Set Of Equations Research Articles

Dynamics of N-elastically longitudinal coupled rotating pendulums with smooth and discontinuous nonlinearities: Generation of chaotic bursting with many orbits as a solution

The dynamics of a mechanical network, consisting of a certain number (N) of cells, in which each unit cell consists of the irrational coupling of a simple pendulum and inclined mass-spring oscillator, is investigated. The single cell was recently introduced by Ning Han and Qingjie Cao, and it has strong irrational nonlinearity, exhibiting smooth and discontinuous characteristics due to the geometry configuration. The obtained network has a large degree of freedom and consequently has richer dynamics as compared to a single one. The set of discrete equations of this network is found, while the first equation is driven by an external sinusoidal force; the variables here are the angles between the vertical direction and the directions of mass in each unit cell. By using the non-driven version of this set of equations, the equilibrium points are found as well as their stability using the Jacobian matrix. Then, by using the multiple time scale method, the solutions of the set of the network equations are found for weak amplitude oscillations of the system, while the large amplitude solutions are found numerically, showing the ability of the network to transform the shape of some solutions at the input to other forms as the signals evolve in the network. One notices the generation of simple chaotic signals as well as the train of impulse signals. What is important here is the generation of the chaotic bursting with many orbits by the system, which are the simultaneous oscillations around several fixed points. For the case of numerous cells, the system can be used as a waveguide; this is why the set of the equation of the network is reduced to the extended complex Ginzburg Landau (ECGL) equation, with complex nonlinear dispersion and nonlocality terms, using the Tanuiti's reduction perturbation methods. The dissipative dark compacton and peakon as solutions for the ECGL equation are then found. The input-output matching of the network is next investigated via the transfer function of the system near the equilibrium points. The inverse Fourier transform of the transfer function multiplied by the Fourier transform of the input sinusoidal force gives the solution of the network equation for low amplitude oscillations. Finally, the supratransmission condition is studied, leading to the fact that the input sinusoidal signal with frequency nearly inside the forbidden gap zone gradually transforms into the train of chaotic bursting.

Modeling microbial growth based on time-dependent kinetic mechanisms of digestion and passage in the ruminoreticulum

The original Cornell Net Carbohydrate and Protein System (CNCPS) model, which was developed in 1992 and revised in 2004, included a rumen submodel with a set of equations that were algebraically programmed in a spreadsheet to predict bacterial N flow to the caudal tract of cattle. In this study, we propose a modification to the original CNCPS rumen submodel based on mathematical concepts that simulate the flow paths of solids and liquid in the rumen, which are known to be essential factors affecting rumen fermentation and microbial growth. This modification allows the quantification of the impact of aging mechanisms on the availability of both soluble and insoluble substrates for microbial growth in the rumen. Additionally, a correction has been proposed on peptide uptake by rumen bacteria. Literature data were used to evaluate the model adequacy of CNCPS versions 1992 and 2004, along with the proposed model, to determine their ability to predict bacterial N flow. The proposed model more accurately predicted the variable of interest, but there was an overall underprediction bias. Despite the rightful critics of the ability of mechanistic models to predict microbial crude protein flow from the rumen, the 1992 and 2004 versions of the CNCPS and the proposed modifications implemented within the CNCPS framework led to predictions that agreed with observational data at a discordance rate of 0.05, tolerance probabilities <0.001 for versions 1992 and 2004, and a tolerance probability equal to 0.13 for the proposed model given a sample size n=39. Therefore, the proposed modifications might improve the ability of CNCPS-based models to predict the bacterial N flow from the rumen, thereby resulting in improved predictive performance compared to the original CNCPS model. Thus, it seems that models built on time-dependent kinetics of food particles and fluid are more fitting than steady-state ones to predict bacterial N flow to the caudal tract.

A bi-component model to assess the rheology of soft cellular aggregates probed using the micropipette aspiration technique

The micro-pipette aspiration technique is a classical experiment used to characterize the physical properties of inert fluids and biological soft materials such as cellular aggregates. The physical parameters of the fluid, as viscosity and interfacial tension, are obtained by studying how the fluid enters the pipette when the suction pressure is increased and how it relaxes when the suction pressure is put to zero. A mathematical model representative of the experiment is needed to extrapolate the physical parameters of the fluid-like matter; however, for biological materials as cells or cell aggregates these models are always based on strong starting hypotheses that impact the significance of the identified parameters. In this article, starting from the bi-constituent nature of the cell aggregate, we derive a general mathematical model based of a Cahn-Hilliard-Navier-Stokes set of equations. The model is applied to describe quantitatively the aspiration-retraction dynamics of a cell-aggregate into and out of a pipette. We demonstrate the predictive capability of the model and highlight the impact of the assumptions made on the identified parameters by studying two cases: one with a non-wetting condition between the cells and the wall of the pipette (classical assumption in the literature) and the second one, which is more realistic, with a partial wetting condition (contact angle θs = 150°). Furthermore, our results provide a purely physical explanation to the asymmetry between the aspiration and retraction responses which is alternative to the proposed hypothesis of an mechano-responsive alteration of the surface tension of the cell aggregate. Statement of SignificanceOur study introduces a general mathematical model, based on the Cahn-Hilliard-Navier-Stokes equations, tailored to model micro-pipette aspiration of cell aggregates. The model accounts for the multi-component structure of the cell aggregate and its intrinsic viscoelastic rheology. By challenging prevailing assumptions, particularly regarding perfect non-wetting conditions and the mechano-responsive alteration of cell surface tension, we demonstrate the reliability of the mathematical model and elucidate the mechanisms at play, offering a purely physical explanation for observed asymmetries between the aspiration and retraction stages of the experiment.

Flow and sensitivity analysis of MHD radiative hybrid nanofluid flow across a permeable wedge with slip condition: Response surface methodology

AbstractThis current paper has been written to investigate the effect of a hybrid Tiwari and Das nanofluid model along with optimizing the flow using sensitivity analysis. Heat transfer rates and skin friction assessments were examined with the supplementation of external effects like heat source, thermal radiation, buoyancy forces, velocity slip as well as magnetic effect with spherical‐shaped nanoparticles. The mathematical modeling was formulated using a system of nonlinear PDE. Similarity transformations were implemented in the governing equations for obtaining the final set of nondimensional equations which were then solved using bvp4c code in MATLAB. The results obtained were in close agreement with published results. Some of the significant results obtained included an increase in Nusselt number by 2.26% for an increase in velocity ratio parameter () from to 0.3 along with an increase in skin friction coefficient by 0.77% with magnetic () parameter to . The heat transfer profile was augmented by 0.79% with increasing radiation parameter to while an increase of 18.84% was observed for skin friction profile values recorded for the hybrid model as compared to the base fluid model. Nusselt number reduced by 0.51% for a transition to nanofluid model from regular fluid and by 1.76% for the hybrid model compared to the nanofluid model The unblocked face‐centered central composite design (CCD) was used for analyzing the sensitivity analysis of independent, continuous input parameters on the response parameters, skin friction coefficient, and Nusselt number. The high values for , and , for Nusselt number and skin friction coefficient values, respectively, validate the ANOVA results obtained using response surface methodology (RSM). Only shows positive sensitivity for both the Nusselt number and skin friction responses. The present hybrid nanofluid model finds extensive applications in bio‐toxicity studies, manufacturing of water heaters, and forging of hot exhaust valve heads.

An Integrated Artificial Intelligence Approach for Building Energy Demand Forecasting

Buildings are complex assets, characterized by environments and uses that change over time, variable occupancies, and long life cycles. They have high operational costs, mostly due to their energy requirements, and account for 30% to 40% of global greenhouse gas emissions. Consequently, substantial effort has been made to forecast their energy needs, with the scope of optimizing their economic and environmental impact. In this regard, the available literature focuses mainly on short-term modeling through the implementation of sets of physics-based equations (i.e., white-box), functional relationships between input and output variables (i.e., black-box), or a combination of both (i.e., grey-box). On the other hand, more research is required on long-term forecast models with the aim of reducing the energy needs. Within this context, this article presents an original automatic procedure for forecasting the energy needs of buildings in short- and long-term time horizons. This is accomplished by scaling an unknown facility from a similar facility that is already known and by executing a black-box approach based on machine learning algorithms. The proposed method is implemented in real case studies in Italy, predicting the energy needs (i.e., heating, cooling, and electricity) of Sant’Anna Hospital in Ferrara using the historical data of Ca’ Foncello Hospital in Treviso. The results show an adjusted coefficient of determination above 0.7 and an average error below 10% for all the energy vectors, demonstrating a feasible forecast performance with a low training set-to-test set ratio.

The renormalization group for large-scale structure: origin of galaxy stochasticity

The renormalization group equations for large-scale structure (RG-LSS) describe how the bias and stochastic (noise) parameters — both of matter and biased tracers such as galaxies — evolve as a function of the cutoff Λ of the effective field theory. In previous work, we derived the RG-LSS equations for the bias parameters using the Wilson-Polchinski framework. Here, we extend these results to include stochastic contributions, corresponding to terms in the effective action that are higher order in the current J. We derive the general local interaction terms that describe stochasticity at all orders in perturbations,and a closed set of nonlinear RG equations for their coefficients. These imply thata single nonlinear bias term generates all stochastic moments through RG evolution. Further, theevolution is controlled by a different, lower scale than the nonlinear scale. This has implications for the optimal choice of the renormalization scale when comparing the theory with data to obtain cosmological constraints.

Convective MHD flow of Casson fluid through porous medium using the Darcy‐Forchheimer model under the impact of prescribed heat sources

AbstractBlood and other non‐Newtonian fluids are most appropriately described by a rheological model known as a Casson fluid. The presence of yield stress in Casson fluids makes them particularly relevant in the fields of biomechanics and polymer industries. In this analysis, a numerical simulation of flow of a Casson nanofluid over a stretching surface was estimated. The flow of fluid was subjected to a magnetic field of varying strength. The governing partial differential equations for momentum and heat transmission are converted into a set of nonlinear ordinary differential equations with the aid of similarity transformations. After that, numerical techniques are used to solve the equations. Figures and tables are utilized to illustrate how non‐dimensional parameters affect energy, concentration, and velocity trends.

Formation of WNL Stars in the MW and LMC Based on the k − ω Model

We adopt a set of second-order differential equations (k − ω model) to handle core convective overshooting in massive stars, simulate the evolution of nitrogen sequence Wolf–Rayet (WNL) stars with different metallicities and initial masses, both rotating and nonrotating models, and compare the results with the classical overshooting model. The results indicate that, under the same initial conditions, the k − ω model generally produces larger convective cores and wider overshooting regions, thereby increasing the mass ranges and extending the lifetimes of WNL stars, as well as the likelihood of forming WNL stars. The masses and lifetimes of WNL stars both increase with higher metallicities and initial masses. Under higher-metallicity conditions, the two overshooting schemes significantly differ in their impacts on the lifetimes of WNL stars, but are insignificant in the mass ranges of the WNL stars. Rotation may drive the formation of WNL stars in low-mass, metal-poor counterparts, with this effect being more pronounced in the overshooting model. The surface nitrogen of metal-rich WNL stars formed during the main-sequence phase is likely primarily from the CN cycle, while it may come from both the CN and NO cycles for relatively metal-poor counterparts. Our model can effectively explain the distribution of WNL stars in the Milky Way, but appears to have inadequacies in explaining the WNL stars in the LMC.

Mass superinflation in the Reissner-Nordström black hole

Ever since Penrose and Simpson's prediction contradicted Novikov's idea that an infalling person would emerge into an asymptotically flat universe, there have been a continued interest in predicting the nature of singularity at the inner horizon of a Reissner-Nordström black hole. This prediction was first confirmed by Poisson and Israel using cross-stream of massless particles, leading to the phenomenon coined as mass inflation. On the other hand, Ori obtained a weaker singularity using a null shell of radiation. Thus it is important to capture the nature of singularity at the inner horizon. We therefore consider in this work a massive scalar field coupled to the Reissner-Nordström spacetime. The ensuing field equations lead to a coupled set of nonlinear dynamical equations. For definiteness, we analytically solve these equations employing the Adomian decomposition method. This facilitates in obtaining a closed form solution that exhibits an unbounded double-exponential growth in the mass function, giving rise to a novel phenomenon coined herein as mass superinflation. The scalar field is also found to undergo a very strong blueshift at the inner horizon.

Impact of atomizer design on slurry fuel atomization behavior

The paper presents experimental findings on the atomization characteristics of coal–water slurries with and without petrochemicals. The fuels were based on flame coal filter cake (slime), which is a typical coal processing waste, flame coal of different particle sizes, wood biomass (sawdust), used transformer oil, and water. The atomized flow characteristics—droplet size and velocity as well as jet angle—were found to depend on the atomizer dimensions and slurry rheology. The experimental data were used to calculate the slurry atomization efficiency factor. The findings were generalized to provide a mathematical description of how the slurry composition and atomizer geometry affect the slurry atomization behavior. Approximations were obtained for atomization characteristics that can be used to predict the jet angle as well as droplet radii and velocities. The developed mathematical tool can be employed to calculate the spraying characteristics when using devices like external-mix twin-fluid atomizers. The approach proposed for data generalization can be applied to adapt the set of approximation equations to other types of nozzles.

Multilayer deep-learning intelligent computing for the numerical analysis of unsteady heat and mass transfer in MHD Carreau Nanofluid model

The objective of this article is to investigate the mechanism of unsteady heat and mass transfer (HMT) in a magnetohydrodynamics Carreau nanofluid flow (MHD-CNFF). To accomplish this, we utilized a novel artificial intelligence-based (AI) method, deep learning predictive technique combined with Levenberg-Marquardt scheme (DLPT-LMS). The similarity transformation is applied to convert the partial differential equations (PDEs) into a set of non-linear ordinary differential equations (ODEs). We numerically solved these resulting ODEs using the Adam numerical approach in sophisticated “Mathematica” software. The influence of Brownian motion parameter, thermophoresis parameters, Prandtl number, Lewis number, and mass transfer parameters on the temperature, velocity, and concentration profile is analyzed using a synthetic dataset. This evaluation is conducted across different variation of MHD-CNFF by leveraging DLPT-LMS. The absolute error (AE) across various iteration of MHD-CNFF demonstrated a progressive drop, indicating the accuracy of DLPT-LMS. The outcomes of this investigation revealed that the AE ranged from 10-3 to 10-8 which shows a minimum error between actual and predicted value. Additionally, the MSE confirmed the accuracy of proposed DLPT-LMS in predicting the behavior of velocity, temperature, and concentration profiles, with lower values showing better predictions. The model accuracy was reinforced by performance results, including regression analysis and error histograms. Error values gradually decreased during training, validation, and testing phases, exhibiting a robust convergence and indicating the highly reliability of AI-based algorithm for investigating intricate fluid dynamics in MHD-CNFF systems.

Probabilistic analysis of tumor growth inhibition models to Support trial design

A large enough sample size of patients is required to statistically show that one treatment is better than another. However, too large a sample size is expensive and can also result in findings that are statistically significant, but not clinically relevant. How sample sizes should be chosen is a well-studied problem in classical statistics and analytical expressions can be derived from the appropriate test statistic. However, these expressions require information regarding the efficacy of the treatment, which may not be available, particularly for newly developed drugs. Tumor growth inhibition (TGI) models are frequently used to quantify the efficacy of newly developed anticancer drugs. In these models, the tumor growth dynamics are commonly described by a set of ordinary differential equations containing parameters that must be estimated using experimental data.One widely used endpoint in clinical trials is the proportion of patients in different response categories determined using the Response Evaluation Criteria In Solid Tumors (RECIST) framework. From the TGI model, we derive analytical expressions for the probability of patient response to combination therapy. The probabilistic expressions are used together with classical statistics to derive a parametric model for the sample size required to achieve a certain significance level and test power when comparing two treatments.Furthermore, the probabilistic expressions are used to generalize the Tumor Static Exposure concept to be more suitable for predicting clinical response. The derivatives of the probabilistic expressions are used to derive two additional expressions characterizing the exposure and its sensitivity. Finally, our results are illustrated using parameters obtained from calibrating the model to preclinical data.

Double-diffusive convection in flow of Carreau fluid with variable density inside converging and diverging channels of rectilinear walls

In this article, the effects of double-diffusive convection have been seen on the flow of Carreau fluid of variable density, maintained inside straight and converging (diverging) channels. Note that the channel walls exhibit variable porosity and undergo stretching or shrinking at non-uniform velocities. In the flow domain, we have taken into account the convective transport of both heat and species mass concentrations. The pseudo-similarity solutions of previous investigation for such type flows are not used here; however, a new set of similarity transformations has been formed, which reduced the governing system (PDEs) into an exact set of ordinary differential equations (ODEs), whereas the longstanding issues of semi-similarity solutions have been successfully resolved in this particular case and in general for all situations of Carreau fluid flow. In a nutshell, a unified mathematical approach has been devised in this paper. We strictly emphasized the simulation of flow of Carreau fluid and double diffusive convection in flow inside a channel with multiple dynamical behaviours and structures (both Jeffery Hammel and Poissuielle flow types) of walls. The boundary layer approximation and stream function assumptions have not been used in the study's execution. The shear thinning and shear thickening features of Carreau fluids with variable density cause them to exhibit lower axial velocity in the centre of a horizontal channel than Newtonian fluids due to variable wall configurations and complex flow conditions. By enhancing buoyancy driven convection, which promotes effective heat dispersion, increasing the solutal and thermal Grashof numbers (GrS=GrT>0) lowers the temperature field. It has been found that in channel with diverging walls (a1=2.0), increasing the modified Reynolds numbers (σ1,σ2>0) increases concentration profiles, i.e. ϕ(η), while decreasing them in channel with rectilinear walls (a1=0.0).

Physics‐Informed Neural Networks for the Augmented System of Shallow Water Equations With Topography

AbstractPhysics‐informed neural networks (PINNs) are gaining attention as an alternative approach to solve scientific problems governed by differential equations. This work aims at assessing the effectiveness of PINNs to solve a set of partial differential equations for which this method has never been considered, namely the augmented shallow water equations (SWEs) with topography. Differently from traditional SWEs, the bed elevation is considered as an additional conserved variable, and therefore one more equation expressing the fixed‐bed condition is included in the system. This approach allows the PINN model to leverage automatic differentiation to compute the bed slopes by learning the topographical information during training. PINNs are here tested for different one‐dimensional cases with non‐flat topography, and results are compared with analytical solutions. Though some limitations can be highlighted, PINNs show a good accuracy for the depth and velocity predictions even in the presence of non‐horizontal bottom. The solution of the augmented system of SWEs can therefore be regarded as a suitable alternative strategy to deal with flows over complex topography using PINNs, also in view of future extensions to realistic problems.

Stable coexistence in indefinitely large systems of competing species

The Lotka–Volterra system is a set of ordinary differential equations describing growth of interacting ecological species. One of the debated questions is understanding how the number of species in the system influences the stability of the model. Robert May studied how large systems may become unstable when species–species interactions do not vanish. This outcome has frequently been interpreted as a universal phenomenon and summarized as ‘large systems are unstable’. By exploring general interaction networks, we show that the competitive Lotka–Volterra system may maintain stability even for large networks, despite non-vanishing interaction strength. We establish sufficient conditions for stability as a threshold on the interspecific interaction strength, formulated in terms of the maximum and minimum degrees (or weights) rather than the network’s size. For values below this threshold, coexistence of all species is attained, regardless of the system size. Our finding generalizes May’s result by showing that it is the outlier nodes with large degree that cause instability rather than the large number of species in the system.

Analysis of heat and mass transfer rates in conducting Casson fluid flow over an expanding surface considering Ohmic heating and Darcy dissipation effects

In a recent scenario, the flow of Casson fluid via porous media has practical applications in various engineering processes, such, as the design of cooling systems for electronic devices, oil recovery in porous reservoirs, and polymer extrusion processes, etc. The proposed investigation illustrates the thermal and solutal transfer rates in a conducting Casson fluid flow over an expanding surface. However, the emphasis goes to the behavior of the Ohmic heating and Darcy dissipation when considering the transverse magnetic field and porous matrix. The governing flow phenomena with their dimensional form are altered into a non-dimensional set of equations with the help of suitable similarity rules. Further, an adequate numerical simulation is adopted to solve the transformed equations using the in-house bvp4c function in MATLAB. The physical parameters involved in the flow problem and their behavior on the governing flow phenomena are presented graphically and described briefly. Prior to this investigation, the conformity of the current numerical output obtained for the heat transfer rate was validated with the earlier work with a good correlation. Moreover, the major outcomes are; the non-Newtonian Casson parameter retards the axial and transverse velocity profile and the shear rate also decreases significantly. The Eckert number caused by the inclusion of the dissipative heat encourages the fluid temperature throughout.

Equation-Oriented Meanline Method for Axial Turbine Performance Prediction Under Choking Conditions

Abstract Meanline models play a crucial role in turbine design and system-level analyses, facilitating rapid evaluation of design concepts and prediction of off-design performance. Most of the existing meanline methods are inadequate in predicting turbine performance under choking conditions. These models either neglect the impact of losses on choking, or increases the computational complexity significantly. This limitation is addressed in this work, presenting a novel meanline model. The choking state at each cascade is determined by maximizing the mass flow rate, while taking into account the effect of losses. Leveraging the method of Lagrange multipliers, the optimization problems are transformed into a set of equations that seamlessly integrate with the rest of the meanline model. The resulting system of equations is then solved simultaneously using efficient root-finding algorithms, resulting in fast and reliable convergence. Validation against experimental data from three different turbines demonstrates the model's ability to accurately predict mass flow rate, torque, and exit flow angles across single- and multi-stage turbines, with errors typically within ±2.5 % and ±5.0 % for mass flow rate and torque respectively, and within ±5 degrees for flow angles. The proposed approach represents a significant advancement in meanline modeling, offering improved accuracy and computational efficiency.

Periodic and Axial Perturbations of Chaotic Solitons in the Realm of Complex Structured Quintic Swift-Hohenberg Equation

This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it.

Iterative Method of Determining Stress Intensity Coefficients Under Dynamic Loading of the Crack System

An elastic isotropic body in a state of plane deformation, which contains a system of randomly placed cracks under the action of a dynamic (harmonic) loading, is considered. The authors set the problem of determining the stress field around the cracks under the conditions of their wave interaction. The solution method is based on the introduction of displacements in the body in the form of a superposition of discontinuous solutions of the equations of motion constructed for each crack. With this in mind, the initial problem is reduced to a system of singular integro-differential equations with respect to unknown displacement jumps on the crack surfaces. To solve this system, a new iterative method, which involves solving a set of independent integro-differential equations that differ only in their right-hand parts at each iteration, is proposed. For the zero approximation, solutions that correspond to individual cracks under the action of dynamic loading are chosen. Such a new approach allows to avoid the difficulties associated with the need to solve systems of integro-differential equations of large dimensions that arise when traditional methods are used. Based on the results of the iterations, formulas for calculating the stress intensity coefficients for each crack were obtained. In the partial case of four cracks, a good agreement between the results obtained during the direct solution of the system of eight integro-differential equations by the mechanical quadrature method and the results obtained by the iterative method was established. In general, numerical examples demonstrate the convergence and stability of the proposed method in the case of systems with a fairly large number of densely located cracks. The influence of the interaction between cracks on the stress intensity factor (SIF) value under dynamic loading conditions was studied. An important and new result for fracture mechanics is the detection of the absolute maximum of the normal stresses at certain frequencies of the oscillating normal loading. The number of interacting cracks and the configuration of the crack system itself affect the values of the frequencies at which SIF reaches a maximum and the maximum values. These maximum values significantly (by several times) exceed the SIF values of single cracks under a similar loading. At the same time, under conditions of static or low-frequency loading, it is possible to reduce the SIF values compared to the SIF for individual cracks. When cracks are sheared, the values of the tangential stresses have a tendency to decrease with increasing frequency, and their values do not significantly differ from the values of the tangential stress for an individual crack.

Thermal analysis of hybrid nano-fluids: Modeling and non-similar solutions

The thermal analysis of hybrid nano-fluids is a significant research area with diverse applications in industries such as paint, electronics, and mechanical engineering. Existing literature provides limited solutions to the governing equations for the flow of these fluids. Modeling and deriving non-similar solutions for these equations pose interesting and challenging mathematical problems. This study focuses on investigating heat transfer in the flow of two types of nano-fluids, specifically Al2O3/H2O micropolar nano-fluid and Al2O3 + Ag/H2O hybrid nano-fluid, near an isothermal sphere. Conservation laws are employed to formulate the mathematical problem, and by normalizing the variables, the governing equations are converted into a set of dimensionless partial differential equations. Non-similar solutions are then obtained using numerical methods. A comparative analysis is carried out to assess the influence of various parameters on different profiles and engineering quantities for both types of nano-fluids. Both linear and rotational velocities fall down near the surface of sphere with rising microstructure in hybrid nanofluid. The micro-rotation parameter rises the temperature profile while reduces the Nusselt number of both traditional Al2O3/water based nanofluid as well as hybrid nanofluid.