Numerical Study on 3D MHD Darcy-Forchheimer Flow Caused by Gyrotactic Microorganisms of the Bio-Convective Casson Nanofluid across a Stretched Sheet

A review of the literature revealed that nanofluids are more effective in transferring heat than conventional fluids.Since there are significant gaps in the illumination of existing methods for enhancing heat transmission in nanomaterials,a thorough investigation of the previously outlined models is essential.The goal of the ongoing study is to determine whether the microscopic gold particles that are involved in mass and heat transmission drift in freely.The current study examines heat and mass transfer on 3D MHD Darcy–Forchheimer flow of Casson nanofluid-induced bio-convection past a stretched sheet.The inclusion of the nanoparticles is a result of their peculiar properties,such as remarkable thermal conductivity,which are important in heat exchangers and cutting-edge nanotechnology.The gyrotactic microorganisms must be included to prevent the potential deposition of minute particles.The proposed flow dynamics model consists of an evolving nonlinear system of PDEs,which is subsequently reduced to a system of dimensionless ODEs utilizing similarity approximations.MATLAB software was utilized to create an effective code for the Runge-Kutta technique using a shooting tool to acquire numerical results.This method is extensively used to solve these issues since it is accurate to fourth order,efficient,and affordable.The influence of submerged factors on the velocity,temperature,concentration,and density of motile microorganisms is shown in the figures.Additionally,tables and bar charts are used to illustrate the physical characteristics of the Nusselt and Sherwood numbers for the densities of both nanoparticles and motile microorganisms.The dimensionless velocities are observed declining when the casson,magnetic,porosity,and forchheimer parameters grow,whereas the dimensionless temperature and concentration rise as the thermophoresis parameter rises.This work provides insights into practical applications such nanofluidic,energy conservation,friction reduction,and power generation.Furthermore,in a concentration field,the Brownian and thermophoresis parameters exhibit very distinct behaviours.However,the work makes a significant point that the flow of a Casson fluid including nanoparticles can be regulated by appropriately modifying the Casson parameter,thermophoresis parameter,and Brownian motion parameter.

裂隙-孔隙双重介质Darcy-Forchheimer耦合流动模拟方法及工程应用

针对裂隙-孔隙双重介质非线性渗流问题,采用压力交换函数描述孔隙Darcy渗流和裂隙Forchheimer渗流耦合特性,推导了渗流方程有限体积的数值格式,并编制了相应的计算程序。通过与单裂隙和相交裂隙渗流的Frih和Arraras解对比,验证了新方法的合理性。对富水深埋裂隙型围岩隧道非线性渗流问题的计算表明,所提算法对复杂裂隙系统问题具有很强的适用性。进一步分析隧道围岩渗流特征:越接近隧道位置,水压梯度越大,流量也越大;隧道周围水压梯度呈现“底部大,顶部小”的特点,最大相差2.5倍,因此隧道底部的流量大于顶部流量;裂隙方向均匀性和密度是影响隧道围岩水力特性的重要因素。在一定水力梯度下,裂隙方向越集中于水力梯度方向且密度越大时,围岩导水性越大,隧道流量越大,越容易发生涌水事故。研究成果为裂隙型围岩隧道防水设计及工程实践提供参考。

Stability analysis and modeling for the three-dimensional Darcy-Forchheimer stagnation point nanofluid flow towards a moving surface

In this research,the three-dimensional(3D)steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed.The disturbance in the porous medium has been characterized by the Darcy-Forchheimer relation.The slip for viscous fluid is considered.The energy equation is organized in view of radiative heat flux which plays an important role in the heat transfer rate.The governing flow expressions are first altered into first-order ordinary ones and then solved numerically by the shooting method.Dual solutions are obtained for the velocity,skin friction coefficient,temperature,and Nusselt number subject to sundry flow parameters,magnetic parameter,Darcy-Forchheimer number,thermal radiation parameter,suction parameter,and dimensionless slip parameter.In this research,the main consideration is given to the engineering interest like skin friction coefficient(velocity gradient or surface drag force)and Nusselt number(temperature gradient or heat transfer rate)and discussed numerically through tables.In conclusion,it is noticed from the stability results that the upper branch solution(UBS)is more reliable and physically stable than the lower branch solution(LBS).

Hybrid nanomaterial flow and heat transport in a stretchable convergent/divergent channel:a Darcy-Forchheimer model

The flow behavior in non-parallel walls is an important factor of any physical model including cavity flow and canals, which is applicable for diverging/converging channel. The present communication explains that the flow of the hybrid nanomaterial subjected to the convergent/divergent channel has non-parallel walls. It is assumed that the hybrid nanomaterial movement is in the porous region. A Darcy-Forchheimer medium of porosity is considered to interpret the porosity features. A useful similarity function is adopted to get the strong ordinary coupled equations. Numerical solutions are achieved through the Runge-Kutta-Fehlberg(RKF) fourth-fifth order method, and they are validated with the existing results. Physical nature of the involving constraints is reported with the help of plots. It is explored that the velocity of divergent channel decreases, and convergent channel enhances for the higher solid volume faction. Further, the presence of inertia coefficient and porosity parameter amplifies the velocity at the wall.

Features of Darcy-Forchheimer flow of carbon nanofluid in frame of chemical species with numerical significance

Present work reports chemically reacting Darcy-Forchheimer flow of nanotubes.Water is utilized as base liquid while carbon nanotubes are considered nanomaterial.An exponential stretchable curved surface flow is originated.Heat source is present.Xue relation of nanoliquid is employed to explore the feature of CNTs (single and multi-wall).Transformation technique is adopted in order to achieve non-linear ordinary differential systems.The governing systems are solved numerically.Effects of involved parameters on flow,temperature,concentration,heat transfer rate (Nusselt number) with addition of skin friction coefficient are illustrated graphically.Decay in velocity is noted with an increment in Forchheimer number and porosity parameter while opposite impact is seen for temperature.Moreover,role of MWCNTs is prominent when compared with SWCNTs.

Computational Analysis of Heat and Mass Transfer in Magnetized Darcy-Forchheimer Hybrid Nanofluid Flow with Porous Medium and Slip Effects

A computational analysis of magnetized hybrid Darcy-Forchheimer nanofluid flow across a flat surface is presented in this work.For the study of heat and mass transfer aspects viscous dissipation,activation energy,Joule heating,thermal radiation,and heat generation effects are considered.The suspension of nanoparticles singlewalled carbon nanotubes(SWCNTs)and multi-walled carbon nanotubes(MWCNTs)are created by hybrid nanofluids.However,single-walled carbon nanotubes(SWCNTs)produce nanofluids,with water acting as conventional fluid,respectively.Nonlinear partial differential equations(PDEs)that describe the ultimate flow are converted to nonlinear ordinary differential equations(ODEs)using appropriate similarity transformation.The ODEs are dealt with numerically by means of MATLAB’s inbuilt routine function bvp4c.Velocity,temperature,and concentration profiles are explained pictorially whereas Sherwood number,local skin friction coefficient,and Nusselt number values are represented through bar charts.Thermal radiation and activation parameters shows direct impact on flow field.Furthermore,hybrid nanofluid admits a higher magnitude of velocity and temperature than nanofluid,but the concentration profile exhibits the opposite trend.The notable findings of the present investigation have significant applications in heat combustion and cooling chambers,space technology,the ceramics industry,paint and conductive coatings,bio-sensors,and many more.

Darcy-Forchheimer flow by rotating disk with partial slip

The viscous dissipation and heat transfer in the Darcy-Forchheimer flow by a rotating disk are examined. The partial slip conditions are invoked. The optimal series solutions are computed via the optimal homotopic analysis method(OHAM). The thermophoresis and Brownian motions are studied. The Darcy-Forchheimer relation characterizes the porous space. The roles of influential variables on the physical quantities are graphically examined. A reduction in the local Nusselt number is observed through thermophoresis and thermal slip parameters. The local Sherwood number depicts an increasing trend for the higher Brownian motion and concentration slip parameters.

Darcy-Forchheimer flow of variable conductivity Jeffrey liquid with Cattaneo-Christov heat flux theory

The role of the Cattaneo-Christov heat flux theory in the two-dimensional laminar flow of the Jeffrey liquid is discussed with a vertical sheet. The salient feature in the energy equation is accounted due to the implementation of the Cattaneo-Christov heat flux. A liquid with variable thermal conductivity is considered in the Darcy-Forchheimer porous space. The mathematical expressions of momentum and energy are coupled due to the presence of mixed convection. A highly nonlinear coupled system of equations is tackled with the homotopic algorithm. The convergence of the homotopy expressions is calculated graphically and numerically. The solutions of the velocity and temperature are expressed for various values of the Deborah number, the ratio of the relaxation time to the retardation time, the porosity parameter, the mixed convective parameter, the Darcy-Forchheimer parameter, and the conductivity parameter. The results show that the velocity and temperature are higher in Fourier＇s law of heat conduction cases in comparison with the Cattaneo-Christov heat flux model.

Numerical treatment for Darcy-Forchheimer flow of carbon nanotubes due to an exponentially stretching curved surface

This article gives a numerical report to two dimensional(2D)Darcy-Forchheimer flow of carbon-water nanofluid.Flow is instigated by exponential extending curved surface.Viscous liquid in permeable space is described by Darcy-Forchheimer.The subsequent arrangement of partial differential equations is changed into ordinary differential framework through proper transformations.Numerical arrangements of governing frameworks are set up by NDSolve procedure.Outcomes of different sundry parameters on temperature and velocity are examined.Skin friction and heat transfer rate are also shown and inspected.

Upshot of ohmically dissipated Darcy-Forchheimer slip flow of magnetohydrodynamic Sutterby fluid over radiating linearly stretched surface in view of Cash and Carp method

The present work concerns the momentum and heat transmission of the electro-magnetohydrodynamic (E-MHD) boundary layer Darcy-Forchheimer flow of a Sutterby fluid over a linear stretching sheet with slip. The nonlinear equations for the proposed model are analyzed numerically. Suitable techniques are used to transform the coupled nonlinear partial differential equations (PDEs) conforming to the forced balance law, energy, and concentration equations into a nonlinear coupled system of ordinary differential equations (ODEs). Numerical solutions of the transformed nonlinear system are obtained using a shooting method, improved by the Cash and Carp coefficients. The influence of important physical variables on the velocity, the temperature, the heat flux coefficient, and the skin-friction coefficient is verified and analyzed through graphs and tables. From the comprehensive analysis of the present work, it is concluded that by intensifying the magnitude of the Hartmann number, the momentum distribution decays, whereas the thermal profile of fluid increases. Furthermore, it is also shown that by aug- menting the values of the momentum slip parameter, the velocity profile diminishes. It is found that the Sutterby fluid model shows shear thickening and shear thinning behaviors. The momentum profile shows that the magnitude of velocity for the shear thickening case is dominant as compared with the shear thinning case. It is also demonstrated that the Sutterby fluid model reduces to a Newtonian model by fixing the fluid parameter to zero. In view of the limiting case, it is established that the surface drag in the case of the Sutterby model shows a trifling pattern as compared with the classical case.

Navier’s slip condition on time dependent Darcy-Forchheimer nanofluid using spectral relaxation method

In industrial applications involving metal and polymer sheets, the flow situation is strongly unsteady and the sheet temperature is a mixture of prescribed surface temperature and heat flux. Further, a proper choice of cooling liquid is also an important component of the analysis to achieve better outputs. In this paper, we numerically investigate Darcy-Forchheimer nanoliquid flows past an unsteady stretching surface by incorporating various effects, such as the Brownian and thermophoresis effects, Navier’s slip condition and convective thermal boundary conditions. To solve the governing equations, using suitable similarity transformations, the nonlinear ordinary differential equations are derived and the resulting coupled momentum and energy equations are numerically solved using the spectral relaxation method. Through the systematically numerical investigation, the important physical parameters of the present model are analyzed. We find that the presence of unsteadiness parameter has significant effects on velocity, temperature, concentration fields, the associated heat and mass transport rates. Also, an increase in inertia coefficient and porosity parameter causes an increase in the velocity at the boundary.

Numerical simulation for Darcy-Forchheimer 3D rotating flow subject to binary chemical reaction and Arrhenius activation energy

Three-dimensional Darcy-Forchheimer nanoliquid flow in the presence of rotating frame and activation energy is inspected.Flow is developed through linearly stretching of the surface.Convection of heat and mass exchange is given due consideration.The novel characteristics in regards to Brownian dispersion and thermophoresis are retained.The variation in partial differential framework (PDEs) to nonlinear ordinary differential framework (ODEs) is done through reasonable transformations.Governing differential frameworks have been computed in edge of NDSolve.Discussion regarding thermal field and concentration distribution for several involved parameters is pivotal part.Physical amounts like surface drag coefficients,transfer of heat and mass rates are portrayed by numeric esteems.It is noticed that impacts of porosity parameter and Forchheimer number on the thermal and concentration fields are quite similar.Both temperature and associated thermal layer thickness are enhanced for larger porosity parameter and Forchheimer number.Temperature and concentration fields exhibit similar trend for the higher values of rotational parameter.Effects of thermal and concentration Biot numbers on the temperature and concentration fields are qualitatively similar.Higher Prandtl and Schmidt numbers correspond to stronger temperature and concentration fields.Larger nondimensional activation energy,temperature difference parameter and fitted rate constant yield weaker concentration field.Brownian motion parameter for temperature and concentration has reverse effects while similar trend is observed via thermophoresis parameter.

Darcy-Forchheimer flow with nonlinear mixed convection

An analysis of the mixed convective flow of viscous fluids induced by a nonlinear inclined stretching surface is addressed.Heat and mass transfer phenomena are analyzed with additional effects of heat generation/absorption and activation energy,respectively.The nonlinear Darcy-Forchheimer relation is deliberated.The dimensionless problem is obtained through appropriate transformations.Convergent series solutions are obtained by utilizing an optimal homotopic analysis method(OHAM).Graphs depicting the consequence of influential variables on physical quantities are presented.Enhancement in the velocity is observed through the local mixed convection parameter while an opposite trend of the concentration field is noted for the chemical reaction rate parameter.

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional(2 D)flow of fourth-grade material over a stretching surface is explored.The flow is created via a stretching surface.A Darcy-Forchheimer(D-F)porous medium is considered in the flow field.The heat transport is examined with the existence of the Cattaneo-Christov(C-C)heat flux.The fourth-grade material is electrically conducting subject to an applied magnetic field.The governing partial differential equations(PDEs)are reduced into ordinary differential equations(ODEs)by appropriate transformations.The solutions are constructed analytically through the optimal homotopy analysis method(OHAM).The fluid velocity,temperature,and skin friction are examined under the effects of various involved parameters.The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter,porosity parameter,and Forchheimer number.The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number,magnetic parameter,and thermal relaxation parameter.Furthermore,the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters,thermal relaxation parameter,and Forchheimer number.

A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS

In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem.The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order.The concentration equation is solved by the method of mixed finite element,where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method.The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation.In actual computations the characteristics adopts a large time step without any loss of accuracy.The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux.In order to derive the optimal 3/2-order error estimate in L^(2) norm,a post-processing technique is included in the approximation to the scalar unknowns.Numerical experiments are illustrated finally to validate theoretical analysis and efficiency.This method can be used to solve such an important problem.

多孔介质Darcy-Forchheimer不可压缩混溶驱动问题的二重网格混合元算法

研究采用二重网格混合有限元法求解多孔介质中不可压缩混相驱替问题,其中,该问题的速度与压力的关系由Darcy-Forchheimer定律描述。主要目的是将在细网格上求解一个大规模非线性系统转换为在粗网格上求解一个小规模非线性系统以及在细网格上求解一个线性系统。求解非线性系统需要用迭代法,而转换为线性系统后,只需要解线性代数方程组,可以大大提高运算的速度。在本文中,我们用混合元逼近速度和压力,用一般的有限元逼近组分浓度。在本文的数值实验中,我们验证了细网格上的误差估计,以及计算效率。

Significance of Darcy-Forchheimer Porous Medium in Nanofluid Through Carbon Nanotubes

This article manages Darcy-Forchheimer 3D flow of water based carbon nanomaterial(CNTs).A bidirectional nonlinear stretchable surface has been utilized to make the flow.Disturbance in permeable space has been represented by Darcy Forchheimer(DF)expression.Heat transfer mechanism is explored through convective heating.Outcomes for SWCNT and MWCNT have been displayed and compared.The reduction of partial differential framework into nonlinear common differential framework is made through reasonable variables.Optimal series scheme is utilized for arrangements advancement of associated flow issue.Optimal homotopic solution expressions for velocities and temperature are studied through graphs by considering various estimations of physical variables.Moreover surface drag coefficients and heat transfer rate are analyzed through plots.

Darcy-Forchheimer relation in Casson type MHD nanofluid flow over non-linear stretching surface

Present article aims to discuss the characteristics of Casson type nanofluid maintained to flow through porous medium over non-linear stretching surface in the perspective of heat and mass transfer developments.A Casson type incompressible viscous nanofluid passes through the given porous medium via Darcy-Forchheimer relation.Slip boundary conditions are used for velocity,temperature and concentration of the nanoparticles.Brownian diffusion and thermophoresis is attended.An induced magnetic field effect is involved to accentuate the thermo-physical characteristics of the nanofluid.The model incorporates boundary layer formulations and small magnetic Reynolds for practical validity.A fourth order Runge-Kutta(RK)scheme is enforced to solve the system numerically.Graphs are prepared for various progressive values of non-dimensionalized parameters whereas;variation in wall drag factor,heat and mass transfer rates is analyzed through numerical data.Results indicate that momentum boundary layer reduces for stronger inertial impact and the resistance offered by the porous media to the fluid flow.Temperature is found as a progressive function for the Brownianmotion factor and thermophoresis.The magnitude of wall drag factor,heat transfer and masstransfer rates shows reduction for progressive values of slip parameters.

Impact of melting heat transfer in the time-dependent squeezing nanofluid flow containing carbon nanotubes in a Darcy-Forchheimer porous media with Cattaneo-Christov heat flux

This study aims to investigate the time-dependent squeezing of nanofluid flow, comprising carbon nanotubes of dual nature, e.g. single-walled carbon nanotubes, and multi-walled carbon nanotubes,between two parallel disks. Numerical simulations of the proposed novel model are conducted,accompanied by Cattaneo-Christov heat flux in a Darcy-Forchheimer permeable media. Additional impacts of homogeneous–heterogeneous reactions are also noted, including melting heat. A relevant transformation procedure is implemented for the transition of partial differential equations to the ordinary variety. A computer software-based MATLAB function, bvp4c, is implemented to handle the envisioned mathematical model. Sketches portraying impacts on radial velocity, temperature, and concentration of the included parameters are given, and deliberated upon. Skin friction coefficient and local Nusselt number are evaluated via graphical illustrations. It is observed that the local inertia coefficient has an opposite impact on radial velocity and temperature field. It is further perceived that melting and radiation parameters demonstrate a retarding effect on temperature profile.

Darcy-Forchheimer mangetized flow based on differential type nanoliquid capturing Ohmic dissipation effects

Hydromagnetic nanoliquid establish an extraordinary category of nanoliquids that unveil both liquid and magnetic attributes.The interest in the utilization of hydromagnetic nanoliquids as a heat transporting medium stem from a likelihood of regulating its flow along with heat transportation process subjected to an externally imposed magnetic field.This analysis reports the hydromagnetic nanoliquid impact on differential type(second-grade)liquid from a convectively heated extending surface.The well-known Darcy-Forchheimer aspect capturing porosity characteristics is introduced for nonlinear analysis.Robin conditions elaborating heat-mass transportation effect are considered.In addition,Ohmic dissipation and suction/injection aspects are also a part of this research.Mathematical analysis is done by implementing the basic relations of fluid mechanics.The modeled physical problem is simplified through order analysis.The resulting systems(partial differential expressions)are rendered to the ordinary ones by utilizing the apposite variables.Convergent solutions are constructed employing homotopy algorithm.Pictorial and numeric result are addressed comprehensively to elaborate the nature of sundry parameters against physical quantities.The velocity profile is suppressed with increasing Hartmann number(magnetic parameter)whereas it is enhanced with increment in material parameter(second-grade).With the elevation in thermophoresis parameter,temperature and concentration of nanoparticles are accelerated.