Boundary Layer Partial Differential Equations Research Articles

Heat and mass transfer analysis of conducting non-Newtonian nanofluid flows over an elongating sheet with a non-uniform heat source

Nanofluids help in many fields to improve the performance of thermal systems by augmenting heat transfer rates through their thermophysical properties. The performance of the nanofluids with various base fluids may be different. Therefore, the study and comparison of behaviors of various nanofluids are useful in several applications such as fuel as a coolant in automobiles, and in medical and electronic equipment to reduce the thermal resistance. This research proposed a novel model to investigate the flow behavior of three different nanofluids over an elongating surface in the presence of a non-uniform heat source and thermal radiation effects. This investigation describes how the considered nanofluids behave in the presence of a transverse magnetic field, and other effects. The proposed governing boundary layer partial differential equations (PDEs) are reformed into a system of nonlinear ordinary differential equations (ODEs) by introducing the proper similarity transformation. The finalized equations are solved numerically with the help of the ND solve package in Mathematica software. We intended how the fluid flow and heat transfer are affected by non-dimensional controlling factors with the help of graphics. Further, the calculations and discussions are accompanied by the numerical values of the skin friction coefficient and heat and mass transfer rates. According to the current findings, the Maxwell nanofluid exhibits superior performance in velocity, and the Oldroyd-B nanofluid shows more concentration and less temperature. As a special case, the results of this investigation are compared with the existing results, and found a good agreement between the results.

Calculating the entropy generation of a Bingham plastic fluid flow due to a heated rotating disk

We present the numerical study of the flow on a heated rotating disk for Bingham plastic fluid. Thermo-physical properties and viscous dissipation are assumed with the influence of entropy generation. This kind of study has a significant attentiveness in the fields of applied engineering for their increased mathematical advantages. We analyzed the solutions of fluid flow and entropy phenomenon for the Bingham fluid. The similarity solution was firstly presented by von Kármán for Newtonian fluid, it is found to be beneficial for the Bingham plastic fluid by transforming the boundary layer partial differential equations into ordinary differential equations. Shooting method is used to crack the dimensionless ordinary differential equations. The Bingham model is momentous for unlimited high shear rates for this we suggest these numerical data provide accurate description of the Bingham fluid and entropy minimization near a rotating disk. The outcomes exhibits that the azimuthal velocity profiles upsurges for upward values of Bingham fluid, and the similar behavior for tangential radial velocity is perceived. The temperature curve is noted to be enhanced for higher magnitudes of Eckert number and thermal conduction parameter. The soret number and diffusivity coefficient creates increment in concentration curve. The entropy function in the flow system enhances for up surging the inputs of Bingham fluid coefficient and thermophoresis diffusion. The results are locally similar for various parameters. We have also included the comparison analysis with published works.

Note on the Numerical Solutions of Unsteady Flow and Heat Transfer of Jeffrey Fluid Past Stretching Sheet with Soret and Dufour Effects

A numerical investigation of unsteady boundary layer flow with heat and mass transfer of non-Newtonian fluid model, namely, Jeffrey fluid subject, to the significance of Soret and Dufour effects is carried out by using the local nonsimilarity method and homotopy analysis method. An excellent agreement in the numerical results obtained by both methods is observed and we establish a new mathematical approach to obtain the solutions of unsteady-state flow with heat and mass transfer phenomenons. Similarity transformation is applied to governing boundary layer partial differential equations to obtain the set of self-similar, nondimensional partial differential equations. Graphical results for different emerging parameters are discussed. The dimensionless quantities of interest skin friction coefficient, Sherwood number, and Nusselt number are discussed through tabulated results. The main novelty of the current work is that the average residual error of the mth-order approximation of the OHAM scheme for steady-state solution is decreased for higher-order approximation. Further, a rapid development of the boundary layer thickness with the increasing values of dimensionless time τ is observed. It is noted that for large values of τ, the steady state in the flow pattern is gained. It is worth mentioning that the magnitude of Sherwood number is increased with the increasing values of Schmidt number Sc and Dufour number Df. The magnitude of local Nisselt number is increased for the increasing values of Soret number, Sr.

Flow and Heat Transfer of Power Law Fluid over Horizontal Stretching Cylinder with Partial Slip Condition and Thermal Radiation

The aim of the present study is to investigate the boundary layer flow of power-law fluid over the horizontal stretching cylinder. The temperature-dependent thermal conductivity of the power-law fluid is considered. Combined effects of constant thermal conductivity and viscous dissipation are analyzed in heat transfer. The relevant boundary layer partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by using suitable transformations. These nonlinear ordinary differential equations are solved by the BVP4C method using MATLAB. The accuracy of computed results is checked by comparing them with existing literature. To discuss the effects of flow parameters on velocity and temperature profiles graphs with numeric values are developed. The constant thermal conductivity and radiation effects are enhanced corresponding to a higher fluid temperature. The thermal boundary layer thickness is enhanced with the increase in Eckert number. The effect of all physical parameters on skin friction coefficient and local Nusselt number are discussed through tables and graphs.

Thermal efficiency and stability of copper-alumina nanoparticles with Darcy-Forchheimer effects

The main objective of the current study is to find dual branches with stability analysis of two-dimensional (2D) flow and heat transport of water-based hybrid nanofluid on a linear shrinking/stretching sheet with effects of the thermal radiation and convective condition. The effect of Darcy-Forchheimer has also been considered. A permeable surface is referred to sustain a shrinking flow by a sufficient mass suction of the wall. The system of governing boundary layer partial differential equations (PDEs) is transformed into ODEs by employing similarity transformation. The bvp4c solver is used as a numerical computing tool in the MATLAB software to solve the resultant ODEs and the outcomes are offered in graphs and tables. Our computational results revealed dual branches of solutions possessed by the shrinking parameter i.e. . The increment number of copper volume fraction can increase the range of two branches, while the thermal radiation and Biot parameters have zero effect on delaying the separation of boundary layers along with porosity and inertia parameters. Increased suction and Biot parameters can raise the rate of heat transfer while the opposite effect is achieved by enhancing the magnitude of .

Numerical investigation of a convective hybrid nanofluids around a rotating sheet

Hybrid nanofluids are formulated with various kinds of base fluids. They are designed to provide good heat transfer performance. They can achieve this by dispersing various kinds of nanoparticles in the base materials. This new technology of formulating hybrid nanofluids has a wide range of applications in various industries such as solar energy, medical equipment, and aerospace. Keeping these applications in view, this study provides an insight into the effects of convective heat transport on a Hybrid nanofluid, across a rotating sheet with a variable heat source. In this investigation, the governing boundary layer partial differential equations were modified into the ordinary differential equations, by using the proper similarity transformations. Later, they were solved numerically, with the support of the Lobatto IIIA technique in MATLAB. The influence of the Richardson number on flow parameters was studied, and it was discovered that increasing Ri increases the velocity while decreasing temperature and concentration profiles. The impact of various other flow parameters on the flow fields and also on the behavior of Nusselt number, coefficient skin friction, and Sherwood number were studied and represented graphically. The outcomes were found to be in excellent accord when compared with quoted studies.

Computational analysis of hydromagnetic boundary layer stagnation point flow of nano liquid by a stretched heated surface with convective conditions and radiation effect

In this study, the boundary layer phenomena for stagnation point flow of water-based nanofluids is being observed with the upshot of MHD and convective heating on a nonlinear stretching surface. To develop a fundamental flow model, a boundary layer approximation is done, which signifies time-dependent momentum, energy, and concentration expressions. Through a proper transformation framework, the modeled boundary layer partial differential equations (PDEs) have been diminished to a dimensionless system of nonlinear ordinary differential equations (ODEs). With the assistance of a built-in algorithm in Mathematica software, the fundamental flow equations are analyzed numerically by imposing a shooting technique explicitly. A stability and convergence analysis were also unveiled, and the ongoing investigation was found to have converged. The effect of mathematical abstractions on velocity, energy, and concentration is plotted and discussed. The influence of skin-friction and Nusselt number on the sheet are debated for the various values of important parameters.

Cattaneo-Christov Heat Flux Model of Eyring Powell fluid along with Convective Boundary Conditions

Heat and mass transfer effects in three-dimensional mixed convection flow of Eyring Powell over an exponentially stretching surface with convective boundary conditions are inspected. Cattaneo-Christov Heat Flux model is a modified version of the classical Fourier’s law that takes into account the interesting aspect of thermal relaxation time. First order chemical reaction effects are also taken into account. Similarity transformations are invoked to reduce the leading boundary layer partial differential equations into the ordinary differential equations. The nonlinear, coupled ordinary differential equations with boundary conditions has been analyzed numerically by using Finite Element Method. Similarity transformations are invoked to reduce the leading boundary layer partial differential equations into the ordinary differential equations.

Thermal Radiation Effects on MHD Unsteady Couette Flow Heat and Mass Transfer Free Convective in Vertical Channels due to Ramped and Isothermal Temperature

Numerical investigations were carried out to study thermal radiation effects on magneto-hydrodynamics (MHD) unsteady Couette flow heat mass transfer free-convective in vertical channels due to ramped and isothermal temperature. The governing coupled non-linear partial differential equations of the flow were transformed into non-dimensional form using suitable dimensional quantities. Finite element method (FEM) was employed to find numerical solution of the dimensionless governing coupled boundary layer partial differential equations. The expressions of velocity, temperature, concentration, skin friction, Nusselt number as well as Sherwood number have been obtained and discussed using line graph. From the outcome of the result it was revealed that, increase of porosity parameter K, ratio of mass transfer parameter N, Time parameter t, Eckert number Ec enhances the velocity and temperature while reverse is the case with the with increase of Magnetic parameter M, Radiation parameter tern R and Prandtl number Pr. At y = 0, the fluid skin friction gets enlarged with increase in porosity parameter K, Nusselt number gets increased with increase of Prandtl number Pr and Sherwood number gets boosted with increase of Eckert number Ec. Similarly, at y = 1 skin friction gets enhanced with increase of porosity parameter K, Nusselt number diminishes with increase of Prandtl number Pr and Sherwood number gets enlarged with increase of Eckert number Ec.

Mixed convective 3D flow of Maxwell nanofluid induced by stretching sheet: Application of Cattaneo-Christov theory

The present study invokes the application of Cattaneo-Christov theory for the thermal analysis in the buoyancy driven three dimensional flow of Maxwell nanofluid. The flow is induced above the vertical bidirectional stretching sheet. The phenomena of thermophoresis and Brownian diffusion of nanoparticles in the flow Maxwell liquid are deliberated with the help of Buongiorno model for nanofluid. The physical problem is formulated in the form of boundary layer partial differential equations (PDEs). Moreover, suitable ansatz for flow mechanism are employed to reduce the governing PDEs into the non-linear ordinary differential equations (ODEs). The flow mechanism of Maxwell fluid along with energy transport is analyzed in the form of homotopic solutions of the governing ODEs. The outcomes are presented graphically and discussed with physical explanation. The analysis revealed that both buoyancy and mixed convection parameters enhanced the [Formula: see text]-component of velocity field but declined the [Formula: see text]-component. Moreover, in assisting mode these two parameters also increased the thermal and solutal energy transport in nanofluid. It is noted that the thermophorectic force boosts up the thermal energy transport in the flow in the presence of thermal relaxation phenomenon. The validation of the present results are confirmed through tabular data with some previous studies.

HEAT AND MASS TRANSFER ON UNSTEADY MHD FLOW THROUGH AN INFINITE OSCILLATING VERTICAL POROUS SURFACE

Analytical solutions of an unsteady magnetohydrodynamic (MHD) flow with heat and mass transfer characteristics of an incompressible, viscous, electrically conducting, and Boussinesq fluid over a vertical oscillating plate embedded in a Darcian porous medium in the presence of a thermal radiation are presented. The fluid considered here is a gray, absorbing/emitting radiating, but nonscattering medium. At time t > 0, the plate temperature and concentration near the plate are raised linearly with time t. The dimensionless governing coupled unsteady boundary layer partial differential equations are solved by the Laplace transform technique. An increase in permeability parameter (K) is found to increase fluid velocities and shear stress in the regime. Also it was found that when the conduction-radiation parameter (R) increased, the fluid velocity and the temperature profiles decreased. Applications of the study are found in materials processing and solar energy collector systems.

A comparison of bivariate pseudospectral methods for nonlinear systems of steady nonsimilar boundary layer partial differential equations

A comparison of bivariate pseudospectral methods for nonlinear systems of steady nonsimilar boundary layer partial differential equations

Model of Casson fluid with Cattaneo–Chirstov heat flux and Hall effect

This work is devoted to study a magnetohydrodynamic flow of the Casson fluid over a stretching surface in the presence of Hall effect. Cattaneo–Christov heat flux model is executed to explore the characteristics of heat transfer. The model is an adjustment of the classical Fourier’s law through thermal relaxation time. Convective boundary condition is used on the boundary for heat transfer. The boundary layer partial differential equations are reduced into a system of nonlinear ordinary differential equations through similarity transformations. An efficient computational procedure based on rational Legendre collocation method is formulated and implemented to simulate the resulting problem, numerically. Influences of the physical parameters on the velocity and temperature profiles are analyzed and discussed. Also, impacts of different values of Casson fluid parameter, Hall parameter, Hartman number and Biot numbers on local Nusselt number are computed and examined.

Theoretical aspects of micropolar nanofluid flow past a deformable rotating cone

The intension of this investigation is to analyze the laminar boundary layer flow of micropolar nanofluid inside a deformable rotating cone. A comparison is carried out between three types of nanofluids with nanoparticles, that is, metal (Cu), metallic oxide (Al2O3), and a semiconductor (TiO2) with water as base fluid. Brinkman and Maxwell's model of effective dynamic viscosity and thermal conductivity is considered. Using suitable similarity transformations, the governing coupled non‐linear boundary layer partial differential equations are reduced to non‐linear ordinary differential equations, which are further solved numerically by bvp4c technique. The results obtained reveal interesting effects of important pertinent parameters, that is, rotation parameter ω, nanoparticle concentration parameter φ, and micropolar parameter Δ on velocity, microrotation, and temperature profiles, which are analyzed graphically. A comparative analysis of the results of numerically calculated skin friction coefficients and Nusselt number for these parameters are also presented. The problem throughout the whole study is analyzed for both the strong and weak concentration of micro‐elements. It is found that the ω and Δ enhance the flow and ω, φ, and Δ boosts the rate of heat transfer in all cases of stretching/shrinking rotating cone except for the case of shrinking rotating cone in which the growth of Δ declines the rate of heat transfer for strong/weak concentration of microelements. Moreover, Cu–water has the highest skin friction and Nusselt number.

Effects of Some Flow Parameters on Unsteady MHD Fluid Flow Past a Moving Vertical Plate Embedded in Porous Medium in the Presence of Hall Current and Rotating System

This paper is on the numerical study of the effects of some flow parameters like Hall current, rotation, thermal diffusion (Soret) and diffusion thermo (Dufour) on unsteady magnetohydrodynamic natural convective heat and mass transfer of a viscous, rotating, electrically conducting and incompressible fluid flow past an impulsively moving vertical plate embedded in porous medium. The fundamental governing dimensionless coupled boundary layer partial differential equations are solved by the method of lines (MOL). Computations are then performed to determine the effects of the governing flow parameters. The results show that an increase in Soret number, Dufour number and Hall current parameter, causes an increase in the primary and secondary velocities of the fluid flow. As rotating parameter increases, the primary velocity of the flow decreases. Similarly, as Dufour and Soret numbers increase, the temperature and concentration profiles of the fluid flow increase. The effects of the flow parameters on primary and secondary velocity, temperature and concentration fields for externally cooling of the plate are shown graphically.

Boundary Layer Flow of a Nanofluid Through a Permeable Medium Due to Porous Plate

In the present article, an attempted have been made to study the behavior of boundary layer viscous fluid flow and heat transfer containing some nanosized solid particles flowing through a permeable porous medium. The problem was first modeled into a coupled system of nonlinear partial differential equations of conservation of mass, momentum and nanoparticle concentration. The system of coupled nonlinear boundary layer partial differential equations governing the flowing fluid momentum and heat transfer characteristics are reduced to a new simplified coupled nonlinear system of ordinary differential equations by means of a suitable similarity transformation. The transformed set of nonlinear coupled ordinary differential equations is than solved numerically by means of the fourth order numerical scheme the Runge-Kutta shooting method. The effects of important involved parameters that control the flow field and heat transfer characteristics, that is the viscosity parameter, the convection parameter, the Porosity parameter, the Prandtl number and the Lewis number have been obtained and discussed. Numerical solutions for velocity and temperature are sketched and graphically analyzed. The graphical results observed are indicating that by increasing the values of the non-dimensional viscosity parameter, the dimension less fluid flow profile increases, while for increasing values of the nanoparticles Brownian motion parameter, the nanoparticle concentration profile increases.

On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations

In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

On the convective heat and zero nanoparticle mass flux conditions in the flow of 3D MHD Couple Stress nanofluid over an exponentially stretched surface

Three dimensional problems reflect more imperative understanding to real world issues in comparison to two dimensional problems. Keeping this fact in mind, a mathematical model is designed to deliberate the 3D magnetohydrodynamic couple stress nanofluid flow with joule heating and viscous dissipation effects past an exponential stretched surface. The analysis is performed keeping in mind the physical effects of Brownian motion and thermophoresis combined with convective heat condition. This paper also distinctly introduces a more realistic boundary constraint for nanoliquid flow model. For instance, zero mass flux condition has been instituted for the first time for 3D couple stress nanofluid model as far as the exponential stretched surface is concerned. Self-similar transformations are engaged to obtain a system of ordinary differential equations possessing high nonlinearity from the system of boundary layer partial differential equations. Analytic solution is constructed in the form of series using Homotopy Analysis Method (HAM). Numerically calculated values of Skin friction and local Nusselt number are also given with suitable analysis. Moreover, the influences of sundry parameters on velocity distribution, and heat and mass transfer rates are deliberated and depicted through relevant graphs. The results obtained clearly show that the Biot number and Hartmann number possess increasing effect on temperature distribution. To authenticate our obtained results, a comparison in limiting case is also given.

Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge

The present article examines the influence of thermal radiation on two-dimensional incompressible magnetohydrodynamic (MHD) mixed convective heat transfer flow of Williamson fluid flowing past a porous wedge. An adequate similarity transformation is adopted to reduce the fundamental boundary layer partial differential equations of Williamson fluid model in to a set of non-linear ordinary differential equations. The solutions of the resulting nonlinear system are obtained numerically using the fifth order numerical scheme the Runge-Kutta-Fehlberg method. The effects of different pertinent physical parameter such as magnetic parameter, Williamson parameter, radiation parameter and Prandtl number on temperature and velocity distributions are observed through graph.

Finite element analysis of magnetohydrodynamic flow over flat surface moving in parallel free stream with viscous dissipation and Joule heating

PurposeThe purpose of this paper is to investigate two-dimensional viscous incompressible magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting fluid over a continuous moving flat surface considering the viscous dissipation and Joule heating.Design/methodology/approachSuitable similarity variables are introduced to reduce the governing nonlinear boundary layer partial differential equations to ordinary differential equations. A numerical solution of the resulting two-point boundary value problem is carried out by using the finite element method with the help of Gauss elimination technique.FindingsA comparison of obtained results is made with the previous work under the limiting cases. Behavior of flow and thermal fields against various governing parameters like mass transfer parameter, moving flat surface parameter, magnetic parameter, Prandtl number and Eckert number are analyzed and demonstrated graphically. Moreover, shear stress and heat flux at the moving surface for various values of the physical parameters are presented numerically in tabular form and discussed in detail.Originality/valueThe work is relatively original, as very little work has been reported on magnetohydrodynamic flow and heat transfer over a continuous moving flat surface. Viscous dissipation and Joule heating are neglected in most of the previous studies. The numerical method applied to solve governing equations is finite element method which is new and efficient.