Stream Function Research Articles

Optimizing transcranial magnetic stimulator coils for minimal influence of individual variability in head geometry

In prior studies focused on transcranial magnetic stimulator (TMS) coils, optimization of coil windings typically occurred on generic curved surfaces, such as planes or cylindrical shapes. However, these surfaces do not consider the challenge of coil misalignment, which arises due to variations in individual head geometries. This misalignment can significantly diminish the coil's effectiveness. To address this issue, we propose a novel coil design approach that identifies the 'best-fit' curved surface, minimizing the likelihood of mismatch with diverse skull geometries. We further enhance the coil winding pattern using the stream function applied to this custom curved surface. Numerical simulations using a hemisphere brain model demonstrate that the coil designed on this 'best-fit' curved surface exhibits a remarkable 22% increase in efficiency and a 41% expansion in stimulation area compared to the conventional butterfly coil. These findings underscore the advantages of our method in crafting high-efficiency and wide-ranging therapeutic TMS coil.

Analysis of a Reiner–Rivlin liquid sphere enveloped by a permeable layer

The present article investigates the axisymmetric flow of a steady incompressible Reiner–Rivlin liquid sphere enveloped by a spherical porous layer using the cell model technique. The Brinkman-extended Darcy model is deployed for the porous medium hydrodynamics, and isotropic permeability is considered. The stream function solutions of the governing equations are obtained, which involves the Gegenbauer functions and the modified Bessel functions. An asymptotic series expansion in terms of the Reiner–Rivlin liquid parameter S has been employed to determine the expression of the flow field for the Reiner–Rivlin liquid. Boundary conditions on the cell surface corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Analytical expressions are derived for dimensionless pressure, tangential stress, and the couple stress components using the method of separation of variables and Gegenbauer functions/polynomial. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the porous zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the Reiner–Rivlin fluid on the porous sphere are derived with corresponding expressions for the drag coefficient. Mathematical expression of the drag coefficient, pressure distribution, velocity profile, and separation parameter is established. On the basis of viscosity ratio, permeability parameter, dimensionless parameter, and the volume fraction, variations of the drag coefficient, velocity profiles, fluid pressure, and separation parameter (SEP) are investigated with their plots. The effects of the streamline pattern make the flow more significant for the Mehta–Morse when compared to the other models. Additionally, the mathematical expression of the separation parameter (SEP) is also calculated, which shows that no flow separation occurs for the considered flow configuration and is also validated by its pictorial depiction. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behavior often arises in addition to porous media effects.

A numerical simulation of natural convection in a porous enclosure with nanofluid

In day-to-day life, heat transfer has several applications and hence we have many research areas of interest to learn new things. Nanoparticles improve the characteristics of heat transfer due to the random motion of particles. In this study, a computational approach is used to study natural convection in a porous enclosure with nanofluid. The convective term is solved by the QUICK scheme, and the finite difference method is used to solve the stream function with a 60 × 60 mesh size. CuO-water nanofluid is used in this work since nanoparticles are widely used in almost all engineering fields. In this study, we considered a two-dimensional complex cavity which is solved mathematically by using the Boussinesq approximation method. For validation of the software code, the obtained results are compared with the existing results in the literature and are found to be in good agreement. We see distortion of isotherms increases with an increase in the Rayleigh number and the nanoparticle content. That is, heat transfer takes place by natural convection very fast. Nusselt number increases with increasing nanofluid particles and the Rayleigh number. Also, it can be concluded that the heat transfer is a strong function of the Darcy number and The Rayleigh number.

Impact of Porous Media on Peristaltic Transport of Tangent Hyperbolic Nanofluid in Asymmetric Channel

The purpose behind this paper is to discuss nanoparticles effect, porous media, radiation and heat source/sink parameter on hyperbolic tangent nanofluid of peristaltic flow in a channel type that is asymmetric. Under a long wavelength and the approaches of low Reynolds number, the governing nanofluid equations are first formulated and then simplified. Associated nonlinear differential equations will be obtained after making these approximations. Then the concentration of nanoparticle exact solution, temperature distribution, stream function, and pressure gradient will be calculated. Eventually, the obtained results will be illustrated graphically via MATHEMATICA software.

Numerical modeling of Carreau–Yasuda fluid with induced magnetic field in a heated curved channel having ciliary walls

Cilia motion is commonly located in mammalian cells of living organisms which continually grow and feature conspicuously. The ciliary apparatus is associated with cell cycle movement and proliferation, and cilia play a dynamic part in human and animal growth and everyday life. The motivation of these applications, this article is to present a metachronal wave analysis for non-Newtonian Carreau–Yasuda fluid inside a curved channel with ciliated walls. Hall effect is encountered in the momentum equations to measure the magnetic field while the viscous dissipation and thermal radiation terms are considered to address the energy analysis. The curvilinear coordinates are used due to the curved nature of flow geometry in the derivation of flow equations. The computations for axial velocity, pressure rise, temperature field, mass concentration, and stream function are carried out under low Reynolds number and long wavelength approximation in the wave frame of reference by utilizing appropriate numerical implicit finite difference method (FDM). The implementation of numerical procedure and graphical representation of the computations are accomplished using MATLAB language. Furthermore, the skin friction and Nusselt number at the channel walls are determined for a variety of critical parameter assessments.

SIMULATION OF A WELL JET PUMP WORKING PROCESS

A mathematical model of the movement of the working flow in the flow part of the jet pump was developed based on the use of a radial function of a complex variable with a shifted leakage center. According to the accepted model, the working fluid radially asymmetrically exits from the displaced leakage center in all directions. The distance to the center of the leak has an inverse effect on the rate of spread of the working fluid. The offset turn of the radial function allows you to take into account the misalignment of the working nozzle and the mixing chamber due to poor manufacturing of jet pump parts. The vector of the complex potential with a shifted origin is defined as the difference of vectors whose origin is at the center of coordinates, and the endpoints characterize the initial and final positions of the shifted function. Using the equation of the potential of velocities and the stream function, the relationship for the complex potential of the plane-radial and spatial working flow with one-way and two-way displacement of the leakage center is obtained. The kinematic picture of the movement of the working medium is determined by a spatial hydrodynamic grid formed by equipotential surfaces and flow surfaces of leakage functions, which has the form of orthogonally placed coaxial spheres and radial meridional planes. The leakage displacement determines the deviation of the axis of the working nozzle from the axis of the displacement chamber of the jet pump. It is established that the velocity profiles of the radial flow with a displaced leak lose their similarity and the series of kinematic curves cannot be replaced by a single dimensionless dependence that would determine the flow kinematics regardless of the distance between the leak point and the inlet cross- section of the mixing chamber of the jet pump. The displacement of the leakage center has an inverse effect on the radial flow rate and decreases with increasing distance to the mixing chamber. As the value of the leakage displacement increases, the asymmetry of the velocity profile increases and its unevenness increases. Superimposition of the obtained characteristic function of the asymmetric radial flow and the vortex function of the complex variable allows to determine the structure of the mixed flow equation when simulating the rotation process of the jet pump in the well.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Effect of variations hollow of octagon porous media on heat and mass transfer

This study investigates stream function, temperature distribution, and concentration distortion of porous media with three types of hollow. The key factors in conductive heat and mass transfer are the number of vanes in hollows and various hollows, including semi-circle, square, and triangle vanes, which are the novelty of this research. Furthermore, the valuable result of the present study is that hollows with semi-circle vanes had a higher conductive heat transfer than hollows with square and triangle vanes. It is a matter to analyze and study porous media in heat and mass phenomena which are used in three mutually coupled equations in FEM. The finite element approach is employed to compare concentration distribution in porous media with small and large square hollows. Based on the obtained numerical results, the most effect was in the reduction of stream function, temperature distribution, and concentration distortion related to square vanes in hollow. On the other hand, the Nusselt phenomena in the bottom wall influence extremely, and the results depended on some parameters such as ε, Diffusion, Radiation, and Rayleigh number. Finally, Stream function, temperature distribution, concentration distortion, and Nusselt phenomena of octagon porous media are analyzed to develop knowledge in conductive heat and mass transfer.

Analysis of radiative heat transfer on electro‐osmotic magneto nanofluid flow through ciliary propulsion

AbstractA novel mathematical investigation is carried out to reveal the significance of thermal radiation on the dissipative magneto hydrodynamic electroosmotic ciliary propulsion of a Newtonian nanofluid (DMHECP‐NNF) in an symmetric micro‐channel by implementing the impact of an axial electrical as well as transverse magnetic fields. The ciliary transport model is explored by using conservation of mass and momentum, heat, nanoparticle concentration, and electric potential expressions along with associated boundary conditions. The pertinent system for the proposed flow problem DMHECP‐NNF consisting of partial differential equations are converted into ordinary differential equations system by incorporating the strengths dimensionless mechanism. Then analytical expressions are found for electrical potential, axial velocity, stream function, pressure gradient, temperature and concentration profiles. Whereas numerical computations are carried out to study transverse velocity and rise of pressure as per wavelength. The impacts of significant physical quantities of DMHECP‐NNF are investigated with the help of graphical manipulations. It is observed in this novel study that axial velocity rises initially and then decline with increase in the magnitude of electro‐osmotic parameter and Helmholtz‐Smoluchowski velocity. Also, it is seen that with the rising value of electric‐osmotic parameter the complexion of the trapped bolus is inflated and surrounded by few streamlines.

Metachronal wave analysis for magnetized Williamson fluid through a ciliated curved channel

Cilia play an important role in many psychological processes such as locomotion, alimentation, circulation, respiration, and reproduction. Therefore, the present investigation is modeled to study the impact of velocity slip on the cilia motion of electrically conducting Williamson fluid in a curved channel. The walls of the channel are carpeted with cilia such that their coordinated beatings produce a metachronal wave. Moreover, the viscous dissipation effect is also observed through the heat transfer mechanism. The governing system of coupled partial differential equations with highly nonlinear terms is simplified using the long wavelength and low Reynolds number approximations. The numerical solutions of simplified normalized equations are obtained using the finite difference method with an incorporating relaxation algorithm. The outcomes regarding the influences of several physical parameters on the temperature, velocity, pumping characteristics, and stream function are examined through graphs. Furthermore, the skin friction and Nusselt number at the channel walls are determined for a variety of critical parameter assessments. It is concluded that the fluid velocity is diminished at the lower wall of the channel and enhanced at the upper wall of the channel by enhancing the values of the Weissenberg number. The Brickman number shows a stronger viscous dissipation effect, leading to an increase in the liquid temperature.

Microbic flow analysis of nano fluid with chemical reaction in microchannel with flexural walls under the effects of thermophoretic diffusion

The current investigation examines the peristaltic flow, in curved conduit, having complaint boundaries for nanofluid. The effects of curvature are taken into account when developing the governing equations for the nano fluid model for curved channels. Nonlinear & coupled differential equations are then simplified by incorporating the long wavelength assumption along with smaller Reynolds number. The homotopy perturbation approach is used to analytically solve the reduced coupled differential equations. The entropy generation can be estimated through examining the contributions of heat and fluid viscosities. The results of velocity, temperature, concentration, entropy number, and stream functions have been plotted graphically in order to discuss the physical attributes of the essential quantities. Increase in fluid velocity within the curved conduit is noticed for higher values of thermophoresis parameter and Brownian motion parameter further entropy generation number is boosted by increasing values of Grashof number.

Land use characteristics affect the sub-basinal scale urban fish community identified by environmental DNA metabarcoding

The heterogeneity of urban landscapes has effects on the environmental characteristics and fish composition of individual urban streams, even within a single water system. It is, therefore, imperative to assess the influence of physiochemical properties on urban streams by analyzing the spatial distribution of fish communities at the local scale. However, conventional fish surveys encounter time and labor constraints when selecting and surveying dense sampling points under 2 km in stream networks. In this study, environmental DNA (eDNA) metabarcoding was used as an innovative survey methodology to identify the effects of land use and stream order on fish composition and tolerance guild in an urban area. The eDNA sampling was conducted in 31 sites of the Anyang stream network in Korea, including part of the stream undergoing ecological restoration. The eDNA survey detected 12 of 17 species (70.6%) that appeared in the historical data, and 12 of 18 species (66.7%) identified in a conventional field survey with kick nets and casting nets. The proportions of urban area, forest and grassland were positively correlated with abundance (p < 0.05) and richness (p < 0.05) in multiple regression analyses, while the proportion of agricultural area showed a negative correlation (p < 0.05). For abundance, richness, and diversity within the fish community from first- to third-order streams, there was a significant decrease in sensitive species (p < 0.05) alongside a significant increase in tolerant species (p < 0.01) across all three indices. The results of this study highlight variations in fish composition across sites within the local scale of the urban stream network, underscoring the need for detailed monitoring to understand the ecological function of urban streams.

Mathematical modelling of couple stress fluid flow around a semi‐permeable sphere enclosing a solid core

AbstractThe focus of this article is to study theoretically the steady‐axisymmetric creeping flow dynamics of a couple stress fluid external to a semi‐permeable sphere containing a solid core. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behaviour often arises in addition to porous media effects. The non‐Newtonian Stokes’ couple stress fluid model features couple stresses and body couples that are absent in the classical Navier–Stokes viscous model. It provides a robust framework for simulating emulsions, complex suspensions and other liquids which possess microstructure. The physical regime is delineated into two zones – the interior of the semi‐permeable zone (region II) which engulfs the solid core and the external couple stress fluid zone (region I). The model is formulated using a spherical polar coordinate system in terms of the stream function ψ. The Brinkman‐extended Darcy model is deployed for the porous medium hydrodynamics and isotropic permeability is considered. Analytical expressions are derived for dimensionless pressure, tangential stress and the couple stress components using the method of separation of variables and Gegenbauer functions of the first kind. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the semi‐permeable zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the couple stress fluid on the semi‐permeable sphere and volumetric flow rate are also derived with corresponding expressions for the drag coefficient and non‐dimensional volumetric flow rate. The influence of permeability (k), separation parameter (l), couple stress viscosity coefficient (η), couple stress inverse length dependent parameter (λ = √(μ/η)) and couple stress viscosity ratio (τ = η/η/) on all key variables is studied graphically. Additionally, streamline contours are computed for a range of parameters including inverse permeability parameter (α = 1/k). The computations show that increasing couple stress inverse length dependent parameter (λ) greatly reduces the dimensionless volumetric flow rate in particular at high values of separation parameter (l). Flow rate is, however, markedly enhanced with permeability (k) and also couple stress viscosity parameter (η). In the presence of the solid core, a much greater drag coefficient is observed at all values of couple stress inverse length dependent parameter (λ) relative to the case without a solid core. A significant distortion in streamlines is computed with increasing separation parameter (l) with a dual vortex structure emanating. With greater couple stress viscosity parameter (τ), no tangible modification is observed in the streamline contours. The present work generalizes the earlier study of Krishnan and Shukla to consider a semi‐permeable (porous medium) outer sphere and furthermore presents detailed streamline visualizations of the creeping flow for a range of emerging parameters of relevance to non‐Newtonian chemical engineering processes including emulsion droplet dynamics in porous materials.

Dieback and Replacement of Riparian Trees May Impact Stream Ecosystem Functioning

Alders are nitrogen (N)-fixing riparian trees that promote leaf litter decomposition in streams through their high-nutrient leaf litter inputs. While alders are widespread across Europe, their populations are at risk due to infection by the oomycete Phytophthora ×alni, which causes alder dieback. Moreover, alder death opens a space for the establishment of an aggressive N-fixing invasive species, the black locust (Robinia pseudoacacia). Shifts from riparian vegetation containing healthy to infected alder and, eventually, alder loss and replacement with black locust may alter the key process of leaf litter decomposition and associated microbial decomposer assemblages. We examined this question in a microcosm experiment comparing three types of leaf litter mixtures: one representing an original riparian forest composed of healthy alder (Alnus lusitanica), ash (Fraxinus angustifolia), and poplar (Populus nigra); one with the same species composition where alder had been infected by P. ×alni; and one where alder had been replaced with black locust. The experiment lasted six weeks, and every two weeks, microbially driven decomposition, fungal biomass, reproduction, and assemblage structure were measured. Decomposition was highest in mixtures with infected alder and lowest in mixtures with black locust, reflecting differences in leaf nutrient concentrations. Mixtures with alder showed distinct fungal assemblages and higher sporulation rates than mixtures with black locust. Our results indicate that alder loss and its replacement with black locust may alter key stream ecosystem processes and assemblages, with important changes already occurring during alder infection. This highlights the importance of maintaining heathy riparian forests to preserve proper stream ecosystem functioning.

Heat and mass transfer characteristics of Al2O3/H2O and (Al2O3+Ag)/H2O nanofluids adjacent to a solid sphere: A theoretical study

A theoretical study is carried out to explore the heat and mass transfer features in the flow of two types nanofluid, i.e., A l 2 O 3 / H 2 O micropolar nanofluid and ( A l 2 O 3 + Ag ) / H 2 O hybrid nanofluid, adjacent to a solid sphere. Furthermore, simultaneous effects of linear thermal radiation and chemical reaction are also explored on the heat and mass transfer characteristics. In mathematical formulation basic conservation laws are utilized along with the Soret and Doufer effects as a coupling agents in the energy and mass concentration equations. The normalized variables are used to transform the governing system of non-linear partial differential equations into the system of dimensionless partial differential equations. Stream function reduces the number of dependent variables and the resulting system is solved numerically by Keller box method. Several plots are presented to explore the impact of involved parameters on the different profiles and engineering interest quantities for both types of fluids. The temperature profile and Nusselt number are large in magnitude for hybrid nanofluid as compared to A l 2 O 3 /water based nanofluid. The heat transfer and mass transfer rates rises with thermal radiation parameter and chemical reaction parameter, respectively.

Electrohydrodynamic peristaltic microfluidic flow with thermal and molecular analysis featuring sloped configurations

This study examines the simultaneous impacts of thermal and molecular transfer of non-Newtonian fluid propelled by the collective impact of electroosmosis and peristaltic pumping via inclined channel. Internal frictional heating and heat generation are also taken into consideration. The electrohydrodynamic phenomenon is described by the nonlinear Poisson–Boltzmann equation. The problem is modulated mathematically through a system of coupled non-linear differential equations in wave frame of reference. Using the regular perturbation method around a small wave number δ, we obtain sequential solutions for stream function ψ, electric potential ϕ, velocity u, pressure gradient dpdx, temperature θ and concentration Ω distributions. The influences of newly arising parameters such as the electroosmotic parameter me, relaxation time parameter λ1, retardation time parameter λ2, wave number δ, Reynold’s number Re, heat generation parameter β, Prandtl number Pr, Schmidt number Sc, Froude number Fr and Soret number Sr on physical characteristics are analyzed graphically. There is an inverse correlation between the parameters of relaxation time and retardation time and their effects on axial velocity, concentration, and temperature distributions. In particular, the temperature of the fluid increases as the EDL parameter, relaxation time parameter, viscous dissipation parameter, and Reynolds number increase, but decreases as the retardation time parameter increases. Moreover, the axial pressure gradient increases as Reynolds number and inclination angle increase, while it decreases as Froude number increases. The current model has applicability in the chemical industry, reservoir engineering, micro-fabrication, and biomedical applications, where electroosmotic effects on heat and mass movement are crucial.

Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 2

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder Ω⊂R3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry, and S2 is perpendicular to it. We have two parts of S2. On S1 and S2, we impose vanishing of the normal component of velocity and the angular component of vorticity. Moreover, we assume that the angular component of velocity vanishes on S1 and the normal derivative of the angular component of velocity vanishes on S2. We prove the existence of global regular solutions. To prove this, the coordinate of velocity along the axis of symmetry must vanish on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The paper is a generalization of Part 1, where the periodic boundary conditions are prescribed on S2. The transformation is not trivial because it needs to examine many additional boundary terms and derive new estimates.

Magnetotactic bacteria and Fe3O4–water in a wavy walled cavity

PurposeThe purpose of this study is to investigate the interaction between magnetotactic bacteria and Fe3O4–water nanofluid (NF) in a wavy enclosure in the presence of 2D natural convection flow.Design/methodology/approachUniform magnetic field (MF), Brownian and thermophoresis effects are also contemplated. The dimensionless, time-dependent equations are governed by stream function, vorticity, energy, nanoparticle concentration and number of bacteria. Radial basis function-based finite difference method for the space derivatives and the second-order backward differentiation formula for the time derivatives are performed. Numerical outputs in view of isolines as well as average Nusselt number, average Sherwood number and flux density of microorganisms are presented.FindingsConvective mass transfer rises if any of Lewis number, Peclet number, Rayleigh number, bioconvection Rayleigh number and Brownian motion parameter increases, and the flux density of microorganisms is an increasing function of Rayleigh number, bioconvection Rayleigh number, Peclet number, Brownian and thermophoresis parameters. The rise in buoyancy ratio parameter between 0.1 and 1 and the rise in Hartmann number between 0 and 50 reduce all outputs average Nusselt, average Sherwood numbers and flux density of microorganisms.Research limitations/implicationsThis study implies the importance of the presence of magnetotactic bacteria and magnetite nanoparticles inside a host fluid in view of heat transfer and fluid flow. The limitation is to check the efficiency on numerical aspect. Experimental observations would be more effective.Practical implicationsIn practical point of view, in a heat transfer and fluid flow system involving magnetite nanoparticles, the inclusion of magnetotactic bacteria and MF effect provide control over fluid flow and heat transfer.Social implicationsThis is a scientific study. However, this idea may be extended to sustainable energy or biofuel studies, too. This means that a better world may create better social environment between people.Originality/valueThe presence of magnetotactic bacteria inside a Fe3O4–water NF under the effect of a MF is a good controller on fluid flow and heat transfer. Since the magnetotactic bacteria is fed by nanoparticles Fe3O4 which has strong magnetic property, varying nanoparticle concentration and Brownian and thermophoresis effects are first considered.

Influence of chemical reaction on electro-osmotic flow of nanofluid through convergent multi-sinusoidal passages

AimIn the present study, the influences of chemical reactions on heat transfer and the peristaltic pumping of nanofluid in a convergent channel are analyzed. This mathematical analysis is looked at under the postulates of greater wavelength and smaller Reynold's number. Research methodologyThe governing equations are initially transformed from fixed to a wave frame by using linear transformations. Furthermore, these transformed equations are non-dimensioned with the help of similarity variables. Due to the complex form of non-dimensional flow equations, numerical solutions for velocity profile, stream function, temperature profile, and nanoparticle concentration are obtained with the help of Mathematica 11.0 software. These numerical solutions are described via graphs in Mathematica software. These numerical solutions are plotted for numerous rheological parameters. The transportation of nanofluid is based upon multi-sinusoidal natures (cosine wave, sine wave, triangular wave, square wave, sawtooth wave) of peristaltic waves. OutcomesIt is found that with an increasing heat source parameter, the channel flow is decelerated due to the magnitudes of velocity profile and stream function are reduced. While sharp enhancements are noticed in both temperature and nanoparticle concentration by increasing the heat source parameter under chemical reaction effects. The high temperature is obtained with larger chemical reactions. The contrast among viscous and non-viscous fluids is also addressed. A significant enhancement is observed in both velocity profile and stream function by increasing the electroosmotic parameter. Significances and applicationsThe study is relevant to nano-cooling systems, drug delivery systems, and microfluidic pumps where laminar flows in convergent domains arise. This model is significant for the thermal enhancement of mechanical and chemical rheological processes.

A novel Lagrangian–Eulerian weighted-least squares scheme coupled with other stable techniques for multi-physical fluid flow around complex obstacle

In this paper, a stable novel meshless coupled method is proposed to simulate the non-isothermal magnetohydrodynamics (MHD) flow problems (multi-physics quantities) inside a lid-driven cavity around complex obstacle. The proposed method is mainly motivated by a Lagrangian–Eulerian (L–E) weighted-least squares (WLS) scheme combined with a stream function-vorticity (SFV) and other stable techniques, and it is further to investigate the non-isothermal MHD flow around an airfoil obstacle at large Hartmann (Ha) or Reynolds (Re) number, for the first time. In the present meshless coupled approach (named L–E WLS–SFV), the traditional MHD equations are derived as another form with an SFV method under divergence-free constraint, which can avoid the tedious treatment of pressure on complex irregular obstacle. Then, a stable L–E WLS coupled algorithm is proposed to approximate the space derivatives of multi-physical quantities (velocity, magnetic, temperature, etc.), in which a corrected particle shifting technique is employed to improve the tensile instability among Lagrangian particles moving inside the domain and a second-order upwind scheme is adopted to stabilize large Re number problem in Eulerian fixed nodes near the boundary. Several benchmarks are simulated to show the numerical accuracy and convergence rates of the proposed WLS scheme for MHD flow at different parameters. Subsequently, the case of the non-isothermal MHD flow around a square obstacle under large parameters is simulated by the proposed L–E WLS–SFV method and compared with other numerical results to demonstrate the validity and capacity of the proposed method for multi-physical flow and the necessity of imposing the above two stable techniques. Finally, the case of non-isothermal MHD flow around the circular or airfoil obstacle is numerically investigated, and the important effects of the Hartmann, Rayleigh, and Reynolds numbers on the multi-physical quantities (stream function, vorticity, temperature, and magnetic field) are discussed. The advantages of the proposed method for the muti-physical flow around irregular obstacles are also exemplified. All the numerical results show that the proposed L–E WLS–SVF method is robust and accurate to simulate the multi-physical fluid flow around complex obstacles.