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This study explores the immobilisation of enzymes within porous catalysts of various geometries, including spheres, cylinders and flat pellets. The objective is to understand the irreversible Michaelis-Menten kinetic process within immobilised enzymes through advanced mathematical modelling.

Mathematical models were developed based on reaction-diffusion equations incorporating nonlinear variables associated with Michaelis-Menten kinetics. This research introduces fractional derivatives to investigate enzyme reaction kinetics, addressing a significant gap in the existing literature. A novel approximation method, based on the independent polynomials of the complete bipartite graph, is employed to explore solutions for substrate concentration and effectiveness factor across a spectrum of parameter values. The analytical solutions generated through the bipartite polynomial approximation method (BPAM) are rigorously tested against established methods, including the Bernoulli wavelet method (BWM), Taylor series method (TSM), Adomian decomposition method (ADM) and fourth-order Runge-Kutta method (RKM).

The study identifies two main findings. Firstly, the behaviour of dimensionless substrate concentration with distance is analysed for planar, cylindrical and spherical catalysts using both integer and fractional order Michaelis-Menten modelling. Secondly, the research investigates the variability of the dimensionless effectiveness factor with the Thiele modulus.

The study primarily focuses on mathematical modelling and theoretical analysis, with limited experimental validation. Future research should involve more extensive experimental verification to corroborate the findings. Additionally, the study assumes ideal conditions and uniform catalyst properties, which may not fully reflect real-world complexities. Incorporating factors such as mass transfer limitations, non-uniform catalyst structures and enzyme deactivation kinetics could enhance the model’s accuracy and broaden its applicability. Furthermore, extending the analysis to include multi-enzyme systems and complex reaction networks would provide a more comprehensive understanding of biocatalytic processes.

The validated bipartite polynomial approximation method presents a practical tool for optimizing enzyme reactor design and operation in industrial settings. By accurately predicting substrate concentration and effectiveness factor, this approach enables efficient utilization of immobilised enzymes within porous catalysts. Implementation of these findings can lead to enhanced process efficiency, reduced operating costs and improved product yields in various biocatalytic applications such as pharmaceuticals, food processing and biofuel production. Additionally, this research fosters innovation in enzyme immobilisation techniques, offering practical insights for engineers and researchers striving to develop sustainable and economically viable bioprocesses.

The advancement of enzyme immobilisation techniques holds promise for addressing societal challenges such as sustainable production, environmental protection and healthcare. By enabling more efficient biocatalytic processes, this research contributes to reducing industrial waste, minimizing energy consumption and enhancing access to pharmaceuticals and bio-based products. Moreover, the development of eco-friendly manufacturing practices through biocatalysis aligns with global efforts towards sustainability and mitigating climate change. The widespread adoption of these technologies can foster a more environmentally conscious society while stimulating economic growth and innovation in biotechnology and related industries.

This study offers a pioneering approximation method using the independent polynomials of the complete bipartite graph to investigate enzyme reaction kinetics. The comprehensive validation of this method through comparison with established solution techniques ensures its reliability and accuracy. The findings hold promise for advancing the field of biocatalysts and provide valuable insights for designing efficient enzyme reactors.

Analyzing and reducing entropy generation is useful for enhancing the thermodynamic performance of engineering systems. This study aims to explore how polymers and nanoparticles in the presence of Lorentz forces influence the fluid behavior and heat transfer characteristics to lessen energy loss and entropy generation.

The dispersion model is initially used to examine the behavior of polymer additives over a magnetized surface. The governing system of partial differential equations (PDEs) is subsequently reduced through the utilization of similarity transformation techniques. Entropy analysis is primarily performed through the implementation of numerical computations on a non-Newtonian polymeric FENE-P model.

The numerical simulations conducted in the presence of Lorentz forces provide significant insights into the consequences of adding polymers to the base fluid. The findings suggest that such an approach minimizes entropy in the flow region. Through the utilization of polymer-MHD (magnetohydrodynamic) interactions, it is feasible to reduce energy loss and improve the efficiency of the system.

This study’s primary motivation and novelty lie in examining the significance of polymer additives as agents that reduce entropy generation on a magnetic surface. The author looks at how nanofluids affect the development of entropy and the loss of irreversibility. To do this, the author uses the Lorentz force, the Soret effect and the Dufour effect to minimize entropy. The findings contribute to fluid mechanics and thermodynamics by providing valuable insights for engineering systems to increase energy efficiency and conserve resources.

This paper aims to propose a hybrid force/position controller based on the adaptive variable impedance.

First, the working space is divided into a force control subspace and a position subspace, the force control subspace adopts the position impedance control strategy. At the same time, the contact force model between the robot and the surface is analyzed in this space. Second, based on the traditional position impedance, the model reference adaptive control is introduced to provide an accurate reference position for the impedance controller. Then, the BP neural network is used to adjust the impedance parameters online.

The experimental results show that compared with the traditional PI control method, the proposed method has a higher flexibility, the dynamic response accommodation time is reduced by 7.688 s and the steady-state error is reduced by 30.531%. The overshoot of the contact force between the end of robot and the workpiece is reduced by 34.325% comparing with the fixed impedance control method.

The proposed control method compares with a hybrid force/position based on PI control method and a position fixed impedance control method by simulation and experiment.

The adaptive variable impedance control method improves accuracy of force tracking and solves the problem of the large surfaces with robot grinding often over-polished at the protrusion and under-polished at the concave.

This study investigates the coupling effects between temperature, permeability and stress fields during the development of geothermal reservoirs, comparing the impacts of inter-well pressure differentials, reservoir temperature and heat extraction fluid on geothermal extraction.

This study employs theoretical analysis and numerical simulation to explore the coupling mechanisms of temperature, permeability and stress fields in a geothermal reservoir using a thermal-hydrological-mechanical (THM) three-field coupling model.

The results reveal that the pressure differential between wells significantly impacts geothermal extraction capacity, with SC-CO2 achieving 1.83 times the capacity of water. Increasing the aperture of hydraulic and natural fractures effectively enhances geothermal production, with a notable enhancement for natural fractures.

The research provides a critical theoretical foundation for understanding THM coupling mechanisms in geothermal extraction, supporting the optimization of geothermal resource development and utilization.

The purpose of this paper is to study the two-dimensional micropolar fluid flow with conjugate heat transfer and mass transpiration. The considered nanofluid has graphene nanoparticles.

Governing nonlinear partial differential equations are converted to nonlinear ordinary differential equations by similarity transformation. Then, to analyze the flow, the authors derive the dual solutions to the flow problem. Biot number and radiation effect are included in the energy equation. The momentum equation was solved by using boundary conditions, and the temperature equation solved by using hypergeometric series solutions. Nusselt numbers and skin friction coefficients are calculated as functions of the Reynolds number. Further, the problem is governed by other parameters, namely, the magnetic parameter, radiation parameter, Prandtl number and mass transpiration. Graphene nanofluids have shown promising thermal conductivity enhancements due to the high thermal conductivity of graphene and have a wide range of applications affecting the thermal boundary layer and serve as coolants and thermal management systems in electronics or as heat transfer fluids in various industrial processes.

Results show that increasing the magnetic field decreases the momentum and increases thermal radiation. The heat source/sink parameter increases the thermal boundary layer. Increasing the volume fraction decreases the velocity profile and increases the temperature. Increasing the Eringen parameter increases the momentum of the fluid flow. Applications are found in the extrusion of polymer sheets, films and sheets, the manufacturing of plastic wires, the fabrication of fibers and the growth of crystals, among others. Heat sources/sinks are commonly used in electronic devices to transfer the heat generated by high-power semiconductor devices such as power transistors and optoelectronics such as lasers and light-emitting diodes to a fluid medium, thermal radiation on the fluid flow used in spectroscopy to study the properties of materials and also used in thermal imaging to capture and display the infrared radiation emitted by objects.

Micropolar fluid flow across stretching/shrinking surfaces is examined. Biot number and radiation effects are included in the energy equation. An increase in the volume fraction decreases the momentum boundary layer thickness. Nusselt numbers and skin friction coefficients are presented versus Reynolds numbers. A dual solution is obtained for a shrinking surface.

Traditional grey models are integer order whitening differential models; these models are relatively effective for the prediction of regular raw data, but the prediction error of irregular series or shock series is large, and the prediction effect is not ideal.

The new model realizes the dynamic expansion and optimization of the grey Bernoulli model. Meanwhile, it also enhances the variability and self-adaptability of the model structure. And nonlinear parameters are computed by the particle swarm optimization (PSO) algorithm.

Establishing a prediction model based on the raw data from the last six years, it is verified that the prediction performance of the new model is far superior to other mainstream grey prediction models, especially for irregular sequences and oscillating sequences. Ultimately, forecasting models are constructed to calculate various energy consumption aspects in Chongqing. The findings of this study offer a valuable reference for the government in shaping energy consumption policies and optimizing the energy structure.

It is imperative to recognize its inherent limitations. Firstly, the fractional differential order of the model is restricted to 0 < a < 2, encompassing only a three-parameter model. Future investigations could delve into the development of a multi-parameter model applicable when a = 2. Secondly, this paper exclusively focuses on the model itself, neglecting the consideration of raw data preprocessing, such as smoothing operators, buffer operators and background values. Incorporating these factors could significantly enhance the model’s effectiveness, particularly in the context of medium-term or long-term predictions.

This contribution plays a constructive role in expanding the model repertoire of the grey prediction model. The utilization of the developed model for predicting total energy consumption, coal consumption, natural gas consumption, oil consumption and other energy sources from 2021 to 2022 validates the efficacy and feasibility of the innovative model.

These findings, in turn, provide valuable guidance and decision-making support for both the Chinese Government and the Chongqing Government in optimizing energy structure and formulating effective energy policies.

This research holds significant importance in enriching the theoretical framework of the grey prediction model.

The highlights of the paper are as follows:

1. A novel grey Bernoulli prediction model is proposed to improve the model’s structure.

2. Fractional derivative, fractional accumulating generation operator and Bernoulli equation are added to the new model.

3. The proposed model can achieve full compatibility with the traditional mainstream grey prediction models.

4. Energy consumption in Chongqing verifies that the performance of the new model is much better than that of the traditional grey models.

5. The research provides a reference basis for the government to formulate energy consumption policies and optimize energy structure.

A novel grey Bernoulli prediction model is proposed to improve the model’s structure.

Fractional derivative, fractional accumulating generation operator and Bernoulli equation are added to the new model.

The proposed model can achieve full compatibility with the traditional mainstream grey prediction models.

Energy consumption in Chongqing verifies that the performance of the new model is much better than that of the traditional grey models.

The research provides a reference basis for the government to formulate energy consumption policies and optimize energy structure.

The influence of the temperature discrepancy parameter and higher order of the time-derivative is discussed. Classical coupled and generalized hygrothermoelasticity models are recovered by considering the various special cases and illustrated graphically.

The theory of integral transformations has been used to study a new hygrothermal model that includes higher-order time derivatives with three-phase-lags and memory-dependent derivatives (MDD). This model considers the microscopic structure’s influence on a non-simple hygrothermoelastic infinitely long cylinder. The generalized Fourier and Fick’s law was adopted to derive the linearly coupled partial differential equations with higher-order time-differential with the two-phase lag model, including memory-dependent derivatives for the hygrothermal field. The investigation of microstructural interactions and the subsequent hygrothermal change has been undertaken as a result of the delay time and relaxation time translations.

These two-phase-lag models are also practically applicable in modeling nanoscale heat and moisture transport problems applied to almost all important devices. This work will enable future investigators to gain insight into non-simple hygrothermoelasticity with different phase delays of higher order in detail.

To the best of my knowledge, and after completing an intensive search of the relevant literature, the author could not learn any published research that presents a general solution for a higher-order time-fractional three-phase-lag hygrothermoelastic infinite circular cylinder with memory memory-dependent derivative.

Understanding the mechanical and thermal behavior of materials is the goal of the branch of study known as fractional thermoelasticity, which blends fractional calculus with thermoelasticity. It accounts for the fact that heat transfer and deformation are non-local processes that depend on long-term memory. The sphere is free of external stresses and rotates around one of its radial axes at a constant rate. The coupled system equations are solved using the Laplace transform. The outcomes showed that the viscoelastic deformation and thermal stresses increased with the value of the fractional order coefficients.

The results obtained are considered good because they indicate that the approach or model under examination shows robust performance and produces accurate or reliable results that are consistent with the corresponding literature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This paper aims to examine the Dufour and Soret combined effects on the study of two-dimensional squeezed flow of copper water nanofluid between parallel plates along with applied (external) magnetic field. Impact of higher order chemical reaction is also considered.

The nonlinear partial differential equations (PDEs) are changed into system of ordinary differential equations (ODEs) by employing suitable similarity transformations. These transformed ODEs are then solved by means of a semianalytical method called differential transform method (DTM). Effects of several changing physical parameters on fluid flow, temperature and concentration have been deliberated through graphs.

It is observed that Dufour and Soret numbers are directly related to temperature profile and a reverse trend was observed in the concentration profile. Temperature enhancement is perceived for the enhanced Dufour number. Enhancement in Dufour number shows a direct association with Sh and Nu for all values of squeezing parameter.

The combined Dufour and Soret effects are used in separation of isotopes in mixture of gases, oil reservoirs and binary alloys solidification. The squeeze nanoliquid flow can be used in the field of composite material joining, rheological testing and welding engineering.

This study is mainly useful for geosciences and chemical engineering.

The uniqueness in this research is the study of the impact of cross diffusion on chemically reacting squeezed nanoliquid flow with the chemical reaction order more than one in the presence of applied magnetic force using a semianalytical procedure, named DTM.

In this study, we investigate the effects of an extended ternary hybrid Tiwari and Das nanofluid model on ethylene glycol flow, with a focus on heat transfer. Using the Cross non-Newtonian fluid model, we explore the heat transfer characteristics of this unique fluid in various applications such as pharmaceutical solvents, vaccine preservatives, and medical imaging techniques.

Our investigation reveals that the flow of this ternary hybrid nanofluid follows a laminar Cross model flow pattern, influenced by heat radiation and occurring around a stretched cylinder in a porous medium. We apply a non-similarity transformation to the nonlinear partial differential equations, converting them into non-dimensional PDEs. These equations are subsequently solved as ordinary differential equations (ODEs) using MATLAB’s bvp4c tools. In addition, the magnetic number in this study spans from 0 to 5, volume fraction of nanoparticles varies from 5% to 10%, and Prandtl number for EG as 204. This approach allows us to examine the impact of temperature on heat transfer and distribution within the fluid.

Graphical depictions illustrate the effects of parameters such as the Weissenberg number, porous parameter, Schmidt number, thermal conductivity parameter, Soret number, magnetic parameter, Eckert number, Lewis number, and Peclet number on velocity, temperature, concentration, and microorganism profiles. Our results highlight the significant influence of thermal radiation and ohmic heating on heat transmission, particularly in relation to magnetic and Darcy parameters. A higher Lewis number corresponds to faster heat diffusion compared to mass diffusion, while increases in the Soret number are associated with higher concentration profiles. Additionally, rapid temperature dissipation inhibits microbial development, reducing the microbial profile.

The numerical analysis of skin friction coefficients and Nusselt numbers in tabular form further validates our approach. Overall, our findings demonstrate the effectiveness of our numerical technique in providing a comprehensive understanding of flow and heat transfer processes in ternary hybrid nanofluids, offering valuable insights for various practical applications.