.

The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.

In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

This paper aims to combine the grooves with an annular air thrust bearing with multi-hole restrictors and discusses the influence of the groove parameters on the bearing performance.

Four models of aerostatic bearings with grooves of different geometries are established. The pressure distribution, load-carrying capacity (LCC), stiffness and flow characteristics of the flow field in the bearing clearances are obtained by computational fluid dynamics simulation.

The numerical and simulation results show that air bearing with grooved restrictors can slow down the pressure drop at the air inlet and increase the LCC and stiffness of the bearing. The gas flow in the aerostatic bearing is also studied, and the air vortex in the recess is analyzed.

This research optimizes the structure of the annular air thrust bearing, analyzes the gas vortex in the recess, improves the LCC and stiffness of the bearing and provides a reference for the bearing in the selection of groove parameters.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-01-2023-0006/

Based on social network theory and the literature of contract governance, the purpose of this study is to explore how distributors’ “banding together” (network intensity) affects contract governance (detailed contracts), which, in turn, influences channel conflict and simultaneously tests the moderating effect of network centrality.

The authors collect the data from the side of 288 manufacturers.

This study finds that, first, distributors’ network intensity positively affects detailed contracts between manufacturers and distributors; second, detailed contracts, in turn, declines channel conflict; and third, network centrality will weaken the positive influence of network density on detailed contracts. In addition, an ex post analysis finds that detailed contracts play a negative mediating role between distributors’ network density and channel conflict and this negative mediating effect would be weakened by distributors’ network centrality.

The current study not only helps to make up for the shortcomings of using the dyadic analysis paradigm to analyze channel behavior but also helps manufacturers to understand and respond to the phenomenon of distributors’ “banding together” comprehensively and deeply.