An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

Experimental and numerical studies on the sluggish diffusion in face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys

On purpose of studying the sluggish diffusion of high-entropy alloys, three different face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys were prepared, and assembled into three groups of sandwich- type diffusion multiple annealed at 1273, 1323, and 1373 K respectively. By means of the electron probe microanalyzer technique and recently developed numerical inverse method, the composition- dependent interdiffusivities at different temperatures were effectively evaluated by minimizing the residual between the model-predicted compositions/interdiffusion fluxes and the respectively experi- mental ones. After that, the tracer diffusivities were predicted based on the assessed mobility parameters and thermodynamic descriptions with the simplified ideal solution model. The comprehensive compari- son between the interdiffusivities/tracer diffusivities in the Co-Cr-Cu-Fe-Ni high-entropy alloys and those in sub-binary, ternary, quaternary and other quinary alloys indicates that the sluggish diffusion exists in interdiffusion instead of tracer diffusion for the present Co-Cr-Cu-Fe-Ni high-entropy alloys.

Investigation on the temperature-dependent diffusion growth of intermetallic compounds in the Mg-Al-Zn system:Experiment and modeling

With the rapid development of Mg alloys,deeper understanding to the thermodynamic and diffusional kinetic behavior of intermetallic compounds(IMCs)is important for studying the effect of alloying elements to the microstructure evolution.Specially,a systematic quantitative investigation on the diffusional growth of IMCs is of great necessity.However,the works studying the elemental diffusion behaviors of multiple-element IMCs are rare in magnesium alloy systems.The current work takes the ternary Mg-Al-Zn system as research target,and combines the diffusion couple technique,phase stability diagrams,in-situ observation technique and numerical inverse method to investigate the temperature-dependent kinetic coefficients.The parabolic growth constant(PGC)and interdiffusion coefficients for Mg solid-solution phase andγ-Mg_(17)Al_(12),β-Mg_(2)Al_(3),ε-Mg_(23)Al_(30),MgZn_(2),Mg_(2)Zn_(3),τ-Mg_(32)(Zn,Al)49 andφ-Mg_(5)Zn_(2)Al_(2) IMCs in the Mg-Al-Zn alloy system are determined.By comparing the current experimental with calculation results,the rate-controlling factor of the temperature-dependent diffusion growth ofφ,τandεternary IMCs in the Mg-Al-Zn system is further discussed in detail.

Diffusion growth of Φ ternary intermetallic compound in the Mg-Al-Zn alloy system:In-situ observation and modeling

Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property.However,there is a very lack of existing data due to difficulties in both experimental and computational aspects.The current work aims at presenting the experimental observation on the diffusion growth behavior of Φ phase at 360℃ as well as calculating its composition-dependent interdiffu sion coefficients.We designed and succes s fully fabricated four Mg-τ ternary diffusion couples annealed at 360℃ for different times,where the diffusion path goes across the Φ phase region and the diffusion growth of ternary intermetallic compound can be solely detected.In-situ observation of the time-dependent growth of Φ phase was performed to accurately determine the parabolic growth constant.The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds,which can reproduce the presently observed diffusion growth behavior of Φ ternary intermetallic compound in Mg-τ diffusion couples.