Dimensionless Stream Function Research Articles

Natural convection with evaporation in a vertical cylindrical cavity under the effect of temperature-dependent surface tension

The effect of surface tension on laminar natural convection in a vertical cylindrical cavity filled with a weak evaporating liquid has been analyzed numerically. The cylindrical enclosure is insulated at the bottom, heated by a constant heat flux from the side, and cooled by a non-uniform evaporative heat flux from the top free surface having temperature-dependent surface tension. Governing equations with corresponding boundary conditions formulated in dimensionless stream function, vorticity, and temperature have been solved by finite difference method of the second-order accuracy. The influence of Rayleigh number, Marangoni number, and aspect ratio on the liquid flow and heat transfer has been studied. Obtained results have revealed that the heat transfer rate at free surface decreases with Marangoni number and increases with Rayleigh number, while the average temperature inside the cavity has an opposite behavior; namely, it growths with Marangoni number and reduces with Rayleigh number.

Convective heat transfer of micropolar fluid in a horizontal wavy channel under the local heating

A numerical analysis of laminar mixed convection of micropolar fluid in a horizontal wavy channel with a local heater has been carried out. Governing equations formulated in dimensionless stream function, vorticity and temperature have been solved by finite difference method. Effects of Rayleigh number, Reynolds number, Prandtl number, vortex viscosity parameter and undulation number on streamlines, isotherms, vorticity isolines as well as horizontal velocity and temperature profiles with average Nusselt number at the heater have been studied. It has been found that the average Nusselt number is an increasing function of Rayleigh, Reynolds and Prandtl numbers and a decreasing function of undulation number while an influence of vortex viscosity parameter is non-monotonic.

Numerical Analysis of Unsteady Conjugate Natural Convection of Hybrid Water-Based Nanofluid in a Semicircular Cavity

Unsteady conjugate natural convection in a semicircular cavity with a solid shell of finite thickness filled with a hybrid water-based suspension of Al2O3 and Cu nanoparticles (hybrid nanofluid) has been analyzed numerically. The governing equations for this investigation are formulated in terms of the dimensionless stream function, vorticity, and temperature and have been solved by the finite difference method of the second-order accuracy. The effects of the dimensionless time, Rayleigh number, thermal conductivity ratio, and the nanoparticles volume fraction on the flow patterns and heat transfer have been studied. The obtained results have revealed essential heat transfer enhancement at solid–fluid interface with addition of nanoparticles. In addition, a comparison of the heat transfer enhancement level due to the suspension of various nanoparticles materials (Al2O3 and Cu) in water as regular nanofluids (Al2O3/water and Cu/water) and as a hybrid Al2O3–Cu/water nanofluid is reported.

Heatline visualization of natural convection in a thick walled open cavity filled with a nanofluid

Natural convection of an alumina-water nanofluid in a partially open rectangular cavity with a left heat-conducting solid wall of finite thickness and conductivity has been studied numerically. Governing equations formulated in dimensionless stream function and vorticity variables on the basis of a single-phase nanofluid model with corresponding boundary conditions have been solved by finite difference method. Analysis of the influence of Rayleigh number (Ra=103–106), thermal conductivity ratio (1⩽K⩽20), solid wall thickness (0.1⩽δ⩽0.3) and nanoparticles volume fraction (0⩽ϕ⩽0.05) on streamlines, heatlines and isotherms as well as average Nusselt number at solid-fluid interface and fluid flow rate has been carried out. It has been revealed that for the considered models for an effective thermal conductivity ratio and dynamic viscosity an increase in the nanoparticles volume fraction leads to the heat transfer reduction and decrease of the fluid flow rate.

Similarity Solutions on Mixed Convection Heat Transfer from a Horizontal Surface Saturated in a Porous Medium with Internal Heat Generation

The aim of this work is to study the mixed convection boundary layer flow from a horizontal surface embedded in a porous medium with exponential decaying internal heat generation (IHG). Boundary layer equations are reduced to two ordinary differential equations for the dimensionless stream function and temperature with two parameters: ε, the mixed convection parameter, and λ, the exponent of <i>x</i>. This problem is numerically solved with a system of parameters using built-in codes in Maple. The influences of these parameters on velocity and temperature profiles, and the Nusselt number, are thoroughly compared and discussed.

Моделирование конвективного теплоперено- са во вращающейся замкнутой полости с ло- кальным источником энергии

Numerical simulation of transient convective heat transfer inside a rotating enclosure with a local heater of constant temperature has been carried out. The boundary-value problem formulated using dimensionless stream function, vorticity and temperature has been solved by finite difference method of the second order accuracy on the basis of uniform mesh. Analysis has been conducted in a wide range of Rayleigh and Taylor numbers as well as time parameter reflecting a formation of periodic flow and heat transfer patterns. The flow and temperature fields illustrating a development of considered process with governing parameters have been shown.Received 19.03.2017; accepted 25.04.2017

Free convection in a porous wavy cavity filled with a nanofluid using Buongiorno's mathematical model with thermal dispersion effect

A numerical study of natural convection inside a porous wavy cavity filled with a nanofluid under the effect of thermal dispersion has been carried out using the Forchheimer–Buongiorno approach. The left boundary of the cavity is a wavy isothermal wall while the rest are flat isothermal walls. All boundaries are assumed to be impermeable to the base fluid and nanoparticles. The governing equations formulated in dimensionless stream function, temperature and nanoparticle volume fraction variables have been solved using implicit finite difference schemes of the second order accuracy. The effects of the Rayleigh number, undulation number, thermal dispersion parameter and flow inertia parameter on the average Nusselt number along the hot bottom wall, as well as on the streamlines, isotherms and isoconcentrations have been analyzed. It has been revealed the heat transfer enhancement with Rayleigh number, undulation number and dispersion parameter. While convective flow is attenuated with a growth of undulation number, dispersion parameter and flow inertia parameter. More essential homogenization of nanoparticles distribution inside the cavity occurs with an increase in Rayleigh number and a decrease in undulation number.

MHD natural convection in a partially open trapezoidal cavity filled with a nanofluid

A numerical study of natural convection in a partially open trapezoidal cavity filled with a CuO nanofluid under the effect of uniform magnetic field of various orientations has been carried out. Inclined wall of the cavity is heated, horizontal walls are adiabatic while cold nanofluid enters into the cavity from an open boundary. To analyze the effects of magnetic field intensity and its inclination angle with nanoparticles volume fraction, the finite difference method has been utilized to solve the governing equations formulated in dimensionless stream function, vorticity and temperature. Analysis has been conducted in a wide range of governing parameters such as Rayleigh number (Ra=103–105), Prandtl number (Pr=7.0), Hartmann number (Ha=0, 10, 50, 100), inclination angle of magnetic field (α=0–π) and nanoparticles volume fraction (0≤ϕ≤0.04). It has been found that an increase in Hartmann number leads to the heat transfer reduction, while an increase in the nanoparticles volume fraction reflects the heat transfer enhancement.

Time-dependent natural convection of micropolar fluid in a wavy triangular cavity

Natural convection of micropolar fluid in a right-angled wavy triangular cavity has been analyzed numerically. Governing equations formulated in dimensionless stream function, vorticity and temperature using the Boussinesq and Eringen approaches with appropriate initial and boundary conditions have been solved by finite difference method of the second-order accuracy. The effects of the dimensionless time, Prandtl number, vortex viscosity parameter, and undulation number on streamlines, isotherms, vorticity isolines as well as average Nusselt number at wavy wall and fluid flow rate inside the cavity have been studied. Obtained results have revealed essential heat transfer reduction and fluid flow attenuation with vortex viscosity parameter.

MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater

A numerical analysis of MHD natural convection in a wavy open porous tall cavity filled with a Cu–water nanofluid in the presence of an isothermal corner heater has been carried out. The cavity is cooled from the left wavy wall and heated from the right bottom corner while the bottom wall is adiabatic. Uniform magnetic field affects the heat transfer and fluid flow with an inclination angle to the axis x¯. Mathematical model formulated using the single-phase nanofluid approach in dimensionless variables stream function, vorticity and temperature has been solved by finite difference method of the second order accuracy in a wide range of governing parameters: Rayleigh number (Ra=100–1000), Hartmann number (Ha=0–100), inclination angle of the magnetic field (γ=0–π) and solid volume fraction parameter of nanoparticles (φ=0.0–0.05). Main efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms and average Nusselt numbers. It has been found heat transfer enhancement with Rayleigh number and heat transfer reduction with Hartmann number, while magnetic field inclination angle leads to non-monotonic changes of the heat transfer rate.

Effect of inclined magnetic field on natural convection melting in a square cavity with a local heat source

MHD natural convection melting in a square cavity with a local heater has been analyzed numerically. The domain of interest is an enclosure bounded by isothermal vertical walls of low constant temperature and adiabatic horizontal walls. A heat source of constant temperature is located on the bottom wall. An inclined uniform magnetic field affects the natural convective heat transfer and fluid flow inside the melt. The governing equations formulated in dimensionless stream function, vorticity and temperature with corresponding initial and boundary conditions have been solved using implicit finite difference method of the second-order accuracy. The effects of the Rayleigh number, Stefan number, Hartmann number, magnetic field inclination angle and dimensionless time on streamlines, isotherms and Nusselt number at the heat source surface have been analyzed.

Existence of the multiple exact solutions for nanofluid flow over a stretching/shrinking sheet embedded in a porous medium at the presence of magnetic field with electrical conductivity and thermal radiation effects

In this research, the author investigated two-dimensional incompressible, laminar and steady boundary-layer flow past a stretching/shrinking sheet which is in a water-based nanofluid containing metallic and nonmetallic nanoparticles; namely, silver, copper, alumina and titania. Theoretical exact solutions for the dimensionless stream function and dimensionless temperature were obtained. Hence, a detailed analysis of the existence of those solutions was effectively deduced for the stretching/shrinking sheet conditionally in order with the values of permeability, magnetic and suction/injection parameters. Via many tables and graphs, the effects of various physical parameters included in dimensionless profiles for the stream function, velocity and temperature were studied. These parameters include the solid volume fraction, magnetic, permeability, radiation, surface convection, stretching/shrinking sheet, reduced skin friction coefficient and reduced Nusselt number.On comparing the present results with those in the literature, a very good agreement was obtained in some special cases for different values of the investigated parameters. In addition, it was shown that the effective electrical conductivity considerably and remarkably affects the nanofluids' flow and, hence, it is mandatory to take this into account and not to ignore it as in some recent papers, especially with its presence in the solutions' conditions, otherwise, a spurious physical sight is to be obtained. It was therefore indicated that the included parameters have to be re-investigated and we must not depend on the published results that dropped this term. Further, the critical values and curves for obtaining one solution (dual solutions) were successfully secured for a stretching sheet (shrinking sheet). For a stretching/shrinking sheet, profiles of the velocity decrease as the solid volume fraction increases in the region (0, 0.2], then, they increase as this parameter increases when its value lies in the region (0.2, 0.4].

Numerical simulation of a conjugate turbulent natural convection combined with surface thermal radiation in an enclosure with a heat source

A detailed numerical analysis of complex heat transfer (turbulent natural convection, conduction and surface thermal radiation) in a rectangular enclosure with a heat source has been carried out. The finite difference method for the solution of the governing equations using the dimensionless stream function, vorticity and temperature variables has been utilized. The effects of Rayleigh number in a range from 108 to 1011, thermal conductivity ratio in a wide range from 10 to 1000, as well as internal surface emissivity 0≤ε˜<1 on the fluid flow and heat transfer have been extensively explored. Detailed results including temperature fields, flow profiles, and average Nusselt numbers have been presented. In this investigation we have tried to study the main regularities of heat and mass transfer in the considered domain.

Free convection in a trapezoidal cavity filled with a micropolar fluid

The present investigation deals with the study of steady laminar natural convective flow and heat transfer of micropolar fluids in a trapezoidal cavity. The bottom wall of the cavity is kept at high constant temperature, the inclined walls is kept at low constant temperatures while the top horizontal wall is adiabatic. Governing equations formulated in dimensionless stream function and vorticity variables has been solved by finite difference method of the second order accuracy. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. Computations have been carried out to analyze the effects of Rayleigh number, Prandtl number and vortex viscosity parameter both for weak and strong concentration cases. Obtained results have been presented in the form of streamlines, isotherms and vorticity profiles as well as the variation of the average Nusselt number and fluid flow rate. It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity.

MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid

A mathematical modelling of MHD free convection in an inclined wavy enclosure filled with a Cu–water nanofluid in the presence of an isothermal corner heater has been carried out. The cavity is heated from the left bottom corner and cooled from the top wavy wall while the rest walls are adiabatic. Uniform magnetic field affects the heat transfer and fluid flow with an inclination angle to the axis x¯. Wavy cavity is inclined to the horizontal direction. Mathematical model formulated using the single-phase nanofluid approach in dimensionless variables stream function, vorticity and temperature has been solved by finite difference method of the second order accuracy in a wide range of governing parameters: Hartmann number (Ha=0–100), inclination angle of the magnetic field (χ=0−π), undulation number (κ=0–4), inclination angle of the cavity (ζ=0−π), solid volume fraction parameter of nanoparticles (φ=0.0–0.05), and dimensionless time (τ=0–0.27). Main efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms and average Nusselt numbers.

A transient free convection study with temperature-dependent viscosity in a square cavity with a local heat source

Unsteady natural convection inside of a differentially-heated square enclosure filled with a fluid of temperature-dependent viscosity has been numerically studied. A mathematical model formulated in the dimensionless stream function and vorticity has been solved by a finite difference method of the second order accuracy. The effect of dimensionless time and Prandtl number on streamlines and isotherms has been investigated for Ra = 105. The results clearly demonstrate an evolution of fluid flow and heat transfer in the case of variable viscosity fluid.

Natural convection in a wavy open porous cavity filled with a nanofluid: Tiwari and Das’ nanofluid model

Natural convective heat transfer and fluid flow in an open porous cavity filled with a nanofluid is studied numerically using the Tiwari and Das nanofluid model. The transport equations for mass, momentum and energy formulated in dimensionless stream function and temperature are solved numerically using a second-order accurate finite difference method. Particular efforts are focused on the effects of the governing parameters on the heat and fluid flow. It is found that an increase in undulation number of the wavy vertical wall leads to an attenuation of convective flow and a decrease in the heat transfer rate.

Study of Melting of a Pure Gallium in a Rectangular Enclosure

Phase change problems with natural convection in the liquid phase are of prime importance in accurate technological applications. In this work a melting of pure gallium in a rectangular cavity heated from the left wall is simulated using the finite difference method. Numerical algorithm based on dimensionless variables stream function and vorticity has been checked by experimental data of previously published papers. The effect of the grid size on results of calculations was identified. Motion of phase front and changing parameters of heat and mass-transfer were studied

Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno's mathematical model

A numerical investigation is performed on the unsteady natural convection of water based nanofluid within a wavy-walled cavity under the influence of a uniform inclined magnetic field using the mathematical nanofluid model proposed by Buongiorno. The left vertical wavy and right vertical flat walls of the cavity are kept at constant but different temperatures whereas the top and bottom horizontal walls are adiabatic. All boundaries are assumed to be impermeable to the base fluid and nanoparticles. The mathematical model formulated in dimensionless stream function, vorticity and temperature variables is solved using implicit finite difference schemes of the second order. The governing parameters are the Hartmann number, undulation number, wavy contraction ratio, inclination angle of the magnetic field relative to the gravity vector and dimensionless time. The effects of these parameters on the average Nusselt number along the hot wavy wall, as well as on the streamlines, isotherms and isoconcentrations are analyzed.

Homotopy analysis method for unsteady mixed convective stagnation-point flow of a nanofluid using Tiwari-Das nanofluid model

Purpose – The current study is mainly motivated by the need to the development of the transient MHD mixed convection stagnation-point flow and heat transfer of an electrically conducting nanofluid over a vertical permeable stretching/shrinking sheet by means of Tiwari-Das nanofluid model. The purpose of this paper is to investigate the effects of the parameters governing the flow i.e. the nanoparticle volume fraction, the unsteadiness parameter, the magnetic parameter, the wall transpiration parameter, the mixed convection parameter and the velocity ratio parameter on dimensionless velocity and temperature distributions, skin friction coefficient and local Nusselt number. Design/methodology/approach – The mathematical model has been formulated based on Tiwari-Das nanofluid model. Three different types of water-based nanofluid with copper, aluminum oxide (alumina) and titanium dioxide (titania) as nanoparticles are considered in this investigation. Using appropriate similarity variables, the governing equations are transformed into nonlinear ordinary differential equations in the dimensionless stream function, which is solved analytically by the well-know homotopy analysis method. The present simulations agree closely with the previous studies in the especial cases. Findings – The results show that by increasing the nanoparticle volume fraction, the unsteadiness parameter, the magnetic parameter, the wall transpiration parameter, the mixed convection parameter or reducing the velocity ratio parameter, the skin friction coefficient enhances. Furthermore, the local Nusselt number enhances with different rates by increasing the nanoparticle volume fraction, the unsteadiness parameter, the magnetic parameter, the wall transpiration parameter, the mixed convection parameter and the velocity ratio parameter. Besides, the skin friction coefficient and the local Nusselt number are highest for copper-water nanofluid compared to the alumina-water and titania-water nanofluids. Originality/value – Tiwari-Das nanofluid model has not been applied for the flow with these characteristics as mentioned in the paper. A comprehensive survey on boundary layer behavior has been presented. There are few studies regarding as analysis on thermal and hydrodynamics boundary layer. All plots presented in the paper are new and did not report in any other study. The effects of the parameters governing the flow on skin friction coefficient and local Nusselt number have been illustrated in the paper while there are some conflicts with previous published article that have been interpreted in details in the paper.