Dimensionless Stream Function Research Articles

Analysis of heat transfer in a parallelogram-shaped cavity with porous medium under non-uniform temperature

This study investigates fluid flow and natural convection heat transfer in a parallelogram-shaped cavity with a porous medium under non-uniform temperature condition. The bottom wall is heated with a non-uniform temperature, the side walls are cooled, and the upper wall is adiabatic. The governing equations, along with corresponding boundary conditions, are formulated in dimensionless stream function and vorticity using the extended Darcy model for the porous medium. The equations are solved by employing the finite difference method. The study presents outcomes for isotherms, streamlines, and Nusselt numbers across range of parameters, including the Rayleigh number (104≤Ra≤108), Darcy number (10−1≤Da≤10−6), Prandtl numbers (Pr=0.71,1,7.1), fluid porosity (ϵ=0.8), and inclination angles (30∘≤ϕ≤90∘), about flow and heat transfer characteristics. The results indicate that an increase in Ra, Pr, and ϕ, along with a decrease in Da, leads to an enhancement in the strength of vortexes. It is observed that for constant Ra⁎=RaDa, Rate of heat transfer increases with an increase in Ra and a decrease in Da. The maximum value observed is 10.15. Also rate of Heat transfer increases with the increase in angle and maximum rate of heat transfer is observed at ϕ=90∘.

Numerical solutions for magneto-convective boundary layer slip flow from a nonlinear stretching sheet with wall transpiration and thermal radiation effects

A mathematical model is presented for magnetohydrodynamic viscous incompressible flow of an electrically conducting Newtonian polymeric fluid from a moving permeable horizontal stretching sheet with variable magnetic field, momentum and thermal slip boundary conditions. The emerging nonlinear ordinary differential boundary value problem is solved numerically by the Runge-Kutta-Fehlberg fourth-fifth order numerical method in Maple symbolic software. The effects of the governing parameters on the dimensionless stream function, dimensionless flow velocity and the dimensionless temperature have been investigated, displayed graphically and discussed. It is found that greater momentum slip and magnetic field parameters reduce the dimensionless velocity. It is further observed that higher values of Prandtl number, thermal slip, and conductionradiation parameter (weaker radiation contribution) reduce the dimensionless temperature whilst stronger magnetic field increases the temperature due to the supplementary work expended in dragging the polymer against the action of the magnetic field which is dissipated as heat. Comparisons of numerical results with previous published results show an excellent agreement. The study finds applications in electro-conductive polymer (ECP) processing systems.

Study of heat transfer in MHD viscoelastic fluid of second grade over a stretching porous sheet with electromagnetic effects and nonuniform source/sink

This work is devoted to the study of non-Newtonian fluid of second grade. This model assumes that the flow and heat transfer in MHD viscoelastic fluid is subject to convective boundary conditions in the presence of electromagnetic field and nonuniform source/sink. The flow is considered over a porous sheet. The governing partial differential equations are converted as ordinary differential equations using suitable similarity transformations. The resulting coupled of nonlinear differential equations are successfully solved by means of Optimal Homotopy Asymptotic Method (OHAM). The effects of different physical parameters such as: the second-grade viscoelastic parameter, thermal radiation parameter or Eckert number are analyzed. Several closed form analytical approximate solutions for dimensionless stream function and temperature are presented for some particular cases. Our procedure is based upon an original construction of the solutions using a moderate number of the convergence-control parameters. These parameters lead to very high precision, comparing our approximate solutions with numerical solutions. Let us note that every nonlinear differential equation is reduced to two linear differential equations. The values of the convergence-control parameters are determined using rigorous mathematical procedures. OHAM is an iterative approach which rapidly converges after only one approximation, is effective, explicit, simple and easy to apply. These remarks are in fact the true power of our procedure to solving nonlinear problems without the presence of small or large parameters in the governing equations or boundary/initial conditions.

Численное исследование влияния неизотермичности на характеристики течения степенной жидкости в трубе с резким расширением

In this paper, a steady laminar non-isothermal flow of a power-law fluid in an axisymmetric sudden pipe expansion is numerically simulated. The rheological behavior of the fluid is described by the modified Ostwald-de Waele law; the apparent viscosity is an exponential function of temperature. The equations are written in terms of dimensionless stream function - vortex - temperature. No-slip conditions and zero temperature are used on the solid wall. At the inlet boundary, the velocity and temperature profiles correspond to a one-dimensional steady non-isothermal flow of the considered fluid. “Soft” boundary conditions are assigned at the outlet boundary. The formulated problem is solved using the finite-difference method. The structure of the flow through a sudden pipe expansion is shown to include one- and two-dimensional flow zones with a recirculation region occurring in the inner corner vicinity. The variation in the two-dimensional flow zone length is analyzed with respect to a power-law index and dimensionless criteria of the problem. Distributions of the velocity, temperature, and apparent viscosity are presented at various Peclet and Reynolds numbers for dilatant and pseudoplastic fluids.

Consequence of Modified Boundary Condition On Natural Convection in a Porous Medium Saturated by Nanofluid – A Computational Approach

In the present numerical simulation, steady, laminar, two-dimensional flow in a porous medium saturated by nanofluid [1] along an isothermal vertical plate is presented. Here we have studied a more realistic situation where the nanoparticle volume fraction at the plate (boundary condition) is passively controlled by assuming that its flux there is zero. We employ Boungiorno model [2] that treats the nanofluid as a two-component mixture, incorporating the effects of Brownian motion and thermophoresis. Darcy model is utilized for the presence of porous medium. With the help of appropriate similarity variables, the governing nonlinear partial differential equations of flow are changed to a bunch of nonlinear ordinary differential equations. Afterwards, they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Dimensionless stream function (s), longitudinal velocity (s’), temperature (θ) and nanoparticle volume fraction (f) are figured and outlined graphically for various values of four dimensionless parameters, namely, Lewis number (Le), buoyancy-ratio parameter (Nr), Brownian motion Parameter (Nb), and thermophoresis parameter (Nt). The dependence of the reduced Nusselt number on these parameters is illustrated.

Entropy generation in concentric annuli of 400 kV gas-insulated transmission line

The object of this study is entropy generation analysis in the concentric annuli of a gas insulation transmission line enclosure (GIL) loaded with air and SF6-N2 mixture gas. The inner and outer cylinders of the cavity are hot and cold, respectively. Because of symmetric conditions the ½ cylinder is considered for modeling. The standard k–ɛ turbulence model, the equation of entropy generation, and the conservation equations are transformed from dimensional form to non-dimensional form utilizing the definition of dimensionless parameters, stream function, and vorticity. Then dimensionless governing equations are solved by a finite volume method (FVM) using an innovative ANSYS Fluent non-dimensionalization scheme. Inflated layers applied to the fine grids to improve the simulation capability for turbulent modeling. The effects of the Rayleigh number at Ra = 2.5 × 106, 1.7 × 109, 2.1 × 109, 4.2 × 109 and 5.7 × 109 are investigated. The results demonstrate that the Nusselt number grows as the Rayleigh number grows and entropy generation number and the Bejan number decreases as the Rayleigh number increases. The GILs have a proper morphology in geometry that reducing the high irreversibility in higher Rayleigh numbers conditions.

Effect of Porous Medium and Copper Heat Sink on Cooling of Heat-Generating Element

Cooling of heat-generating elements is a challenging problem in engineering. In this article, the transient free convection of a temperature-dependent viscosity liquid inside the porous cavity with copper radiator and the heat-generating element is studied using mathematical modeling techniques. The vertical and top walls of the chamber are kept at low constant temperature, while the bottom wall is kept adiabatic. The working fluid is a heat-conducting liquid with temperature-dependent viscosity. A mathematical model is developed based on dimensionless stream function, vorticity, and temperature variables. The governing properties are the variable viscosity, geometric parameters of the radiator, and size of thermally insulated strip on vertical surfaces of the cavity. The effect of these parameters on the energy transport and circulation patterns are analyzed numerically. Based on the numerical results obtained, recommendations are given on the optimal values of the governing parameters for the effective operation of the cooling system. It is shown that the optimal number of radiator fins for the cooling system configuration under consideration is 3. In addition, the thermal insulation of the vertical walls and the increased thickness of the radiator fins have a negative effect on the operation of the cooling system.

Influence of the chamber inclination angle and heat-generating element location on thermal convection of power-law medium in a chamber

PurposeThis paper aims to study the mathematical modeling of passive cooling systems for electronic devices. Improving heat transfer is facilitated by the correct choice of the working fluid and the geometric configuration of the engineering cavity; therefore, this work is devoted to the analysis of the influence of the position of the heat-generating element and the tilted angle of the electronic cabinet on the thermal convection of a non-Newtonian fluid.Design/methodology/approachThe area of interest is a square cavity with two cold vertical walls, while the horizontal boundaries are adiabatic. An element of constant volumetric heat generation is placed on the lower wall of the chamber. The problem is described by Navier–Stokes partial differential equations using dimensionless stream function and vorticity. The numerical solution is based on the developed computational code using the finite difference technique and a uniform rectangular grid.FindingsThe key conclusions of this work are the results of a detailed analysis of streamlines and isotherms, the average Nusselt number and profiles of the average heater temperature. It was found that more intensive cooling of the heat-generating element occurs when the cavity is filled with a pseudoplastic fluid (n < 1) and not inclined (α = 0). The Rayleigh number of Ra = 105 and the thermal conductivity ratio of k = 100 are characterized by the most positive effect.Originality/valueThe originality of the research lies in both the study of thermal convection in a square chamber filled with power-law fluid under the influence of a volumetric heat production element and the analysis of the influence of geometric and thermophysical parameters characterizing the considered process.

Aiding and opposing mixed convection of water with density inversion about a wall of varying temperature

Mixed convection of water over a vertical surface of varying temperature with density inversion is studied for aiding and opposing convection configurations. The temperature of the surface is assumed to be an arbitrary function of vertical distance. The governing equations are transformed using dimensionless stream function and temperature. The temperature differentials of varying wall temperature are used as perturbation functions. The dimensionless stream function and temperature are expanded in power series of perturbation variables and coefficient functions. The obtained coefficient functions are valid for any arbitrary wall temperature function, and hence they are ’universal functions’. Power law wall temperature variation is chosen to show the usefulness of universal functions. The results are presented for velocity and temperature distributions in the boundary layer, velocity and thermal boundary layer thicknesses, skin friction coefficient and heat transfer rates for various values of governing parameter and wall temperature power law index for both aiding and opposing flows. It is found that for the range of \(Gr_y/Re_y^2\) values considered in the study, the skin friction coefficient and heat transfer rates vary almost linearly with wall temperature power law index value for a given \(Gr_y/Re_y^2\) value for both aiding and opposing flows. For special wall temperature cases, the present results are compared to benchmark solutions available in literature and good agreement is found.

Effect of Internal Heat Generation and Concentration Change on Free Convection Boundary Layer from Vertical Flat Plate Embedded in Porous Medium

The effects of exponentially decaying internal heat generation (IHG) and internal mass generation (IMG) over specific components are presented numerically on coupled heat and mass transfer by a free convection boundary layer over a vertical flat plate embedded in a fluid-saturated porous medium. Corresponding similarity solutions are used to reduce the governing partial nonlinear differential equations to three ordinary differential equations for the dimensionless stream function, temperature, and concentration with the following parameters: buoyancy force N, exponent of x, λ, and Lewis number Le. Media with and without IHG and IMG are compared in context with the help of graphs and tables. Computations are performed with a system of parameters using built-in codes in Maple. The influences of these parameters on velocity, temperature and concentration profiles, and Sherwood and Nusselt numbers are thoroughly compared and graphically illustrated.

Flow and heat transfer of non-Newtonian power-law fluids over a stretching surface with variable thermal conductivity

PurposeThe purpose of this paper is to investigate the flow and heat transfer of power-law fluids over a non-linearly stretching sheet with non-Newtonian power-law stretching features.Design/methodology/approachThe governing non-linear partial differential equations are reduced to a series of ordinary differential equations by suitable similarity transformations and the numerical solutions are obtained by the shooting method.FindingsAs the temperature power-law index or the power-law number of the fluids increases, the dimensionless stream function, dimensionless velocity and dimensionless temperature decrease, while the velocity boundary layer and temperature boundary layer become thinner for other fixed physical parameters. The thermal diffusivity varying as a function of the temperature gradient can be used to present the characteristics of flow and heat transfer of non-Newtonian power-law fluids.Originality/valueUnlike classical works, the effect of power-law viscosity on the temperature field is considered by assuming that the temperature field is similar to the velocity field with modified Fourier’s law heat conduction for power-law fluid media.

Natural Convection of Non-Newtonian Power-Law Fluid in a Square Cavity with a Heat-Generating Element

Development of modern technology in microelectronics and power engineering necessitates the creation of effective cooling systems. This is made possible by the use of the special fins technology within the cavity or special heat transfer liquids in order to intensify the heat removal from the heat-generating elements. The present work is devoted to the mathematical modeling of thermogravitational convection of a non-Newtonian fluid in a closed square cavity with a local source of internal volumetric heat generation. The behavior of the fluid is described by the Ostwald-de Waele power law model. The defining Navier–Stokes equations written using the dimensionless stream function, vorticity and temperature are solved using the finite difference method. The effects of the Rayleigh number, power-law index, and thermal conductivity ratio on heat transfer and the flow structure are studied. The obtained results are presented in the form of isolines of the stream function and temperature, as well as the dependences of the average Nusselt number and average temperature on the governing parameters.

A computational framework for natural convective hydromagnetic flow via inclined cavity: An analysis subjected to entropy generation

One of the most interested and essential subjective in mechanical science is natural convection analysis in a cavity. Thus, natural convective flow and entropy generation are scrutinized numerically subjected to magnetic field effect in a semi-annulus enclosure which known as the effect of magneto-hydrodynamic (MHD). Firstly, the governing expressions are rewritten in non-dimensional form utilizing the definition of dimensionless parameters, stream function, and vorticity. Secondly, the entropy generation equation is expressed in non-dimensional form. Then governing non-dimensional expressions are computed by control finite element method (CVFEM). Furthermore, the governing expressions are computed employing finite volume method (FVM) utilizing ANSYS Fluent CFD code. A novel criterion for determination of thermal characteristics of cavity based on thermodynamics second relation is introduced that is called ecological coefficient of performance (ECOP). Flow and heat transport features in addition to entropy generation number are examined for distinct values of the Rayleigh number (Ra = 103,104,105), the orientation of the magnetic field (β = 0°,15°,30°,45°,60°,75°,90°), and Hartmann number (Ha = 0,5,10,15,20). For validation, the entropy generation number and average Nusselt number are compared with those available in literature and excellent agreement is observed. Isotherms and streamlines are calculated using CVFEM and FVM. Some correlations for entropy generation number are proposed. The results show that for constant Rayleigh number, the entropy generation number decays with increasing Hartmann number. Also, for each Hartmann number, there is an optimum inclination angle of magnetic field that gives a minimum for entropy generation number and a maximum for ECOP at each Rayleigh number.

Unsteady natural convection in a partially porous cavity having a heat-generating source using local thermal non-equilibrium model

PurposeThe purpose of this study is a numerical analysis of transient natural convection in a square partially porous cavity with a heat-generating and heat-conducting element using the local thermal non-equilibrium model under the effect of cooling from the vertical walls. It should be noted that this research deals with a development of passive cooling system for the electronic devices.Design/methodology/approachThe domain of interest is a square cavity with a porous layer and a heat-generating element. The vertical walls of the cavity are kept at constant cooling temperature, while the horizontal walls are adiabatic. The heat-generating solid element is located on the bottom wall. A porous layer is placed under the clear fluid layer. The governing equations, formulated in dimensionless stream function, vorticity and temperature variables with corresponding initial and boundary conditions, are solved using implicit finite difference schemes of the second order accuracy. The governing parameters are the Darcy number, viscosity variation parameter, porous layer height and dimensionless time. The effects of varying these parameters on the average total Nusselt number along the heat source surface, the average temperature of the heater, the fluid flow rate inside the cavity and on the streamlines and isotherms are analyzed.FindingsThe results show that in the case of local thermal non-equilibrium the total average Nusselt number is an increasing function of the interphase heat transfer coefficient and the porous layer thickness, while the average heat source temperature decreases with the Darcy number and viscosity variation parameter.Originality/valueAn efficient numerical technique has been developed to solve this problem. The originality of this work is to analyze unsteady natural convection within a partially porous cavity using the local thermal non-equilibrium model in the presence of a local heat-generating solid element. The results would benefit scientists and engineers to become familiar with the analysis of convective heat transfer in enclosures with local heat-generating heaters and porous layers, and the way to predict the heat transfer rate in advanced technical systems, in industrial sectors including transportation, power generation, chemical sectors and electronics.

Natural convection of Al2O3/H2O nanofluid in a cavity with a heat-generating element. Heatline visualization

Cooling of electronic devices is a very important topic. The present paper deals with numerical simulation of natural convection cooling of heat-conducting and heat-generating source using an alumina-water nanofluid under the effect of cold vertical walls. Governing equations formulated in dimensionless stream function, vorticity and temperature variables have been solved by the finite difference method. The effects of nanoparticles concentration, heat source location, internal heat generation flux and heat source material on fluid flow and heat transfer have been studied. It has been found that an addition of nanoparticles can intensify the cooling process when the heat is located near the vertical cold wall.

Melting of nano-enhanced PCM inside finned radiator

The two-dimensional melting process of paraffin enhanced by Al2O3-nanoparticles inside a copper radiator with various frequency of finning is studied numerically. The governing equations have been formulated in dimensionless stream function, vorticity and temperature. The obtained system of partial differential equations has been solved using the finite difference method. The influence of the frequency of the fins location on the melting process of paraffin with different nanoparticles volume fraction has been studied.

Magneto-hydrodynamic natural convection of CuO-water nanofluid in complex shaped enclosure considering various nanoparticle shapes

PurposeThe purpose of this study is to peruse natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method.Design/methodology/approachGoverning equations formulated in dimensionless stream function, vorticity and temperature variables using the single-phase nanofluid model with the Koo–Kleinstreuer–Li correlation for the effective dynamic viscosity and the effective thermal conductivity have been solved numerically by control volume finite element method.FindingsEffects of various pertinent parameters such as Rayleigh number, Hartmann number, volume fraction of nanofluid and shape factor of nanoparticle on the convective heat transfer characteristics are analysed. It was observed that local and average heat transfer rates increase for higher value of Rayleigh number and lower value of Hartmann number. Among various nanoparticle shapes, platelets were found to be best in terms of heat transfer performance. The amount of average Nusselt number reductions was found to be different when nanofluids with different solid particle volume fractions were considered due to thermal and electrical conductivity enhancement of fluid with nanoparticle addition.Originality/valueA comprehensive study of the natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method is addressed.

MHD natural convection of Cu/H2O nanofluid in a horizontal semi-cylinder with a local triangular heater

Purpose The purpose of this study is to investigate free convection of copper-water nanofluid in an upper half of circular horizontal cylinder with a local triangular heater under the effects of uniform magnetic field and cold cylinder shell using control volume finite element method (CVFEM). Design/methodology/approach Governing equations formulated in dimensionless stream function, vorticity and temperature variables using the single-phase nanofluid model with Brinkman correlation for the effective dynamic viscosity and Hamilton and Crosser model for the effective thermal conductivity have been solved numerically by CVFEM. Findings The impacts of control parameters such as the Rayleigh number, Hartmann number, nanoparticles volume fraction, local triangular heater size, shape factor on streamlines and isotherms as well as local and average Nusselt numbers have been examined. The outcomes indicate that the average Nusselt number is an increasing function of the Rayleigh number, shape factor and nanoparticles volume fraction, while it is a decreasing function of the Hartmann number. Originality/value A complete study of the free convection of copper-water nanofluid in an upper half of circular horizontal cylinder with a local triangular heater under the effects of uniform magnetic field and cold cylinder shell using CVFEM is addressed.

Steady free convection flow within a titled nanofluid saturated porous cavity in the presence of a sloping magnetic field energized by an exothermic chemical reaction administered by Arrhenius kinetics

In this work steady natural convection flow energized by an exothermic chemical reaction administered by Arrhenius kinetics inside a tilted nanofluid saturated porous square cavity under the action of a sloping magnetic field has been deliberated numerically following Buongiorno nanofluid model. It is assumed that there is a local thermal equilibrium state between the considered nanofluid and homogeneous porous medium. The mathematical model governing the dimensionless stream function for flow, temperature for heat, and nanoparticles volume fraction for concentration are simulated numerically by means of Galerkin weighted residual type of finite element method. The numerical code has been corroborated compared with the formerly available works, and very good agreement is noticed among the outcomes. Disseminations of streamlines, isotherms, isoconcentrations, and average Nusselt number at wide-ranging key parameters are acquired and discussed in details. The results show that Rayleigh and Frank-Kamenetskii numbers strongly control the convective flows. The average Nusselt number increases with the Frank-Kamenetskii number while it decreases with the Rayleigh number. Heat generation due to a strong exothermic reaction (higher Frank-Kamenetskii number) can blow up the bounded solution. For higher Frank-Kamenetskii number, the rate of heat transfer can be controlled tilting the cavity anticlockwise. Magnetic field slope overwhelms the heat transfer rate when the Frank-Kamenetskii number is high. A strong magnetic field caused the nanoparticles separation for the slow flow. Effects of Brownian diffusion and thermophoresis on average Nusselt number are minimal while their effects on average Sherwood number are quite accountable.

Natural convection of an alumina-water nanofluid inside an inclined wavy-walled cavity with a non-uniform heating using Tiwari and Das’ nanofluid model

The present study is devoted to numerical analysis of natural convective heat transfer and fluid flow of alumina-water nanofluid in an inclined wavy-walled cavity under the effect of non-uniform heating. A single-phase nanofluid model with experimental correlations for the nanofluid viscosity and thermal conductivity has been included in the mathematical model. The considered governing equations formulated in dimensionless stream function, vorticity, and temperature have been solved by the finite difference method. The cavity inclination angle and irregular walls (wavy and undulation numbers) are very good control parameters for the heat transfer and fluid flow. Nowadays, optimal parameters are necessary for the heat transfer enhancement in different practical applications. The effects of the involved parameters on the streamlines and isotherms as well as on the average Nusselt number and nanofluid flow rate have been analyzed. It has been found that the heat transfer rate and fluid flow rate are non-monotonic functions of the cavity inclination angle and undulation number.