In this paper,we introduce and analyze an augmented mixed discontinuous Galerkin(MDG)method for a class of quasi-Newtonian Stokes flows.In the mixed formulation,the unknowns are strain rate,stress and velocity,which are approximated by a discontinuous piecewise polynomial triplet ■for k≥0.Here,the discontinuous piecewise polynomial function spaces for the field of strain rate and the stress field are designed to be symmetric.In addition,the pressure is easily recovered through simple postprocessing.For the benefit of the analysis,we enrich the MDG scheme with the constitutive equation relating the stress and the strain rate,so that the well-posedness of the augmented formulation is obtained by a nonlinear functional analysis.For k≥0,we get the optimal convergence order for the stress in broken ■(div)-norm and velocity in L^(2)-norm.Furthermore,the error estimates of the strain rate and the stress in-norm,and the pressure in L^(2)-norm are optimal under certain conditions.Finally,several numerical examples are given to show the performance of the augmented MDG method and verify the theoretical results.Numerical evidence is provided to show that the orders of convergence are sharp.
Due to a prolonged operation time and low mass transfer efficiency, the primary challenge in the aeration process of non-Newtonian fluids is the high energy consumption, which is closely related to the form and rate of impeller, ventilation, rheological properties and bubble morphology in the reactor. In this perspective, through optimal computational fluid dynamics models and experiments, the relationship between power consumption, volumetric mass transfer rate(kLa) and initial bubble size(d0) was constructed to establish an efficient operation mode for the aeration process of non-Newtonian fluids. It was found that reducing the d0could significantly increase the oxygen mass transfer rate, resulting in an obvious decrease in the ventilation volume and impeller speed. When d0was regulated within 2-5 mm,an optimal kLa could be achieved, and 21% of power consumption could be saved, compared to the case of bubbles with a diameter of 10 mm.
A prediction framework based on the evolution of pattern motion probability density is proposed for the output prediction and estimation problem of non-Newtonian mechanical systems,assuming that the system satisfies the generalized Lipschitz condition.As a complex nonlinear system primarily governed by statistical laws rather than Newtonian mechanics,the output of non-Newtonian mechanics systems is difficult to describe through deterministic variables such as state variables,which poses difficulties in predicting and estimating the system’s output.In this article,the temporal variation of the system is described by constructing pattern category variables,which are non-deterministic variables.Since pattern category variables have statistical attributes but not operational attributes,operational attributes are assigned to them by posterior probability density,and a method for analyzing their motion laws using probability density evolution is proposed.Furthermore,a data-driven form of pattern motion probabilistic density evolution prediction method is designed by combining pseudo partial derivative(PPD),achieving prediction of the probability density satisfying the system’s output uncertainty.Based on this,the final prediction estimation of the system’s output value is realized by minimum variance unbiased estimation.Finally,a corresponding PPD estimation algorithm is designed using an extended state observer(ESO)to estimate the parameters to be estimated in the proposed prediction method.The effectiveness of the parameter estimation algorithm and prediction method is demonstrated through theoretical analysis,and the accuracy of the algorithm is verified by two numerical simulation examples.
