Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equ...Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.展开更多
Environmental issues in the process of rapid urbanization are critical to the sustainability of China. Compact cities, characterized by relative high density, mixed land-use and pedestrian-oriented habitation, have be...Environmental issues in the process of rapid urbanization are critical to the sustainability of China. Compact cities, characterized by relative high density, mixed land-use and pedestrian-oriented habitation, have been proposed as one solution for sustainable urban planning. However, given the fact that Chinese cities are characterized by high population densities, the applicability of a more compact solution to expanding cities in China remains questionable. Essential to the decision-making is a deeper understanding about the relationship between urban compactness and the sustainable environment of the cities. This study is proposed for an evaluation about the urbanization and urban development in China’s context from environmental perspective, especially with respect to atmospheric environment, water resources, waste pollution, and noise problems.展开更多
The first research and experimental results obtained in China of high-accuracy radiometric calibration based on cryogenic radiometer are reported. Uncertainties of cryogenic radiometer and trap detectors at 7 waveleng...The first research and experimental results obtained in China of high-accuracy radiometric calibration based on cryogenic radiometer are reported. Uncertainties of cryogenic radiometer and trap detectors at 7 wavelengths in the visible spectrum (488-786 nm) were less than 0.023% and 0.035% respectively, which proved the reasonability and possibility of establishing and transferring high-accuracy radiometric standards based on detectors.展开更多
基金the National Natural Science Foundation of China(Nos.12172197,12171284,12120101001,and 11672163)the Fundamental Research Funds for the Central Universities(No.2019ZRJC002)。
文摘Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.
文摘Environmental issues in the process of rapid urbanization are critical to the sustainability of China. Compact cities, characterized by relative high density, mixed land-use and pedestrian-oriented habitation, have been proposed as one solution for sustainable urban planning. However, given the fact that Chinese cities are characterized by high population densities, the applicability of a more compact solution to expanding cities in China remains questionable. Essential to the decision-making is a deeper understanding about the relationship between urban compactness and the sustainable environment of the cities. This study is proposed for an evaluation about the urbanization and urban development in China’s context from environmental perspective, especially with respect to atmospheric environment, water resources, waste pollution, and noise problems.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.12172197,12171284,and 12120101001)the Natural Science Foundation of Shandong Province(Grant No.ZR2021ZD03).
文摘The first research and experimental results obtained in China of high-accuracy radiometric calibration based on cryogenic radiometer are reported. Uncertainties of cryogenic radiometer and trap detectors at 7 wavelengths in the visible spectrum (488-786 nm) were less than 0.023% and 0.035% respectively, which proved the reasonability and possibility of establishing and transferring high-accuracy radiometric standards based on detectors.
