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In this paper, a non-isothermal study of the calendering processes is presented using Carreau–Yasuda model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. The final equations are solved numerically using "bvp4c" to find stream function and velocity profiles, while the hybrid numerical method which is the combination of shooting and finite difference methods is used to solve the energy equation. Graphs show the impact of the concerned material parameters on various quantities of interest. The pressure distribution decreases with the increasing values of the slip parameter and Weissenberg number. The mechanical variables show an increasing trend with the increasing values of the slip parameter and Weissenberg number. The temperature distribution increases with an increase in the Brinkman number, while temperature shows declining trend near the roll surface with the increasing values of the slip parameter. The force separating the two rollers, total power input into both rolls, increase with the increasing values of the Weissenberg number and slip parameters. The results show that the Newtonian model predicts higher pressure in the nip zone than the Carreau–Yasuda model. It is interesting to note that for the case of shear thinning, the Carreau–Yasuda model predicts 30% less pressure in the nip region when compared to the Newtonian model. [ABSTRACT FROM AUTHOR]

In this paper, the study investigates the problem of the creeping flow of a non-Newtonian couple stress fluid through a linearly porous walled slit within a Darcy porous medium. The well-established approach, the Inverse Method approach, was employed to attain the precise solution of the governing equations. The inverse technique is utilized to transform the momentum equations into stream functions. The physical parameters are all computed: longitudinal velocity, transverse velocity, fractional reabsorption, leakage flux, axial pressure, volume flow rate, and mean pressure. The data was also graphed, indicating that the initial flow rate affects longitudinal velocity but not transverse velocity. The streamlines are greatly affected by the initial flow rate, resulting in straighter and more uniform patterns as the flow rate increases. Parameters like permeability and the couple stress parameter also have an impact on physical quantities. Because of the porosity parameter, backward flow occurs at the slit's end. The novelty of this research lies in the investigation of couple stress fluid flow within both linear channel walls and a porous medium. The present research focuses on how kidney disease affects fluid flow in renal tubules. It's critical because fibers, lipids, and waste particles can clog or disrupt these channels, lowering kidney performance. Understanding this helps in the management of kidney disease and provides insights for better biomedical engineering in the development of improved renal medicines. [ABSTRACT FROM AUTHOR]

This research aims to investigate the main features of the ciliary flow of fourth‐grade fluid in a curved channel. The fluid is considered electrically conducting with a radial magnetic field effect. The constitutive relation for energy is formulated with the addition of viscous dissipation and thermal radiation. The governing system of coupled partial differential equations with extremely nonlinear expressions is simplified using the long wavelength and low Reynolds number approximations. The numerical outcomes of simplified normalized equations are obtained using the finite difference method incorporating the relaxation algorithm. The numerical outcomes regarding the influences of several physical parameters on the temperature, velocity, pumping characteristics, and stream function are examined through graphs. The outcomes reveal that fluid velocity diminishes by enhancing the magnetic parameter. Also, the temperature is enhanced by enhancing the values of the Brickman number. The current model has been used in bioengineering processes, microfluidics, and drug delivery systems. [ABSTRACT FROM AUTHOR]

To evaluate the impact of different momentum control variable (CV) schemes (CV5, the momentum control variable option with ψχ and CV7, the momentum control variable option with UV) on radar data assimilation (DA) in weather research and forecasting model data-assimilation (WRFDA) systems, a heavy snowfall in central and eastern regions of China, which started on 6 February 2022, was taken as a case in this study. The results of the wind-field increments from the single observation tests indicated that the wind-field increments had a larger range of influence when stream function and velocity potential (ψχ) were used as momentum control variables in CV5. Some spurious increments were also generated in the wind-field analysis, since CV5 tended to maintain the integrated value of the wind field. When U-wind and V-wind were used as control variables in CV7, the wind-field increments had a smaller impact range, and there was less dependence among different locations on the wind increments. For the heavy snow case, the CV7 schemes displayed some improvements in simulating the composite reflectivity compared to the other two experiments, since the composite reflectivity in the CV5 and control experiments were overestimated to some level. It was also found that the RMSEs were lower in the CV7 compared to those in the CV5 in the short-term forecasts during the data-assimilation cycles. Results also indicated that the CV7 had a more significant effect on the 6 h accumulated precipitation forecasts. Meanwhile, the experiment Exp_CV7 achieved the best ETS and FSS scores among the three groups of experiments, while Exp_CV5 appeared to be generally superior to the CTRL. In summary, the precipitation of Exp_CV7 yielded the rainfall intensity and location most close to the observation compared to those from both the CTRL and Exp_CV5 experiments. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A boundary element scheme for the problem of potential flow over an axisymmetric toroidal body is considered. An integral equation for the velocity distribution on the body is derived. It is shown that the numerical scheme for solving the considered equation converges exponentially. [ABSTRACT FROM AUTHOR]

Objective: This study aims to introduce a configurable standard-type (CST) coil design methodology that can accommodate different stimulation needs through the overlapping of standard coil components, while ensuring the performance of CST coil approaches the existing optimal physical limit as closely as possible. Approach: The study utilizes planar, spherical, and hemispherical stream function to represent the spatial geometry of the coils. Singular value decomposition is employed to design the standard coil components while minimizing the number of coil components. Initially, this study delineates the relationship between singular vectors and coil geometry, defining a multi-objective optimization problem for the research. By contrasting the decay characteristics of singular values in the inductance and electric field matrices of the stream functions, the singular vectors of the electric field matrix are utilized as the foundation for CST coil optimization. Subsequently, the relationships between the number of singular vectors and the coil's Pareto fronts are determined through the optimization algorithms, and the geometric patterns of the singular vectors, i.e., the standard coil components, are analyzed. Main results: By constructing the CST coil components using seven singular vectors and comparing with the Pareto front of the performance of over 50 existing coil types, the stimulation depth range of the CST coil based on the planar stream function decreased by only 7.7%; the focality performance of the CST coil based on the spherical stream function improved by 22.4%; the focality performance of the CST coil based on the hemispherical stream function decreased by only 1.6%. Significance: The construction of CST coils with a minimal number of components is explored to approximate the performance at any discrete point on the physical limit curve. This approach overcomes the limitation of one coil associating with one performance point in traditional designs, presenting a novel perspective on coil design. [ABSTRACT FROM AUTHOR]

Main causes of ground disturbances are geological events such as earthquakes, underwater gravity mass flows, volcanic eruptions, and bottom explosions. This paper investigates the effects of transitory ground disturbance on the generation of flexural waves in the presence of a thin floating elastic plate. The problem is formulated using the Stokes stream function and wave potential boundary value problems in a viscous fluid. The expression of the plate deflection is obtained as multiple infinite integrals using the Fourier and Laplace transforms, which is further solved by the steepest descent method. Three types of ground disturbances such as H 0 (x) = e (− x 2 / 2) , H 0 (x) = e − | x | , and H 0 (x) = δ (x) are considered. The deflection of the floating elastic plate is investigated in connection to ground disturbances, fluid viscosity, and structural parameters such as mass per unit length of the plate and flexural rigidity. The study reveals that with higher viscosity and flexural rigidity, the amplitude of the plate deflection is reduced. Moreover, these two parameters play a great role in the reduction of plate deflection. [ABSTRACT FROM AUTHOR]

Mixed convective characteristics of the combustion of a nanoparticles-laden fuel (n-butanol nanofluid) in a vented cavity are investigated. The nanofluid and the oxidizer enter the cavity through the inlets on the left and right vertical walls, respectively. However, the resulting product produced from the oxidation process of the fuel exits the cavity through the outlet at the bottom wall. Heat generated from the oxidation process causes natural convection within the cavity. The conjugate effect of natural and forced convection finally gives rise to mixed convection phenomena. In this regard, a mathematical model for mixed convection flow in a vented cavity is formulated with no-slip and isothermal boundary conditions. Having transformed the model into a dimensionless form, the stream function-vorticity formulation is used. The resulting equations are then solved numerically using the finite difference method. Numerical results are illustrated with the streamlines, isotherms, and isolines of fuel and oxidizer concentrations. The maximum values of the stream function (ψmax) and the temperature (θmax) are found to increase with an increase in the Frank–Kamenetskii number (Λ), volume fraction of nanoparticles (φ), and stoichiometric ratio (χ). On the contrary, they decrease with the increase in the Reynolds number (Re). When the Grashof number (Gr) is increased, ψmax increases and θmax decreases. The remaining concentrations of fuel, (CF)min, and oxidizer, (CO)min, are higher for an increase in Gr, whereas the opposite is recognized for increasing Λ. With the increase in Gr and Λ, the steady-state flow in the cavity tends to be oscillating and then chaotic. [ABSTRACT FROM AUTHOR]

The transport of complex rheological fluids in physiological ducts is often facilitated by the dynamic phenomenon of peristalsis. Additionally, peristaltic transport assisted by cilia plays a significant role in various natural processes such as respiration, circulation, locomotion, and reproduction. This study focuses on magnetically induced flow bounded by non-uniform curved walls, motivated by the importance of peristalsis and micro-organism motility. To characterize the complex rheology of the fluid liner, a viscoelastic model described by the constitutive equation of Jeffrey's fluid is employed. The flow problem is mathematically formulated using curvilinear coordinates. Subsequently, linear transformations and scaling factors are applied to convert the equations into dimensionless form, while considering biotic restrictions such as creeping transport and long wavelength to reduce dependent variables. By utilizing the stream function and cross-differentiation, a fourth-order equation is obtained and numerically approximated using the shooting method. The effects of various parameters on the flow are illustrated through graphs, and a physical interpretation of the graphical results is provided. It is observed that ciliated walls of the channel enhance the velocity and pumping, while trapping phenomena are more pronounced in a non-uniform channel compared to a uniform channel. [ABSTRACT FROM AUTHOR]

In this study, a fluid flow with an incompressible axisymmetric steady couple stress translated through a porous media is analyzed between a hollow sphere and a concentric rigid sphere. In the permeable region, the flow field is regulated by Brinkman's equation. The slip and spin slip conditions are applied on both the rigid sphere and spherical cavity surfaces. Modified Bessel functions provide a systematic approach to the problem by utilizing the principle of a stream function. On the inner sphere, the wall correction factor that an incompressible couple stress fluid encounters is calculated. The effects of the slip, spin slip, coupling stress parameter, separation distance, and permeability parameter on the field functions and the normalized drag force are also graphically shown. The corresponding results are contrasted with the outcomes reported for particular cases of couple stress fluid and viscous fluid flow in two permeability-free concentric circles. Furthermore, graphs of the streamlines for various values of the relevant parameters have been included. [ABSTRACT FROM AUTHOR]

Agricultural land use alters nitrate (NO3–) uptake dynamics in streams, but the specific mechanisms linking individual agricultural stressors to benthic and hyporheic uptake remain unclear. Using stream-side mesocosms and 15N-nitrate additions, we examined the individual and combined effects of fine sediment (FS) and augmented light and phosphorus levels (L&P) on benthic and hyporheic NO3– uptake rates. In absence of FS, L&P stimulated uptake of autotrophic and heterotrophic biofilms, leading to a 12- and 7-fold increase in the benthic and hyporheic compartments, respectively. Under ambient light and nutrient conditions, FS reduced by 3-fold benthic uptake, but effects were not significant. Conversely, in the hyporheic compartment, FS induced anoxic conditions, likely stimulating denitrification and causing a 14-fold increase in hyporheic uptake. When these stressors were combined, they did not interact in the benthic compartment. Conversely, in the hyporheic compartment they interacted antagonistically, with L&P diminishing the increase in uptake induced by FS. Our results indicate that the previously observed increase of whole-stream NO3– uptake in agricultural streams is attributable to nutrients and light stimulating benthic uptake, while fine sediment effects and the role of the hyporheic compartment to total uptake are modest. Moreover, the finding that stressor interactions vary with ecosystem compartments calls for a consideration of all compartments and their contribution to whole-system functioning in multiple stressor studies. We are beginning to understand how multiple interacting stressors affect stream functioning, but more mechanistic evidence is needed to disentangle whether additive or non-additive effects prevail in human-altered ecosystems. [ABSTRACT FROM AUTHOR]

IN THIS PAPER, STOKES FLOW PAST A CONTAMINATED FLUID SPHERE embedded in a porous medium is considered using interfacial slip on the boundary. The stream functions and drag are computed analytically. Special cases are deduced for drag force and a satisfactory agreement is reached with available data in the literature. It was observed that in viscous fluid and couple stress fluid cases with an increase in the viscosity ratio, the slip parameter, the porous parameter there is an increase in the values of the drag coefficient for varying different parameters, respectively. Also noticed that coefficient of drag values for a uniform flow of the viscous fluid flow over a contaminated viscous fluid sphere in a porous medium with the slip condition are superior to those of a couple stress fluid (CSF) flow. [ABSTRACT FROM AUTHOR]

The issue of vascular stenosis has recently received significant innovative interest among researchers. Due to the buildup of adipocytes and other constituents of atherosclerotic, arteries can generate an abnormal condition called arterial stenosis. It optimizes blood circulation to prevent myocardial illness. In light of this, the current scientific report assembled a Casson fluid model to assess the MHD peristaltic transit of blood supply via a mildly stenotic artery when copper nanoparticles are added. A relevant simulation model is often used to outline the issue, acquiring non‐dimensional variables and approximations for the stenosis. Temperature has quantitative approaches, and the result for velocity is derived numerically in MATHEMATICA. The expressions for Nusselt number, wall shear stress, flow resistance distribution and stream function are attained. The under action of copper nanoparticles, the consequences of several new physiological measures on arterial blood circulation properties are visually displayed and explained briefly. A pictorial explanation of the trapping phenomena is also provided. In comparison to arteries devoid of stenosis, arteries with stenosis have a higher temperature distribution throughout the fluid flow and velocity. In addition, high temperature and velocity increase in the middle of the artery and decrease as one gets closer to the arterial wall. Further, the concentration, Grashof number and heat source and sink parameters of the nanoparticles reduce the wall shear stress of the blood in the arteries since the arterial wall contributes heat to the flow zone. The discovery could have medical applications as well as bioinformatics. Furthermore, the current work contributes to the cure of different cardiac illnesses. [ABSTRACT FROM AUTHOR]

In this study, numerical methods for the 3D vorticity equation and Euler equation, which preserve the structure of Nambu mechanics for fluid dynamics, are investigated. Discrete vector fields of the stream function (or vector potential), velocity, and vorticity, as well as discrete counterparts of the gradient, curl, and divergence operators acting on them, are defined such that the structure of the de Rham complex in 3D Euclidean space is preserved. The inner products of the discrete vector fields are defined such that discrete counterparts of integration-by-parts formulae for the gradient, curl, and divergence operators hold. In addition, cross products of the discrete vector fields are introduced to define a skew-symmetric trilinear form. A discrete Nambu bracket, as well as the (kinetic) energy, helicity, and enstrophy of the discrete flow field, are defined straightforwardly. They are employed to derive a discrete vorticity equation in the same way as in the continuum setting. A discrete Euler equation is derived from the discrete vorticity equation based on the discrete counterpart of the Poincaré lemma, which holds under some typical conditions. It is proved that any solution to these discretized equations satisfies discrete analogues of the balances of energy, helicity, and enstrophy. Numerical experiments on a periodic array of rolls are conducted to examine the effectiveness of the method. [ABSTRACT FROM AUTHOR]

This investigation is an approach to setup an analytical solution of steady plane allied MHD fluid flow having infinite electrical conductivity in a rotating frame through porous media by Martin's method. The governing non-linear equations of the fluid flow are transformed into a new form called Martin's form by employing differential geometry where the curvilinear co-ordinates (Φ, Ψ) in the plane of flow shows that, the co-ordinate lines Ψ are the streamlines of flow and the co-ordinate lines Φ are arbitrary constants. Exact solution is obtained and velocity, vorticity, current density magnetic field and pressure distribution are found out. Also, the diagrams have been plotted to sketch the streamline patterns and to study variation of pressure function with angular velocity. [ABSTRACT FROM AUTHOR]

The wave–current–seabed interaction problem is studied by using a coupled-mode system developed for modeling wave scattering by non-homogeneous, sheared currents in variable bathymetry regions. The model is based on a modal series expansion of wave velocity based on vertical eigenfunctions, dependent on local depth and flow parameters, including propagating and evanescent modes. The latter representation is able to accurately satisfy the wave flow continuity condition and the no-entrance boundary condition on the sloping parts of the seabed. A new derivation of a simplified nonlinear system is introduced using decomposition to a mean flow and a perturbative wave field. To force the system to consider incoming waves at the inlet, boundary knowledge of periodic, travelling nonlinear water waves over a flat bottom is required. For this purpose, specific solutions are derived using the semi-analytical method based on the stream function formulation, for cases of water waves propagating above linearly and exponentially sheared currents. Results obtained by the application of the CMS concerning the propagation of waves and currents—in particular, examples characterized by depth inhomogeneities—are presented and discussed, illustrating the applicability and performance of the method. [ABSTRACT FROM AUTHOR]

The current study investigates the influence of MHD on the natural heat transfer considering the irreversibility in a complex-shaped cavity with the existence of inner roundish heater with four attached fins utilizing finite element formulation. Three different cases are considered to figure out the major characteristics of temperature, stream function and total generated entropy, Nusselt number in addition to Bejan number. Concentrated attention is directed to the enclosure geometry modifications and the varied length of attached fins and their impacts. The parameters of study are ranged as follows; Rayleigh number 10 3 ≤ Ra ≤ 10 6 , Hartmann number 0 ≤ Ha ≤ 60 , fins' length 0.1 ≤ H ≤ 0.3 . The novelty of the present work is on studying all of these parameters in the complex enclosure considering three different cases of the shape of the outer walls so that the star-shaped enclosure is classified as case 1. With increasing the length that separated between the outer arc of the star enclosure, the octagonal-shaped enclosure will appear as denoted in case 2 and 3. It had been seen that at low Rayleigh number Ra = 10 4 , case one is the best choice in heat transfer bettering while it is the lowest case at high Rayleigh number Ra = 10 6 . Additionally, the influence of Hartmann number on the total entropy generation reduction percentage for case three is 63.57 % while it is 51.11% for case two while Hartmann number had negligible impact of entropy generation reduction for case one. Lastly, the highest Bejan number is at fin length H = 0.3 is recorded for case one. [ABSTRACT FROM AUTHOR]

This paper presents MQGeometry, a multi-layer quasi-geostrophic (QG) equation solver for non-rectangular geometries. We advect the potential vorticity (PV) with finite volumes to ensure global PV conservation using a staggered discretization of the PV and stream function (SF). Thanks to this staggering, the PV is defined inside the domain, removing the need to define the PV on the domain boundary. We compute PV fluxes with upwind-biased interpolations whose implicit dissipation replaces the usual explicit (hyper-)viscous dissipation. The discretization presented here does not require tuning of any additional parameter, e.g., additional eddy viscosity. We solve the QG elliptic equation with a fast discrete sine transform spectral solver on rectangular geometry. We extend this fast solver to non-rectangular geometries using the capacitance matrix method. Subsequently, we validate our solver on a vortex-shear instability test case in a circular domain, on a vortex–wall interaction test case, and on an idealized wind-driven double-gyre configuration in an octagonal domain at an eddy-permitting resolution. Finally, we release a concise, efficient, and auto-differentiable PyTorch implementation of our method to facilitate future developments on this new discretization, e.g., machine-learning parameterization or data-assimilation techniques. [ABSTRACT FROM AUTHOR]

We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on the external forces. Weak-strong uniqueness criteria based on various growth conditions at the infinity of weak solutions are also given. This is done by employing an energy estimate and a Hardy's inequality. Several estimates of stream functions are carried out and two density lemmas with suitable weights for the homogeneous Sobolev space on 2-dimensional space are proved. [ABSTRACT FROM AUTHOR]

In this work, we study boundary layer flows for thermo-dependent Newtonian fluids. We first show that the governing equations admit a self-similar solution. Furthermore, an asymptotic model is used to obtain correlations on heat transfer and the skin-friction coefficient. In addition, the numerical study of the complete problem enables us to specify the domain of validity of these correlations. Finally, the obtained numerical results are used to analyze the influence of the Prandtl number on the profiles of the stream function and the temperature field, and also on parameters such as the local Nusselt number and the thickness of the boundary layer. [ABSTRACT FROM AUTHOR]

This study investigates the behaviour of subtropical high-pressure systems and the Hadley cell, which affect the weather of South Africa, using the ERA-Interim database and ensemble of 14 global circulation models from Phase 6 of the Coupled Model Intercomparison Project (CMIP6). Mass stream function was used to represent the Hadley cell. To analyse the behaviour of the subtropical anticyclones, monthly sea level pressure, the 1018 hPa isobar and the maximum isobar in the study area were used. The seasonal variation of the anticyclones and Hadley circulation is consistent with rainfall over South Africa. During austral summer, a less intense, narrow mass stream function, South Atlantic Subtropical Anticyclone and Mascarene High are located more southwards, causing rainfall over the eastern parts of South Africa. During the austral winter, Hadley circulation, as well as the anticyclones, is stronger and located more northwards, causing rainfall over the southern and southwestern parts of South Africa. [ABSTRACT FROM AUTHOR]

The peristaltic flow of Walter's B fluid through an asymmetric channel is analyzed. The constitutive equation for the balance of mass, momentum, trapped phenomena, and velocity distribution is modeled. Consideration of low Reynolds number and long wavelength assumption are used in the flow analysis to get an exact solution by using the perturbation method. The impact of impotent parameters on stream function and axial velocity has been gained. The set of figures is graphically analyzed of these parameters. [ABSTRACT FROM AUTHOR]

This focused study investigates the peristaltic motion of a micropolar fluid within an uneven channel filled with a porous medium, incorporating an orthogonal magnetic field to the flow. This research diverges from the traditional assumptions of lubrication theory. The governing equations, encompassing the physical characteristics of micropolar fluid peristalsis, are transformed into nonlinear coupled partial differential equations. These equations are solved using the finite element method, considering inertial effects, such as non-zero wave and Reynolds numbers. This study delves into the influence of various crucial parameters on axial velocity, pressure gradient, microrotation, and stream function, presenting graphical representations. Notably, the incremental phase shift causes an intermingling of upper and lower streamlines within both halves of the channel. As the Reynolds number increases, there is an observed reduction in bolus size, particularly at maximum phase shifts, with a tendency to move toward the central region. An increase in Hartmann number leads the bolus formation to vanish in both channels, reduces microrotation, and leads to increased pressure. Vorticity lines intensify and incline toward the peristaltic walls. An increase in the permeability parameter amplifies velocity, microrotation, volume, and bolus formation regardless of phase differences while countering pressure elevation per wavelength. Reduced concavity is observed as vorticity lines disperse across the entire area. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder Ω ⊂ R 3 with the axis of symmetry. S 1 is the boundary of the cylinder parallel to the axis of symmetry, and S 2 is perpendicular to it. We have two parts of S 2 . On S 1 and S 2 , 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 S 1 and the normal derivative of the angular component of velocity vanishes on S 2 . 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 S 2 . The transformation is not trivial because it needs to examine many additional boundary terms and derive new estimates. [ABSTRACT FROM AUTHOR]

We seek to understand how copper and cadmium act on leaf litter decomposition by their effects on microbial conditioning and litter fragmentation by invertebrates. In this study, we evaluated, in an integrated manner, different biological elements responsible for functioning of streams. Thus, we performed a microcosm assay with different concentrations for the two metals and their combination, evaluating their effects on fungi sporulation rate, consumption rate by shredders, and, consequently, the leaf litter decomposition rates. Sporulation rates were affected by all copper concentrations tested 10 × = 16 µg L−1 and 25 × = 40 µg L−1) but significantly reduced only at the highest concentration of cadmium (25 × = 22.5 µg L−1). Increased copper and cadmium concentrations reduced the consumption of leaf litter by Phylloicus at 60%. The concentrations (10 × and 25 ×) of both metals resulted in a reduction in decomposition rates. When combined, copper and cadmium negatively affected microbial conditioning, consumption by shredders, and leaf litter decomposition. Increases in concentrations of copper and cadmium directly affected organic matter decomposition in aquatic environments. Thus, the presence of a high concentration of heavy metals in aquatic environments alters the functioning of ecosystems. As trace-elements occur in a combined manner in environments, our results show that the combined effects of different metals potentiate the negative effects on ecosystem processes. [ABSTRACT FROM AUTHOR]

El Niño–Southern Oscillation (ENSO), the leading mode of global interannual variability, usually intensifies the Hadley Circulation (HC), and meanwhile constrains its meridional extension, leading to an equatorward movement of the jet system. Previous studies have investigated the response of HC to ENSO events using different reanalysis datasets and evaluated their capability in capturing the main features of ENSO-associated HC anomalies. However, these studies mainly focused on the global HC, represented by a zonal-mean mass stream function (MSF). Comparatively fewer studies have evaluated HC responses from a regional perspective, partly due to the prerequisite of the Stokes MSF, which prevents us from integrating a regional HC. In this study, we adopt a recently developed technique to construct the three-dimensional structure of HC and evaluate the capability of eight state-of-the-art reanalyses in reproducing the regional HC response to ENSO events. Results show that all eight reanalyses reproduce the spatial structure of HC responses well, with an intensified HC around the central-eastern Pacific but weakened circulations around the Indo-Pacific warm pool and tropical Atlantic. The spatial correlation coefficient of the three-dimensional HC anomalies among the different datasets is always larger than 0.93. However, these datasets may not capture the amplitudes of the HC responses well. This uncertainty is especially large for ENSO-associated equatorially asymmetric HC anomalies, with the maximum amplitude in Climate Forecast System Reanalysis (CFSR) being about 2.7 times the minimum value in the Twentieth Century Reanalysis (20CR). One should be careful when using reanalysis data to evaluate the intensity of ENSO-associated HC anomalies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Stirling pulse tube cryocoolers (PTC) are widely used in aerospace applications for the cooling of infrared sensors and for filtering background thermal noise in the astro-imaging devices, etc. Present investigation aims to use numerical methods to demonstrate the nonlinear fluid flow, heat transfer, and vortex generation phenomena in a two-stage coaxial type inertance pulse tube cryocooler. The numerical simulation is conducted using commercially available Fluent® code for both single-stage and multi-stage configurations to show nonlinear processes with varying heat load conditions. It has been noticed that the width of the vortex produced inside the pulse tube grows with an increase in heat load capacity. This undesirable flow conditions yields an adverse effect in the cooling behavior and reduces overall performance of cryocooler with higher heat load. Additionally, streamlines, stream function, pressure and temperature variation plots are given for both stages with different heat load capacity to substantiate our results. [ABSTRACT FROM AUTHOR]

This study attempts to analyze the effect of diverse parameters of the peristaltic flow of second grade dusty fluid through a curved configuration. Stream function conversions are used to model the separate system of equations for the dust particles and fluid. The Perturbation technique is used to get the analytical solutions and the results are verified through graphs. It is noticeable that the trapped bolus, compresses both the dust particles and fluid with an increase in 'the second grade parameter'. Moreover, a surge in 'the Reynolds number' Re effects the bolus trapped in the lower portion of the channel for fluid. Decrement in the velocity is observed with a rise in 'the wave number' - and 'the second grade parameter' α1. [ABSTRACT FROM AUTHOR]

Natural convection of a chemically reacting hybrid nanofluid in a closed wavy‐walled cavity embedded in a porous medium is investigated with an inclined magnetic field. The left wall of the cavity is assumed to be wavy and the walls are maintained at the surrounding temperature. Governing equations are transformed into dimensionless equations which are solved using the finite difference method. To validate the solving procedure, a grid sensitivity test and a comparison with published results have been carried out. Streamlines, isotherms, and isolines of concentration are discussed for varying Rayleigh number (Ra), Hartmann number (Ha), Frank‐Kamenetskii number (Fk), Darcy number (Da), combined buoyancy parameter (N), and nanoparticle volume fractions (φ1 and φ2). Streamlines show clockwise and anticlockwise vortices irrespective of the parameters. For Fk = 0.5, the maximum stream function (ψmax) is 0.64 and the maximum temperature (θmax) is 0.20 while for Fk = 2, ψmax and θmax are 4.08 and 1.36, respectively. Besides, for Ha = 0, ψmax and θmax are 1.61 and 0.379, however, for Ha = 100, ψmax is 0.90 and θmax is 0.377. Maximum temperature is increased with an increase in Ra, N, and Fk, whereas it is decreased with the augmentation of Ha and Da. Isolines of concentration show reverse characteristics of temperature. An increase in Ra, Da, and Fk enhances the intensity of streamlines but the opposite is observed for higher Ha, N and volume fractions. Moreover, the eyes of the vortices are distorted in the direction of the magnetic field. [ABSTRACT FROM AUTHOR]

Mathematical simulation of biological fluids is of upmost significance due to its numerous medical uses. Interpreting various biological flows necessitates a thorough knowledge of the peristaltic mechanism. This paper presents a computational study for the peristaltic pumping within vertical asymmetric channels filled with BN‐EG nanofluid under the influence of temperature‐dependent electrical conductivity and thermal radiation. Experimental study showed that the nanofluid created by suspending Boron Nitride particles in a combination of Ethylene Glycol exhibited non‐Newtonian characteristics. Further, the Carreau's fluid model provides accurate predictions about the rheological properties of BN‐EG nanofluid. Various configurations of the outer boundaries are considered, namely, square wave, multi‐sinusoidal wave, trapezoidal wave, and triangular wave. A uniform magnetic field together with nanoparticles and mass concentrations, joule heating, first‐order chemical reaction as well as viscous dissipation are considered. Influences of the Dufour and Soret numbers are examined, and the cases of biological scientific assumptions which is known as low Reynolds number and long wavelength are applied. All the computations are obtained numerically using Mathematica symbolical software (ND‐Solve), and the obtained results are presented in terms of the axial velocity u, heat transfer rate Z, concentration profile Ω, temperature profile θ, extra stress tensor Sxy${S}_{xy}$, pressure gradient dpdx$\frac{{dp}}{{dx}}$, pressure rise Δpλ${\mathrm{\Delta p}}_{{\lambda}}$ and stream function ψ. The major outcomes revealed that the maximizing in electrical conductivity coefficient, variable viscosity coefficient and magnetic field parameter is better to obtain a higher rate of the heat transfer while the increase in thermo‐diffusion effects as well as linear thermal radiation coefficient causes a reduction in the rate of heat transfer. [ABSTRACT FROM AUTHOR]

Bifurcations of streamlines in peristaltic flow through a channel are investigated without lubrication approximation, that is, when the inertial and streamline curvature effects are present. The stream function for the considered flow is first established and then utilized to locate the stagnation/ equilibrium points. The classification of equilibrium points is made by employing the theory of dynamical systems. It is observed that the inertial effects modify the location and number of equilibrium points and hence give rise to additional bifurcations. This research focuses on the alterations in the topology of flow within a pump when there is partial occlusion. The insights gained from this research could be beneficial in guaranteeing the smooth flow of fluids through the pump with no risk of entrapment from entry to exit. Moreover, if trapping does occur, this information can be helpful in identifying the exact region within the pump where the fluid is being trapped. [ABSTRACT FROM AUTHOR]

Many problems in fluid mechanics describe the change in the flow under the effect of electromagnetic forces. The present study explores the behaviour of an electric conducting, Newtonian fluid flow applying the magnetohydrodynamics (MHD) and ferrohydrodynamics (FHD) principles. The physical problems for such flows are formulated by the Navier–Stokes equations with the conservation of mass and energy equations, which constitute a coupled non-linear system of partial differential equations subject to analogous boundary conditions. The numerical solution of such physical problems is not a trivial task due to the electromagnetic forces which may cause severe disturbances in the flow field. In the present study, a numerical algorithm based on a finite volume method is developed for the solution of such problems. The basic characteristics of the method are, the set of equations is solved using a simultaneous direct approach, the discretization is achieved using the finite volume method, and the solution is attained solving an implicit non-linear system of algebraic equations with intense source terms created by the non-uniform magnetic field. For the validation of the overall algorithm, comparisons are made with previously published results concerning MHD and FHD flows. The advantages of the proposed methodology are that it is direct and the governing equations are not manipulated like other methods such as the stream function vorticity formulation. Moreover, it is relatively easily extended for the study of three-dimensional problems. This study examines the Hartmann flow and the fluid flow with FHD principles, that formulate MHD and FHD flows, respectively. The major component of the Hartmann flow is the Hartmann number, which increases in value the stronger the Lorentz forces are, thus the fluid decelerates. In the case of FHD fluid flow, the major finding is the creation of vortices close to the external magnetic field source, and the stronger the magnetic field of the source, the larger the vortices are. [ABSTRACT FROM AUTHOR]

This research investigates the numerical analysis of magnetohydrodynamic (MHD) mixed convection flow and heat transfer within a bottom lid-driven cavity filled with water-alumina (Al2O3) nanofluid. The cavity's sidewalls exhibit a wavy profile and are maintained at distinct temperatures. Cavity domain exhibit distinct free and force convections. These wavy walls, characterized by zigzag shapes determined by various wave amplitudes and their ratios (wave form), create a dynamic thermal environment. The top and bottom surfaces remain flat and well-insulated, while forced convection is induced by the drag of the bottom wall from left to right at a constant speed. Additionally, the bottom wall is subjected to a vertical magnetic field. The system of equations is discretized using the finite difference method. The numerical solutions are derived by the Gauss-Seidel iterative method. The study primarily focuses on investigating the effects of key parameters, including the wavy wall geometry, solid volume fraction (0 ≤ φ ≤ 0.0003), Rayleigh number (10³≤ Ra ≤105), and Hartmann number (0 ≤ Ha ≤0.6). Numerical solutions are computed across different ranges of these parameters, and the obtained results are successfully validated against previous numerical studies. The findings reveal that higher Hartmann numbers and solid volume fractions lead to lower circulation rates and Nusselt numbers. Convection is markedly enhanced with higher amplitude and its ratios of the wavy sidewalls. The combined two-sinusoidal function with the wave amplitudes of 2.5 and 0.47 of provides the highest mean Nusselt numberof3.204 with the highest dimensionless stream function of 1.638. These results highlight the significant influence of the wave form on both flow and temperature distributions. [ABSTRACT FROM AUTHOR]

We study connections among polynomials, differential equations, and streams over a field , in terms of algebra and coalgebra. We first introduce the class of (F,G)-products on streams, those where the stream derivative of a product can be expressed as a polynomial function of the streams and their derivatives. Our first result is that, for every (F,G)-product, there is a canonical way to construct a transition function on polynomials such that the resulting unique final coalgebra morphism from polynomials into streams is the (unique) commutative -algebra homomorphism—and vice versa. This implies that one can algebraically reason on streams via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G)-product. Finally, we extend this algorithm to solve a more general problem: finding all valid polynomial equalities that fit in a user specified polynomial template. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new concept of the stream function and derive the equation for the stream function in the three-dimensional case. To construct a basis in the space of solutions of the NavierStokes system, we solve an auxiliary spectral problem for the bi-Laplacian with Dirichlet conditions on the boundary. Then, using the formulas employed for introducing the stream function, we find a system of functions forming a basis in the space of solutions of the Navier-Stokes system. It is worth noting that this basis can be utilized for the approximate solution of direct and inverse problems for the Navier-Stokes system, both in its linearized and nonlinear forms. The main idea of this work can be summarized as follows: instead of changing the boundary conditions (which remain unchanged), we change the differential equations for the stream function with a spectral parameter. As a result, we obtain a spectral problem for the bi-Laplacian in the domain represented by a three-dimensional unit sphere, with Dirichlet conditions on the boundary of the domain. By solving this problem, we find a system of eigenfunctions forming a basis in the space of solutions to the Navier-Stokes equations. Importantly, the boundary conditions are preserved, and the continuity equation for the fluid is satisfied. It is also noteworthy that, for the three-dimensional case of the Navier-Stokes system, an analogue of the stream function was previously unknown.

In order to explore the role of the changes in the strength and weakness of the South Asian monsoon on the vertical circulation of the Qinghai-Xizang （Tibetan） Plateau， the South Asian monsoon index WYI was calculated using the high-resolution JRA55 reanalysis data for the past 30 years from 1990 to 2019 and the strong and weak years of the South Asian monsoon were selected， and the effects of the changes in the strength and weakness of the South Asian monsoon on the vertical circulation of the Qinghai-Xizang （Tibetan） Plateau were analyzed by comparing the three different periods of the pre-monsoon outbreak， outbreak period and late outbreak.The following main conclusions were obtained： （1） The strength and extent of updrafts on the southern side of the plateau in May in WYI strong years are greater than in weak years， the intensity of updrafts on the main and southern sides of the plateau increases in July， and the difference between strong and weak years is not obvious.（2） In May and October， both mass and energy converge to the plateau from the south and north sides of the plateau， while in July， it turns to dissipate mass and energy from the plateau to the north and south sides.the intensity of the upper-level radiation dispersion in the south side of the plateau in the WYI strong year is greater than that in the weak year， and the difference of the irradiation intensity in the lower level is smaller.The difference in the intensity of energy and mass transfer between WYI strong and weak years is greatest in May and least in July.（3） In the WYI strong year， the 100 hPa South Asian high pressure shifts to the western type in October， and the 500 hPa West Pacific subalpine lifts northward， so the wind field on the southern side of the plateau shifts and raises the high altitude temperature， while in the WYI weak year this phenomenon occurs in May， indicating that the South Asian monsoon changes the horizontal circulation situation at high altitude and thus affects the vertical circulation of the plateau.The surface temperature of the main body of the plateau in May， July and October in WYI strong years is greater than that in weak years， indicating that under the influence of the warm and humid airflow brought by the South Asian monsoon， the updraft from the southern side of the plateau lifts to the plateau， and the surface temperature of the plateau increases significantly， and the surface heat of the plateau increases which in turn affects the vertical circulation of the plateau.