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Maxwell fluid flow over a curved surface with the impacts of nonlinear convection and radiation, temperature‐dependent properties, and magnetic field are investigated. The governing equations of the physical system are solved using wavelet based physics informed neural network, a machine learning technique. This is an unsupervised method, and the solutions have been obtained without knowing the numerical solution to the problem. Given the nonlinearity of the coupled equations, the methodology used is flexible to implement, and the activation function used improves the accuracy of the solution. We approximate the unknown functions using different neural network models and determine the solution by training the network. The special case of the obtained results is examined with the available results in the literature for validation of the proposed methodology. It is observed that the proposed approach gives reliable results for the analyzed problem of study. Further, an analysis of the influence of flow parameters (deborah number, variable thermal conductivity and viscosity parameter, velocity slip parameter, temperature ratio parameter, suction parameter, and convection parameters) on temperature and fluid flow velocity is carried out. It is observed that as the flow parameter Deborah number, velocity slip parameter, and viscosity parameter increase, there is a decline in velocity and an enhancement in temperature. This study of fluid flow over a curved surface has applications in the polymer industry, which plays an important role in the manufacturing of contact lenses. [ABSTRACT FROM AUTHOR]

This study investigates the dynamics of heat and mass transfer, focusing on the impact of nonlinear convection and thermophoresis in the flow of viscous nanofluid towards a shrinking sheet embedded with a porous matrix. To facilitate understanding of the flow behavior, the study employs similarity variables to simplify the resulting nonlinear partial differential equations. Numerical analysis utilizing the collocation method with Legendre polynomials as basis functions reveals distinct distributions of velocity, temperature, and concentration. The results demonstrate that the dimensionless suction coefficient amplifies nanoparticle volume fraction and momentum along the stretching surface. In addition, an increase in internal frictional force leads to heightened injection of heat energy, accelerates the temperature field, and impedes nanoparticle diffusion due to heat absorption. Elevation of convection terms augments momentum, reduces energy dispersion, and enhances nanoparticle diffusion. Moreover, an increase in the dimensionless Eckert number (Ec) correlates with intensified energy and diminished diffusion concentration of the nanoparticle volume fraction [ABSTRACT FROM AUTHOR]

Present analysis deals with the impact of nonlinear convection on MHD non-Newtonian fluid flow considering exponentially stretched sheet through porous media. Additionally, the effects of radiation, Joule heating and energy dissipation have been taken into account. The considered flow situation of nonlinear PDEs is transmuted to nonlinear ODEs using similarity analysis. The numerical computations are derived using a fourth-order Runge–Kutta scheme with a shooting technique. Meaningful observations are discussed for velocity and energy profiles. Further, the impact of pertinent parameters considering pseudoplastic fluids (n = 0. 6) and dilatant fluids (n = 1. 3) is briefed through graphs. It is noted that the escalation in the values of the power-law index declines the velocity and energy profile. Higher values of the nonlinear parameter decline the energy profile. [ABSTRACT FROM AUTHOR]

We present three results on stability of rolls in Boussinesq convection in a plane horizontal layer with rigid boundaries that is rotating about an inclined axis with the angular velocity $ \mbox{\boldmath${\varOmega}$}=(\varOmega _1,\varOmega _2,\varOmega _3) $ Ω = (Ω 1 , Ω 2 , Ω 3). (i) We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary $ \mbox{\boldmath${\varOmega}$} $ Ω , and by the reduced problem the set of equations for the angular velocity $ (\varOmega _1,0,\varOmega _3) $ (Ω 1 , 0 , Ω 3). Here x,y are horizontal Cartesian coordinates in the layer and z is the vertical one. We prove that a y-independent solution to one of the two problems is also a solution to the second one. (ii) We calculate the critical Rayleigh number for the monotonic onset of convection. The instability mode in the form of rolls (a flow independent of a horizontal direction) is assumed. Let β be the angle between the horizontal projection of $ \mbox{\boldmath${\varOmega}$} $ Ω and the rolls axes. We show that $ \beta =0 $ β = 0 for the least stable mode. When the axis of rotation is horizontal, this is proven analytically, and for $ \varOmega _3\ne 0 $ Ω 3 ≠ 0 , the result is obtained numerically. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of $ \varOmega _1 $ Ω 1 and $ \varOmega _2 $ Ω 2 and the emerging flow are rolls with axis aligned with the horizontal component of the rotation vector. (iii) We study the behaviour of convective flows by integrating numerically the three-dimensional equations of convection for $ \mbox{\boldmath${\varOmega}$}=(0,\varOmega _2,\varOmega _3) $ Ω = (0 , Ω 2 , Ω 3) and a range of the Rayleigh numbers, other parameters of the problem being fixed. We assume square horizontal periodicity cells, whose sides are equal to the period of the most unstable mode. The computations indicate that, in general, in the nonlinear regime convective rolls become more stable as $ \varOmega _2 $ Ω 2 increases. Namely, on increasing $ \varOmega _2 $ Ω 2 , the interval of the Rayleigh numbers for which convective rolls are stable increases. [ABSTRACT FROM AUTHOR]

This article investigates the nonlinear convection transport of heat transfer in a nanoparticle laden two-layer flow in a vertical channel under the additional effects of magneto hydrodynamic (MHD) and radiation. Water and compressed air are chosen as the fluids in two-layer flow. This type of flow is commonly seen in the transportation of oil and gas in pipelines, therefore, correct understanding of flow dynamics including convective transport phenomenon, layers interaction, and graphical interpretation of flow variables is important for better designing of equipments. Titanium dioxide (Ti O 2 ) nanoparticles are added into the air to magnify the transport mechanism. The mathematical model incorporating buoyancy, nonlinear convection, MHD, viscous dissipation, and radiation effects is presented in the form of nonlinear Ordinary Differential Equation (ODE's). Then utilizing homotopy analysis method, these interconnected differential equations are resolved. A visual illustration shows how different physical factors affect fluid rate and temperature. Tabulated values of Nusselt number at both walls are presented to discuss the numerical data of convection and found that it decreases toward the left boundary but rises toward the right wall by increasing Eckert number and Prandtl number. The results of this two-layer flow will help in understanding this complex flow which has practical applications in various engineering processes and further enhance the scientific knowledge related to multilayer problems. [ABSTRACT FROM AUTHOR]

Entropy generation analysis combined with hybrid nanofluid flow principles contribute to the development of more efficient drug delivery systems for cancer treatment. By using nanofluids, it is possible to improve the transport and targeted release of therapeutic agents to specific sites in the body, allowing for better control and efficiency in treating cancerous cells. Keeping these important applications in view, in the current analysis, the production of irreversibility and stagnant point hybrid nanofluid flow has been considered on a Riga plate. The impacts of nonlinear convection and solar radiations have also been used in this study. Glycol (C3H8O2) is taken as base fluid, while nanoparticles of copper (Cu) and aluminum oxide (Al2O3) have been mixed in it to obtain a hybrid nanofluid. The leading equations for the study have converted to dimensionless form by employing a set of suitable variables and then have been solved by using an artificial neural network (ANN). In order to evaluate the effectiveness of the least mean square neural network algorithm (LMS-NNA), statistical neural network techniques are employed, encompassing error analysis and curve-fitting graphs. It has been revealed in this work that an upsurge in EMHD Riga plate factor and the Grashof number escalates the velocity distribution for both nanoparticles as well as hybrid nanoparticles and is opposed by augmentation in width factor for electrode/magnet. The increase in the nanoparticle volume fraction from 0.01 to 0.05 escalates the heat transfer rate up to 7.6% in the case of nanofluid with Cu-nanoparticles while this increase is 9.3% using hybrid nanofluid Cu + Al2O3. These results show that HNF are more efficient in improving the HT rate. [ABSTRACT FROM AUTHOR]

The amalgamation of motile microbes in nanofluid (NF) is important in upsurging the thermal conductivity of various systems, including micro-fluid devices, chip-shaped micro-devices, and enzyme biosensors. The current scrutiny focuses on the bioconvective flow of magneto-Williamson NFs containing motile microbes through a horizontal circular cylinder placed in a porous medium with nonlinear mixed convection and thermal radiation, heat sink/source, variable fluid properties, activation energy with chemical and microbial reactions, and Brownian motion for both nanoparticles and microbes. The flow analysis has also been considered subject to velocity slips, suction/injection, and heat convective and zero mass flux constraints at the boundary. The governing equations have been converted to a non-dimensional form using similarity variables, and the overlapping grid-based spectral collocation technique has been executed to procure solutions numerically. The graphical interpretation of various pertinent variables in the flow profiles and physical quantities of engineering attentiveness is provided and discussed. The results reveal that NF flow is accelerated by nonlinear thermal convection, velocity slip, magnetic fields, and variable viscosity parameters but decelerated by the Williamson fluid and suction parameters. The inclusion of nonlinear thermal radiation and variable thermal conductivity helps to enhance the fluid temperature and heat transfer rate. The concentration of both nanoparticles and motile microbes is promoted by the incorporation of activation energy in the flow system. The contribution of microbial Brownian motion along with microbial reactions on flow quantities justifies the importance of these features in the dynamics of motile microbes. [ABSTRACT FROM AUTHOR]

This study investigates the dynamics of heat and mass transfer, focusing on the impact of nonlinear convection and thermophoresis in the flow of viscous nanofluid towards a shrinking sheet embedded with a porous matrix. To facilitate understanding of the flow behavior, the study employs similarity variables to simplify the resulting nonlinear partial differential equations. Numerical analysis utilizing the collocation method with Legendre polynomials as basis functions reveals distinct distributions of velocity, temperature, and concentration. The results demonstrate that the dimensionless suction coefficient amplifies nanoparticle volume fraction and momentum along the stretching surface. In addition, an increase in internal frictional force leads to heightened injection of heat energy, accelerates the temperature field, and impedes nanoparticle diffusion due to heat absorption. Elevation of convection terms augments momentum, reduces energy dispersion, and enhances nanoparticle diffusion. Moreover, an increase in the dimensionless Eckert number (Ec) correlates with intensified energy and diminished diffusion concentration of the nanoparticle volume fraction.

In this work, the flow of ternary hybrid nanomaterials filling porous spaces is investigated in the presence of magnetohydrodynamics (MHD) and nonlinear convection. Three dissimilar nanoparticles copper (Cu), alumina (Al2O3), and magnesium oxide (MgO) are employed. Various features including Ohmic heating, heat generation, and variable thermal conductivity are studied, along with nonlinear radiation and entropy analysis. Solutions are developed using the ND Solve (shooting) method in Mathematica software. Analysis of parameters of interest is conducted, with conclusions highlighting key outcomes. Results show an increase in the thermal field for Eckert number and radiation, while a decrease in liquid flow is observed with increasing the magnetic parameter. The heat transport rate exhibits an opposite trend between heat generation and radiation. Temperature ratio parameter and Prandtl number show opposite trends in the thermal field, while the temperature gradient decreases against variable thermal conductivity. The drag force follows a similar trend against nonlinear convective and porosity variables. The entropy rate increases with the Brinkman number, and both porosity and radiation have an increasing impact on entropy rate. An increase in liquid flow is noted with the nonlinear convection variable. A comparative study of heat transport rate through the Prandtl number shows good agreement.

The motivation for this comparative examination is to verify and account for the appropriate source of heat energy needed in production industries, and to expunge the contradictions between Fourier's law and the theory of relativity. In heat management such as in engineering systems, the required heat for effective melting, production rate, etc. dictates the appropriate heat generation/injection module to be deployed. However, the material medium of heat transmission can be a hindering factor for a successful convection process, to this, an investigation of the instantaneous heat propagation paradox is required. An assumption of induced flow due to stretching characteristics is made, and an appropriate similarity transformation is deployed for the governing systems. By means of the Galerkin-weighted residual method, numerical solutions to the system of equations are approximated and validated. In a limited case, the solution compares favourably with existing literature, while the novel comparative investigation predicts the dominance of the Temperature-Dependent Heat Source (TDHS) capable of weakening the fluid cohesive force and upsurging the energy rate in short time in contrast to the Space-Dependent Heat Source (SDHS) module. The material relaxation phenomenon indicated that more time will be needed for prosperous/equal heat energy transfer. Further heat transfer enhancement is called for higher discharge of radiation and boosting of thermal conductivity, while TDHS module is strongly recommended in highly required heat systems. [ABSTRACT FROM AUTHOR]

AbstractDue to practical applications, we are motivated in the present study to analyze the effects of nonlinear natural convection in a nanofluid flow past a vertical plate in the presence of a first-order chemical reaction and thermophoresis (the normal flux of the nanoparticles is zero at the boundary). The two-phase model for the nanofluid is also considered. The governing partial differential equations (PDEs) are transformed to ordinary differential equations (ODEs) using similarity transformations. Numerical solutions for the transformed equations are attained by using a Runge-Kutta method with a shooting technique. The effects of the relevant physical parameters on the velocity, the temperature, and the nanoparticle volume fraction are analyzed. Also, the consequences of the parameters on the flow, heat, and mass transport phenomena are observed and analyzed explicitly. The results of this study are interesting and motivating for further investigations on the problem for different situations and with different geometries. [ABSTRACT FROM AUTHOR]

The investigation of the physical processes determining the melting of the lithospheric rocks is of crucial importance for understanding the volcanic dynamics and its related consequences. Rock melting begins when a sufficiently high temperature is experienced by the rock solidus. The heat transfer from the asthenosphere to the lithosphere can be assumed as the main mechanism accountable for the partial melting of rocks, and initiating magma generation. The heat transfer to the lithosphere is considered to be governed mainly by the convective motion inside the asthenosphere. In order to mathematically describe this process, a generalization of a nonlinear convective 1D model, possibly representing a useful though simplified model for the birth of a volcano, and already analyzed from an analytical viewpoint (Godano et al., 2022), is investigated; here, we solve numerically some physically meaningful initial and boundary value problems, and discuss the results.

The amalgamation of motile microbes in nanofluid (NF) is important in upsurging the thermal conductivity of various systems, including micro-fluid devices, chip-shaped micro-devices, and enzyme biosensors. The current scrutiny focuses on the bioconvective flow of magneto-Williamson NFs containing motile microbes through a horizontal circular cylinder placed in a porous medium with nonlinear mixed convection and thermal radiation, heat sink/source, variable fluid properties, activation energy with chemical and microbial reactions, and Brownian motion for both nanoparticles and microbes. The flow analysis has also been considered subject to velocity slips, suction/injection, and heat convective and zero mass flux constraints at the boundary. The governing equations have been converted to a non-dimensional form using similarity variables, and the overlapping grid-based spectral collocation technique has been executed to procure solutions numerically. The graphical interpretation of various pertinent variables in the flow profiles and physical quantities of engineering attentiveness is provided and discussed. The results reveal that NF flow is accelerated by nonlinear thermal convection, velocity slip, magnetic fields, and variable viscosity parameters but decelerated by the Williamson fluid and suction parameters. The inclusion of nonlinear thermal radiation and variable thermal conductivity helps to enhance the fluid temperature and heat transfer rate. The concentration of both nanoparticles and motile microbes is promoted by the incorporation of activation energy in the flow system. The contribution of microbial Brownian motion along with microbial reactions on flow quantities justifies the importance of these features in the dynamics of motile microbes.

The objective of this study examines the Maxwell hybrid nanofluid flow model over an inclined, stretched cylinder, incorporating magnetic, thermal stratification, nonlinear convection, heat source/sink, viscous and Ohmic dissipation effects using a modified Fourier heat flux model. Flow analysis is conducted for inclined stretching and shrinking cylinders, with a velocity slip condition on the cylinder’s surface. Thermal stratification is considered when a higher temperature is assumed across the cylinder’s surface than the surrounding fluid. A Maxwell hybrid nanofluid is created by dispersing Cu and Al2O3 nanoparticles in H2O. The mathematical formulation of a nonlinear PDE’s are transformed into dimensionless ODEs, solved numerically using MATLAB’s bvp4c function. The results of temperature and velocity profiles are graphically discussed. The results show that the impacts of the relevant parameters are statistically significant, showing their essential influence on the flow model’s heat transfer rate. Magnetic and Maxwell parameters show lower velocity profiles. The thermal relaxation and heat generation/absorption parameters enhance the thermal profile. Additionally, the stretching cylinder exhibits 13% and 21% greater heat transfer rates than the shrinking cylinder across various values of the nonlinear thermal Grashof number and thermal stratification parameter. These findings have applications in heat transfer, enhanced oil recovery, advanced cooling systems, renewable energy, and biomedical engineering. Validation and comparison with previous findings ensure the research’s validity and correctness, showing notable agreement. [ABSTRACT FROM AUTHOR]

In this work, the flow of ternary hybrid nanomaterials filling porous spaces is investigated in the presence of magnetohydrodynamics (MHD) and nonlinear convection. Three dissimilar nanoparticles copper (C u) , alumina (Al 2 O 3) , and magnesium oxide (M g O) are employed. Various features including Ohmic heating, heat generation, and variable thermal conductivity are studied, along with nonlinear radiation and entropy analysis. Solutions are developed using the ND Solve (shooting) method in Mathematica software. Analysis of parameters of interest is conducted, with conclusions highlighting key outcomes. Results show an increase in the thermal field for Eckert number and radiation, while a decrease in liquid flow is observed with increasing the magnetic parameter. The heat transport rate exhibits an opposite trend between heat generation and radiation. Temperature ratio parameter and Prandtl number show opposite trends in the thermal field, while the temperature gradient decreases against variable thermal conductivity. The drag force follows a similar trend against nonlinear convective and porosity variables. The entropy rate increases with the Brinkman number, and both porosity and radiation have an increasing impact on entropy rate. An increase in liquid flow is noted with the nonlinear convection variable. A comparative study of heat transport rate through the Prandtl number shows good agreement. [ABSTRACT FROM AUTHOR]

This study aims to investigate heat transfer and entropy production in the non-similar flow of an incompressible fluid, with nonlinear convection and viscous dissipation. To obtain precise solutions, the Sparrow-Quack-Boerner local non-similarity method is implemented. The non-similarity arises from the nonlinear convection term present in the momentum equation. The non-similarity arises from the nonlinear convection term present in the momentum equation. Consequently, the inclusion of non-similarity terms into the energy equation is achieved through the interconnection of momentum and energy equations. Entropy generation is explored by employing the second law of thermodynamics. Numerical results from several truncation levels are presented in tabular form, and equations for both first and second-level truncations are generated. The one- and two-equation models are solved by implementing the Generalized Differential Quadrature Method (GDQM). Comparison with a second level of truncation reveals significantly greater inaccuracies in numerical results obtained from the first level. This discrepancy arose due to the omission of non-similar terms in the primary governing equations at the first truncation stage. The validity and accuracy of the derived numerical solutions are further demonstrated by the application of the GDQM and the midpoint method with Richardson extrapolation. Additionally, a plot of the numerical data produced from the second level of truncations is presented together with a discussion of the various physical factors. Increasing the mixed convection parameter accelerates fluid velocity, also when the viscous dissipation parameter increases, temperature and the Bejan number increase. The Prandtl number and the thickness of the thermal boundary layer are observed to be inversely related. Moreover, the quantity of entropy generated at the stretching surface and inside the boundary layer rises in proportion to the increases in the Prandtl and Eckert numbers. [ABSTRACT FROM AUTHOR]