Fundamental Differential Equations Research Articles

Exploring the Bio-Mechanisms of Rod-Shaped Bacteria Moving on Rigid Substrate Coated with Non-Newtonian Williamson Slime

Bacteria are single-celled organisms that come in various shapes and sizes, often classified based on characteristics like shape, DNA, and movement mechanisms. Among them, gliding bacteria have a unique way of moving: they slide along surfaces without external structures like flagella or cilia. Instead, they create wave-like motions along their surface to glide across slimy areas. How fast can bacteria move, and does their speed change on different surfaces? Do they glide faster on smooth surfaces or slow down on rough or sticky ones? And when they move through thicker, more complex fluids, how do they adapt, and does this impact their speed? Researchers have investigated these questions by studying how surface properties and fluid dynamics affect bacterial movement. This study also aims to analyze the locomotion of gliding bacteria over a layer of Williamson fluid with a rigid substrate using the dynamics of an undulating sheet. The methodology involves converting the fundamental partial differential equations into a fourth-order nonlinear ordinary differential equation through the Stokes flow approximation and solving it by the regular perturbation technique. The impacts of different physical parameters on gliding speed, flow rate, energy loss, and slime velocity are visually illustrated and discussed. The results show that the glider's speed decreases with a lower viscosity ratio, while higher Weissenberg numbers and occlusion ratios improve the glider's motility. Additionally, the flow rate of the Williamson fluid decreases as the viscosity increases, and power loss experienced by the glider rises with an increase in viscosity. These findings highlight how fluid properties influence bacterial movement and energy efficiency, providing insights for developing microfluidic devices and studying microbial motility.

Free vibration and buckling behavior of porous orthotropic doubly-curved shallow shells subjected to non-uniform edge compression using higher-order shear deformation theory

Due to the continuous enhancement of manufacturing technology for porous materials, it is expected to be increasingly used in aerospace, aviation, and other engineering applications, where the necessity for lightweight structures is paramount. Also, the non-uniform edge loads caused by service conditions, complex adjacent structures, and external loads lead to localized stress concentrations within the shells. Localized stress concentrations result in a loss of load-carrying capacity in specific regions, causing step-by-step failure of the shell. This study investigates the buckling and free vibration behavior of porous orthotropic doubly-curved shallow shells with four different porosity distribution patterns subjected to non-uniform edge compressive loads. The equations of motion of doubly-curved shallow shells are established by using higher-order shear deformation theory and Hamilton’s principle. Then, the pre-buckling in-plane stress distributions of doubly-curved shallow shells are determined and appended to the equations of motion. These fundamental partial differential equations are solved by Galerkin’s method, and the critical buckling load and natural frequency formulations are achieved. By comparing with the results in published literature, the feasibility and accuracy of present formulations are validated. Finally, systematic parametric studies are carried out to discuss the effects of different non-uniform edge compression patterns, porosity distribution patterns, porosity coefficients, aspect ratios, arc length-to-thickness ratios, radius-to-arc length ratios, orthotropy ratios, and shell types on the buckling and free vibration responses of porous orthotropic doubly-curved shallow shells. Parametric studies indicate that the reduction or increment in critical buckling loads (CBLs) and fundamental natural frequencies (FNFs) of spherical shells (SSs) are more sensitive than those of hyperbolic paraboloidal shells (HPSs). The CBLs and FNFs of SSs are larger than those of HPSs. The maximum and minimum CBLs are achieved for the triangular and uniform edge compression, respectively. The porosity effect is independent of edge compression pattern types.

A Qualitative Analysis of Leukemia Fractional Order SICW Model

Using a series of fundamental differential equations, including the Caputo derivative, which makes it easier to specify the initial conditions of the differential equations, we present a fractional order concept of leukemia in this study. The universality, positivity, and boundedness of solutions are first established. The local stability properties of the equilibrium are studied using the fractional Routh-Hurwitz stability criteria. The differential equation system has been solved using unconventional finite difference techniques. The Leukemia Fractional Order SICW model introduces several innovative elements compared to traditional epidemiological and disease models. This stands out due to its integration of fractional-order differential equations, inclusion of leukemic cells and immune cells compartments, simulation of treatment strategies, consideration of waning immunity, and its application to leukemia-specific scenarios. These elements collectively make it a valuable tool for studying leukemia dynamics, exploring treatment options, and improving our understanding of how the immune system interacts with cancer cells in leukemia patients. Numerical simulations of the model are shown at the conclusion to interpret our theoretical outcomes in support of various fractional orders of derivative options. From there, we can observe how the evolution of the system components is impacted by the fractional derivative .

Heat and Mass Transfer within Unsteady Nanofluid Movement in the Presence of Sustained Solar Radiation

The unsteady movement of nanofluid on porous inclined media is essential for absorbing and transferring heat from solar radiation. From renewable energy sources, solar is limitless, sustainable and universally accessible without creating conflict. In this study, heat and mass transfer have been explored by unsteadily moving nanofluid with the occurrence of Sun rays and viscous dissipation. Tiwari-Das and Darcy-Forchheimer models are encompassed with convective heat transfer and mass suction/injection. Then, the non-linear higher-order set of ordinary differential equations was obtained from fundamental non-linear partial differential equations by using similarity transformation. Both semi-analytical and numerical strategies have been adopted. Comparisons with published articles have detected and observed similar outcomes. Accordingly, thermal Grashof number elevates nanofluid motion while postponing drag force creation. Permeability and Darcy’s number have publicized a contradictory trend in the nanofluid’s movement and temperature. Nanofluid’s temperature expands by incident solar radiation and Eckert number but not by absorption. There is less heat transfer rate by convective than conductive through magnifying magnetic field and nanoparticles’ concentration. Nanofluid constructed by Cu–H2O produces more drag force and less heat transfer rate than that of Cu–C3H8O2. Heat transfer from solar energy is applicable for cooking, heating water and producing electricity.

On the vibration responses of orthotropic laminated cylindrical panels with non-uniform porosity distributions using higher-order shear deformation theory

The present study investigates free vibration analysis of cylindrical panels made of porous orthotropic layers. Porous material properties are modeled using uniform and two non-uniform porosity distribution patterns through the thickness. The theoretical formulations are expressed based on the higher-order shear deformation theory. The equations of motion of laminated cylindrical panels are established by performing Hamilton’s principle. These fundamental partial differential equations are solved analytically for the laminated cylindrical panels, which are simply supported on all edges. Galerkin’s solution procedure is applied to obtain the natural frequency equation. The natural frequencies are compared to results in the literature to verify the obtained frequency equation. Afterward, novel parametric studies are developed to discuss the influence of porosity parameters, porosity distribution patterns, geometrical parameters, orthotropy, and lamination sequences on the free vibration response of porous laminated cylindrical panels.

A new methodology in evaluating nonlinear electrohydrodynamic azimuthal stability between two dusty viscous fluids

The objective of the current paper is to scrutinize the azimuthal stability of a circular interface between two viscous fluids across porous medium in the presence of fine dust. Dusty fluid models are employed to comprehend the dynamics of aerosols in the atmosphere, a crucial aspect in the investigation of air quality, climate change, and the dissemination of pollutants. Furthermore, the comprehension and management of azimuthal nonlinear instabilities are essential in diverse engineering applications, where turbulence is a prominent factor. This includes aerospace engineering, where it is necessary for developing more efficient aircraft, as well as fluid dynamics, where it is critical for optimizing industrial processes. The system is subjected to the influence of a uniform electric field (EF) that is directed in the azimuthal direction. Electrospray trustees utilized in spaceship propulsion systems make use of the principles of Electrohydrodynamics (EHD). These representatives employ the process of ionization on a liquid propellant and utilize EFs to accelerate the resulting ions, so producing thrust. Assuming a simplified situation, the prototype is assumed to have a two-dimensional structure. Both fluids are presumed as rotating. An examination of rotating fluids is crucial for comprehending the dynamics of the atmosphere and oceans. Obtaining this knowledge is essential for accurately predicting weather, simulating climate patterns, and investigating oceanic currents. The mathematical approach is simplified by considering the viscous potential theory (VPT). The nonlinear technique relies on the use of linear fundamental partial differential equations (PDEs) and the application of the appropriate nonlinear boundary conditions (BCs). This method produces a nonlinear PDE that controls the displacement of the interface. By disregarding the nonlinear elements, a dispersion equation that is linear in nature is obtained. The stability criteria are graphed using several dimensionless physical quantities. The non-perturbative analysis (NPA) achieves nonlinear stability. As well-known, the primary objective of NPA is to convert a linear ordinary differential equation (ODE) into a linear equation. The NPA distinguishes itself from conventional perturbation methods by its ability to accurately analyze the dynamics of highly nonlinear oscillators. This novel methodology is developed utilizing the He's frequency formula (HFF) for the purpose of designing the displacement of the interface. The computational computations demonstrated that the NPA is efficacious, encouraging, and powerful. The linear and nonlinear techniques demonstrate that the effects of various dimensionless physical parameters remain constant. It is found that the Weber number We and Capillary number Cabecome zero as crucial consequences. The Ohnesorge number, Oh which represents the ratio of viscosity and the mass concentration in dusty particles, has a destabilizing effect.

Analysis of irreversibility on nanofluid flow through porous medium considering shape, dispersion and melting effects

This study is conducted to explore the impact of thermal dispersion and melting on the natural convective nanofluid (nickel zinc ferrite + SAE 20W-40 motor oil) flow over a stretching sheet in a Darcy porous medium. An irreversibility analysis has also been performed using the second law of thermodynamics. The fundamental nonlinear partial differential equations (PDEs) are converted into a system of dimensionless nonlinear ordinary differential equations (ODEs) through Lie group similarity transformations and then solved by employing the spectral local linearization method implemented in MATLAB software. It is noteworthy to highlight that the validation results indicate a strong concurrence with existing findings in previously published literature. After simulation, it is specified that the thermal dispersion disrupts the entropy (by 63.7%) and enhances the surface friction. Likewise, the heat transfer rate improves by 1.5% with the melting effect and 58.7% with the Darcy number. The results indicate that the brick-shaped particles exhibit a higher rate of heat transfer and lower entropy production. Further, this work has many uses such as, core, microwave, and biomedical applications like tumor treatments, bone plate surgeries, EMI (electromagnetic interfaces), electronics cooling, and aerodynamic extrusion processes, in the design of passive ventilation systems.

A Hard-Constraint Wide-Body Physics-Informed Neural Network Model for Solving Multiple Cases in Forward Problems for Partial Differential Equations

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs.

Impacts of entropy generation in second-grade fuzzy hybrid nanofluids on exponentially permeable stretching/shrinking surface

The present investigation aims to use entropy analysis to analyze the unsteady Magnetohydrodynamic (MHD) flow in a second-grade fuzzy hybrid Al2O3-Cu/SA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {{\ ext{Al}}_{{2}} {\ ext{O}}_{{3}} - {\ ext{Cu/SA}}} \\right)$$\\end{document} nanofluid over an exponentially shrinking/stretching surface. The model for hybridization of the mixture of alumina Al2O3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {{\ ext{Al}}_{{2}} {\ ext{O}}_{{3}} } \\right)$$\\end{document} and copper (Cu) nanoparticles in the sodium alginate (SA) base fluid under heat source/sink, nonlinear thermal radiation, and viscous dissipation. The fundamental partial differential equations (PDEs) are simplified using an appropriate similarity conversion to generate the ordinary differential equations (ODEs). The analytical computation occurs in the MATHEMATICA program implementing the homotopy analysis method (HAM). In terms of code validity, our results are preferable to previous findings. The features of several parameters against the velocity, surface friction coefficient, entropy, temperature, and Nusselt number are described through graphs. According to our findings, the rise in the Brinkman and Reynolds numbers enhanced the total entropy of the system. Furthermore, the nanoparticle volume fraction and viscus dissipation magnifies the fluid temperature while retards the flow profile throughout the domain. Fluid velocity declined due to the Lorentz force using magnetic impact applications. The imprecision of nanofluid and hybrid nanofluid volume fractions was modelled as a triangular fuzzy number (TFN) [0%, 1%, 2%] for comparison. The double parametric approach was applied to deal with the fuzziness of the associated fuzzy parameters. The nonlinear ODEs convert into fuzzy differential equations (FDEs) and use HAM for the fuzzy solution. From our observation, the hybrid nanofluid displays the maximum heat transfer compared to nanofluids. This important contribution will support industrial growth, particularly in the processing and manufacturing sectors. The percentage increase in skin friction factor is 18.3 and 15.0 when α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} take input in the ranges of 0 ≤ α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} ≤ 0.8 and 0 ≤ β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} ≤ 1, respectively.

Momentum, heat and mass transfer in the hydrodynamic electrically conducting fluid flow over a stretching sheet

This study looks at the numerical results of magnetohydrodynamic liquid flowing across a stretching sheet at its stagnation point while also experiencing chemical reaction, viscous dissipation, thermal radiation and variable magnetic field. The fundamental partial differential equations that regulate physical phenomena are converted into nonlinear ordinary differential equations by using the appropriate similarity transformations. Later, resolved by numerically using Mathematica. The effects of various non-dimensional parameters like velocity ratio parameter ([Formula: see text]), porosity (k), magnetic parameter (M), radiation parameter (R), chemical reaction parameter ([Formula: see text]), etc. on the velocity, temperature and concentration profiles as well as on the local skin-friction and the local Nusselt number are discussed in detail and displayed through graphs. The numerical evaluation of the physical quantities was provided in tabular form for the various values of the relevant stream parameters. It is important to note that magnetic field interaction increases concentration distribution and fluid temperature while decreasing velocities at all domain flow locations.

Numerical analysis for tangent-hyperbolic micropolar nanofluid flow over an extending layer through a permeable medium

The principal purpose of the current investigation is to indicate the behavior of the tangent-hyperbolic micropolar nanofluid border sheet across an extending layer through a permeable medium. The model is influenced by a normal uniform magnetic field. Temperature and nanoparticle mass transmission is considered. Ohmic dissipation, heat resource, thermal radiation, and chemical impacts are also included. The results of the current work have applicable importance regarding boundary layers and stretching sheet issues like rotating metals, rubber sheets, glass fibers, and extruding polymer sheets. The innovation of the current work arises from merging the tangent-hyperbolic and micropolar fluids with nanoparticle dispersal which adds a new trend to those applications. Applying appropriate similarity transformations, the fundamental partial differential equations concerning speed, microrotation, heat, and nanoparticle concentration distributions are converted into ordinary differential equations, depending on several non-dimensional physical parameters. The fundamental equations are analyzed by using the Rung-Kutta with the Shooting technique, where the findings are represented in graphic and tabular forms. It is noticed that heat transmission improves through most parameters that appear in this work, except for the Prandtl number and the stretching parameter which play opposite dual roles in tin heat diffusion. Such an outcome can be useful in many applications that require simultaneous improvement of heat within the flow. A comparison of some values of friction with previous scientific studies is developed to validate the current mathematical model.

Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions

Grid-free Monte Carlo methods based on the walk on spheres (WoS) algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain or approximating functions in a finite basis. Such methods hence avoid aliasing in the solution, and evade the many challenges of mesh generation. Yet for problems with complex geometry, practical grid-free methods have been largely limited to basic Dirichlet boundary conditions. We introduce the walk on stars (WoSt) algorithm, which solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary conditions. The key insight is that one can efficiently simulate reflecting Brownian motion (which models Neumann conditions) by replacing the balls used by WoS with star-shaped domains. We identify such domains via the closest point on the visibility silhouette, by simply augmenting a standard bounding volume hierarchy with normal information. Overall, WoSt is an easy modification of WoS, and retains the many attractive features of grid-free Monte Carlo methods such as progressive and view-dependent evaluation, trivial parallelization, and sublinear scaling to increasing geometric detail.

Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel

Abstract Homogeneous and heterogeneous reactions play a decisive role in biological procedures such as burning, polymer creation, ceramic construction, distillation, and catalysis. The magnetic properties of hemoglobin molecules are organic. Magnetic resonance imaging (MRI) and electronic components with an electromagnetic field are now readily available, allowing for the explanation of fundamental biological processes. These ideas form the foundation of an ongoing study that attempts to look into the impact of both homogeneous and heterogeneous reactivity on the peristaltic transport of magnetohydrodynamics Oldroyd-B fluid. When convective and partial sliding conditions are present, the configuration changes to a non-uniform vertical channel. The fundamental partial differential equations are resolved utilizing the Homotopy Analysis Method. Entropy optimization has been carried out. The primary limits entering the problem are investigated, and then a graph is used to show the influences of temperature, velocity, skin fraction, Nusselt number, and pressure increase against mean circulation, trapping phenomena, homogeneous reactions, and heterogeneous way to respond. When magnetic parameter rises, the velocity of Oldroyd-B fluid and Bejan number decrease, while temperature, entropy generation, and pressure gradient increase. The tables show that the skin friction coefficient rises for accumulative values of the Grashof number and magnetic parameter, while the skin friction coefficient drops for rising values of the velocity slip parameter and Reynolds number. The Nusselt number increases for large values of Eckert, Grashof numbers, and magnetic parameters.

Probabilistic Descriptions of Fluid Flow: A Survey

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier–Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.

A chemo-thermo-mechanical coupled phase field framework for failure in thermal barrier coatings

The failure process in a thermal barrier coating (TBC) system can be complicated due to complex microstructure, material property mismatch, high temperature change, chemical reactions during the failure process, etc. It brings considerable difficulties to the modeling technique. A complete failure process in TBC contains multiple failure types, e.g., bulk fracture, delamination of TC (top coating)/TGO (thermally grown oxide) interface and TGO/BC (bond coating) interface, and the interaction among them. However, existing approaches mainly focus on a single failure type and thus are faced with limits on exploring the complete failure mechanism. In this work, we propose a chemo-thermo-mechanical coupled phase field model based on thermodynamic laws, such that the complete TBC fracture process can be captured. The present contribution provides a step forward in comparison with the existing phase field models. Couplings among thermal conduction, material oxidation and damage are considered, and the fundamental differential equations are derived strictly follow the thermodynamic laws. The proposed model is first validated through several benchmarks, and then followed by several simulations of the failure process of a TBC system. The simulations and experiments show that the bulk fracture occurs when the TGO is thin at the cooling process, the growth effect of TGO layer causes damage initiation at the interface and finally transformed into the interfacial delamination, the interaction of interface cracking and bulk material cracking is captured. The proposed model is proven to be a powerful tool in solving the failure process of a TBC system and is promising to handle other similar problems.

Particle shapes effects on Casson hybrid nanofluid flow over a cylinder with Cattaneo–Christov thermal model with Entropy optimization

In this study, we investigate the effect of entropy generation on a Casson hybrid nanofluid over a stretching cylinder in the presence of linear thermal radiation and Cattaneo–Christov heat flux. We assumed [Formula: see text] and [Formula: see text] to be the nanoparticles suspended in the blood’s basic fluid for our model. Targeted drug delivery is one of the most proficient ways to diagnose and treat cancer. This is because attractive nanoparticles can be used as beneficial agents in the occurrence of both heat and an angled magnetic field. In addition, several form aspects have been taken into account. By making sure that the self-similarity transformations are accurate, the fundamental Partial Differential Equations (PDEs) are converted into Ordinary Differential Equations (ODEs). The Runge–Kutta fourth-order and firing approach are used to solve the ODEs. For the situations of cylinder and plate, homotopy perturbation method (HPM) and numerical method (NM) solutions on behalf of the nonlinear structure are obtained to compare one another. In this model, we compared the shapes of the sphere, the cylinder, the blade, the platelet and the lamina, which are all graphically represented. Additionally, the results are compared to those that have already been published and are found to be in great agreement. The performance of biological applications, particularly Radio-Frequency Identification (RFA), cancer therapy, MRI, tumor therapy and malaria disease, is improved by this kind of theoretical research.

Roll of partial slip on Ellis nanofluid in the proximity of double diffusion convection and tilted magnetic field: Application of Chyme movement

The current study aims to investigate the effect of partial slip conditions over double diffusion convection. The phenomenon is applied with the inclined magnetic flux of peristaltic flow on Ellis nanofluid model in an asymmetric channel. The fundamental differential equations are constructed and solved through lubrication approximation that gives the coupled system of ordinary differential equations. This resultant system is further solved numerically, and graphic representation are used to physically comprehend the flow quantity data. The whole procedure is carried out under the trapping mechanism by drawing contour streamlines. The knowledge gained from this work will be useful in the creation of intelligent magneto-peristaltic pumps for specific heat and medication delivery phenomena.

A novel method to investigate nonlinear advection–diffusion processes

In this study, a new approach called the reversed fixed point iteration method (RFPIM) is implemented to solve a class of fundamental partial differential equations, namely viscous and inviscid Burgers equations. The results confirm that the current method has the potential to effectively solve problems governed by nonlinear PDEs representing the phenomena of advection and diffusion. Derivations reveal that even in challenging situations such as advection dominance, the RFPIM is highly capable of capturing the natural behaviour of the problem under study. Another important finding, depending on the considered problem, is that the RFPIM can allow the use of relatively large time steps. This feature is of great importance in reducing the storage burden and CPU time for the computational society. The observations reveal that the current approach leads to a significant improvement in the understanding of inverse problems and inverse optimal control problems.

Energy–time optimal path planning in dynamic flows: Theory and schemes

We obtain, solve, and verify fundamental differential equations for energy–time path planning in dynamic flows. The equations govern the energy–time reachable sets, optimal paths, and optimal controls for autonomous vehicles navigating to any destination in known dynamic environments, minimizing both energy usage and travel time. Based on Hamilton–Jacobi theory for reachability and the level set method, the resulting methodology computes the Pareto optimal solutions to the multi-objective path planning problem, numerically solving the exact equations governing the evolution of reachability fronts and optimal paths in the augmented energy and physical-space domain. Our approach is applicable to path planning in various dynamic flow environments and energy types. We first validate the methodology through a benchmark case of crossing a steady jet for which we compare our results to semi-analytical optimal energy–time solutions. We then consider unsteady flow environments and solve for energy–time optimal missions in a quasi-geostrophic double-gyre flow field. Results show that our theory and schemes can provide all the energy–time optimal solutions and that these solutions can be strongly influenced by unsteady flow conditions.

Magnetohydrodynamic hybrid nanofluid flow over a decelerating rotating disk with Soret and Dufour effects

PurposeThe investigation of fluid flow over a rotating disk has been increasing due to the spread of machine technology. Because of this development, we scrutinized the Magnetohydrodynamic (MHD) flow of hybrid nanofluid caused by a decelerating rotating disk with Ohmic heating, Soret and Dufour effects. The disk's angular velocity is taken to be an inversely time-dependent linear function. Moreover, the temperature-dependent viscosity of hybrid nanofluid is incorporated in the present investigation. Methanol is considered as base fluid, while copper oxide (CuO) and magnesium oxide (MgO) are nanoparticles.Design/methodology/approachEstimated fundamental partial differential equations of flow problems are altered as a dimensionless system of ordinary differential equations using appropriate similarity transformation and solved using a numerical technique: BVP Midrich scheme in Maple software. The impression of emerging non-dimensional parameters is portrayed graphically. All outcomes are shown in the velocity, temperature and concentration profiles.FindingsThe developed flow problem involves a non-dimensional parameter (A) that reveals the deceleration of the disk. For larger values of A, the disk decelerates faster and for some fixed time, the fluid surrounding the disk revolves more rapidly than the disk itself. The radial velocity of fluid diminishes and axial velocity becomes uniform when the disk is subjected to wall suction velocity (B).Originality/valueThis analysis is significant in biomedical engineering, cancer therapeutic, manufacturing industries and nano-drug suspension in pharmaceuticals. The novelty of the current study is the hybrid nanofluid flow with Ohmic heating, Soret and Dufour effects on a decelerating rotating disk. To the best of the author's knowledge, no such consideration has been published in the literature.