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Selective laser sintering (SLS) of a two-component metal powder layer on the top of multiple sintered layers by a moving Gaussian laser beam is modeled. The loose metal powder layer is composed of a powder mixture with significantly different melting points. The physical model that accounts the shrinkage induced by melting is described by using a temperature-transforming model. The effects of the porosity and the thickness of the atop loose powder layer with different numbers of the existing sintered metal powder layers below on the sintering process are numerically investigated. The present work will provide a better understanding to simulate much more complicated three-dimensional SLS process.

A three-dimensional numerical model of Selective Laser Sintering (SLS) of the metal powders for a single scan line induced by a moving laser beam interacted with a loose powder layer on top of the sintered metal layers is presented. The problem is modeled using a temperature-transforming model and the partial shrinkage induced by melting is accounted for. The heat losses at the top surface due to the natural convection and radiation are taken into account. The liquid flow of the molten low melting point metal powders, which is driven by capillary and gravity forces, is also considered and formulated by using Darcy’s law. The effects of the dominant processing parameters, such as the moving heat source intensity, scanning velocity and number of the existing sintered layers underneath are investigated. A parametric study is performed and the best combination of the processing parameters is recommended.Copyright © 2005 by ASME

Laser sintering of a two-component metal powder layer on top of sintered layers, with a moving circular Gaussian laser beam is modeled numerically. The overlap between the adjacent scanning lines, as well as the binding between the newly sintered layer and existing sintered layers underneath through melting, are also considered. The governing equation is formulated by a temperature-transforming model, with partial shrinkage induced by melting taken into account. The liquid flow of the molten low melting point metal powders driving by capillary and gravity forces is formulated by Darcy’s law. The effects of the dominant processing parameters, including the moving laser beam intensity, scanning speed, and the number of the existing sintered layers underneath on the shape of the heat affected zone (HAZ) are investigated.

Melting of a subcooled two-component metal powder layer is investigated analytically. The powder bed considered consists of a mixture of two metal powders with significantly different melting points. Shrinkage induced by melting is taken into account in the physical model. The temperature distributions in the liquid and solid phases are obtained using an exact solution and an integral approximate solution, respectively. The effects of porosity, Stefan number, and subcooling on the surface temperature and solid–liquid interface are also investigated. The present work provides a strong foundation upon which the investigation of complex three-dimensional selective laser sintering (SLS) process can be based.

A partial shrinkage model for selective laser sintering of a metal powder mixture that contains two kinds of metal powders with significantly different melting points is developed. Laser-induced melting accompanied by partial shrinkage, liquid metal flow driven by capillary and gravitational forces, and resolidification of the metal powder layer are modeled using a temperature transforming model. The effect of volume fraction of the gas in the sintered region on the sintering process is investigated.

Laser sintering of a metal powder mixture that contains two kinds of metal powders with significantly different melting points under a moving Gaussian laser beam is investigated numerically. The continuous-wave laser-induced melting accompanied by shrinkage and resolidification of the metal powder layer are modeled using a temperature-transforming model. The liquid flow of the melted low-melting-point metal driven by capillary and gravity forces is also included in the physical model. The numerical results are validated by experimental results, and a detailed parametric study is performed. The effects of the moving heat source intensity, the scanning velocity, and the thickness of the powder layer on the sintering depth, the configuration of the heat affected zone, and the temperature distribution are discussed.

Selective laser sintering (SLS) of metal powder is an emerging technology by which metal parts can be fabricated in a layer by layer fashion. SLS of the first layer is modeled as melting and resolidification of a metal powder layer subject to a moving heat source on top, while the bottom is adiabatic. SLS of the consecutive layer is modeled as melting and resolidification of a metal powder layer on top of the existing multiple resolidified layers. The results indicate that the thicknesses of the loose metal powder layer, the moving heat source intensity, and the scanning velocity have significant effects on the sintering process in both the first layer and each subsequent layer. A parametric study is performed, and the best combination of processing parameters is recommended.

Measurements of heat transfer and fluid flow of turbulent boundary-layer air flow in natural and mixed convection over an isothermal two-dimensional, vertical forward-facing step are reported. The upstream and downstream walls and the step itself were heated to a uniform and constant temperature. Air velocity and temperature distributions and their turbulent fluctuations are measured simultaneously using a two-component laser-Doppler velocimeter (LDV) and a cold wire anemometer, respectively. The present study treats buoyancy-dominated mixed convection over a vertical forward-facing step and examines the effect of a small free stream velocity on turbulent natural convection. The experiment was carried out for a step height of 22 mm, for a range of free stream air velocities 0 m/s ⋚ u∞ ⋚ 0.55 m/s (corresponding to a range of Reynolds numbers of 0 ⋚ Re\abinf{s} ⋚ 712), and a temperature difference, ΔT, of 30°C between the heated walls and the free stream air (corresponding to a local Grashof number Grxi = 6...

The non-Darcy effects on the vortex instability of natural convection e ow over a horizontal e at plate embedded in a high-porosity medium are examined. These effects include the form-drag, viscous diffusion, and convective terms in the conservation equations. The instability analysis is conducted by employing the linear theory in conjunction with a nonparallel e ow model, which takes into account both the streamwise and transverse variations of thedisturbanceamplitude functions. Theresulting system of partial differential equations forthedisturbance amplitudefunctionsisconverted to asystemofhomogeneouslinearordinary differentialequationswithhomogeneous boundary conditions by the local nonsimilarity method. This results in an eigenvalue problem that is solved by an implicit e nite difference method. The numerical results indicate that all of the three non-Darcy effects examined in the present study reduce the surface heat transfer rate as compared to the case of pure Darcy e ow. These effects also renderthee ow to become more stable to the vortex mode ofinstability. Representative neutral stability curves and critical Grashof numbers are presented. It is found that the parallel e ow model underpredicts the onset of thermal instability, that is, it provides lower critical Grashof numbers than the nonparallel e ow model.

Effect of the backward-facing step heights on turbulent mixed convection flow along a vertical flat plate is examined experimentally. The step geometry consists of an adiabatic backward-facing step, an upstream wall and a downstream wall. Both the upstream and downstream walls are heated to a uniform and constant temperature. Laser–Doppler velocimeter and cold wire anemometer were used, respectively, to measure simultaneously the time-mean velocity and temperature distributions and their turbulent fluctuations. The experiment was carried out for step heights of 0, 11, and 22 mm, at a free stream air velocity, u∞, of 0.41 m/s, and a temperature difference, ΔT, of 30 °C between the heated walls and the free stream air. The present results reveal that the turbulence intensity of the streamwise and transverse velocity fluctuations and the intensity of temperature fluctuations downstream of the step increase as the step height increases. Also, it was found that both the reattachment length and the heat transfer rate from the downstream heated wall increase with increasing step height.

The inertia effect on the onset of thermal instability in natural convection flow over heated horizontal and inclined flat plates embedded in fluid-saturated porous media is analyzed. The linear non-parallel flow model is employed in the instability analysis, which takes into account the streamwise variation as well as the transverse variation of the disturbance amplitude functions. The set of partial differential equations for the disturbance amplitude functions are converted to a system of homogeneous linear ordinary differential equations with homogeneous boundary conditions by the local non-similarity method. The resulting eigenvalue problem is then solved by an implicit finite-difference method. Representative neutral stability curves and critical Rayleigh numbers are presented. It has been found that as the angle of inclination relative to the horizontal increases, the surface heat transfer rate increases, whereas the flow becomes more stable to the vortex mode of instability. Also, as the inertia effect, expressed in terms of Forchheimer number, Fr, increases, the heat transfer rate decreases, but the flow becomes more stable. It is demonstrated that the non-parallel flow model predicts a more stable flow than the parallel flow model.

Measurements of turbulent mixed convection flow over a two-dimensional, vertical backward-facing step are reported. Laser-Doppler velocimeter (LDV) and cold wire anemometer were used, respectively, to measure simultaneously the time-mean velocity and temperature distributions and their turbulent fluctuations. The experiment was carried out for a range of free stream air velocities 0 m/s⩽ u ∞ ⩽0.41 m/s for a step height of 22 mm, and a temperature difference, ΔT , of 30°C between the heated walls and the free stream air. The step geometry consists of an adiabatic backward-facing step, an upstream wall and a downstream wall. Both the upstream and downstream walls are heated to a uniform and constant temperature. The present results reveal that the introduction of a small free stream velocity causes a decrease in the turbulence intensity of both the streamwise and transverse velocity and temperature fluctuations. Also, it was found that the reattachment length increases while the heat transfer rate from the downstream heated wall decreases as the free stream velocity increases.

A detailed measurement of heat transfer and fluid flow of turbulent boundary-layer air flow in natural and mixed convection adjacent to an isothermal vertical flat plate are reported. A cold-wire anemometer and laser Doppler velocimeter were used, respectively, to measure simultaneously the time-mean turbulent temperature and velocity distributions and their turbulent fluctuations. In this experimental study, measurements of the flow and thermal fields were carried out at one streamwise location, x

Measurements of turbulent natural convection boundary-layer air flow over a two-dimensional, vertical, backward-facing step are reported. The step geometry consists of an adiabatic backward-facing step, an upstream wall, and a downstream wall. Both the upstream and downstream walls are heated to a uniform and constant temperature. Laser-Doppler velocimeter and cold-wire anemometer were used, respectively, to measure simultaneously the time-mean velocity and temperature distributions and their turbulent fluctuations. The experiment was carried out for step heights of 0, 11, and 22 mm and a temperature difference, Delta T, of 30 degrees C between the heated walls and the ambient air. The present results reveal that the turbulence intensity of both the streamwise and transverse velocity fluctuations, and the intensity of temperature fluctuations downstream of the step, increase as the step height increases. Also, it was found that both the reattachment length and the heat transfer rate from the downstream he...

An analysis for condensation from an isothermal vertical flat plate in mixed convection is reported. The entire mixed convection regime is divided into two regions. One region covers the forced-convection-dominated regime, and the other covers the free-convection-dominated regime. The governing system of equations is first transformed into a dimensionless form by the nonsimilar transformation, separately for each regime, and then solved using the local nonsimilarity method along with a finite difference scheme. Two nonsimilarity parameters are introduced. The parameter {xi}{sub f} = Gr{sub x}/Re{sub x}{sup 2} characterizes the effect of buoyancy force on forced convection, while the parameter {xi}{sub n} = Re{sub x}/Gr{sub x}{sup 1/2} characterizes the effect of forced flow on free convection. Numerical results for pure steam and refrigerant R-134a are presented for both saturated and superheated cases. It is found that the buoyancy force significantly increases the wall shear stress and condensate mass flux. To a lesser degree, the buoyancy force also increases the wall heat flux. Superheating is found to have an insignificant effect on wall heat flux for a pure vapor.

Measurements are reported for buoyancy-opposing laminar mixed convection flow over a vertical, two-dimensional forward-facing step, in which the upstream wall of the step simulated an adiabatic surface, while both the step and the downstream wall from the step were heated to a uniform temperature that was higher than the approaching air temperature. Results presented in this paper are for a step height of 0.8 cm, over a range of free stream velocities 0.39 m s −1 ≤ u ∞ ≤ 0.71 m s −1 and a range of temperature differences 0°C ≤ ΔT ≤ 35°C between the heated surface and the free stream. Laser-Doppler velocimeter and cold-wire anemometer were used to measure, respectively, the air velocities and the temperature distributions simultaneously. Flow visualizations were also used to observe the nature of the flow and to measure the length of the recirculation region downstream of the forward-facing step. The results reveal that the buoyancy-opposing force due to the downstream wall heating affects significantly the velocity and temperature distributions, the local Nusselt number, and the location and the size of the recirculation region downstream of the step.

An analysis is performed for mixed convective flow through a fluid-saturated porous medium adjacent to a vertical surface with the heating condition of power-law variation in the wall temperature. The entire mixed convection regime is covered by the single parameter χ = [1+( Ra x / Pe x ) 0.5 ] −1 from the pure forced convection limit ( χ = 1) to the pure free convection limit ( χ = 0). In modeling the flow through porous media, non-Darcy effects such as high-flow-rate inertia forces, no-slip boundary condition, near-wall porosity variation and thermal dispersion are taken into consideration. Because of the porosity variation in the near wall region, the stagnant thermal conductivity also varies accordingly. The aims of the present work are to examine the effects of non-Darcian flow phenomena on mixed convective transport and to demonstrate the variation in heat transfer predictions based on different flow models. A finite difference scheme was used to solve the transformed system of equations. Numerical results show that non Darcian and thermal dispersion effects have significant influences on velocity profiles, temperature profiles and heat transfer rates from the vertical surface.

Measurements and predictions of laminar boundary-layer air flow in natural convection over a vertical two-dimensional backward-facing step are reported. The upstream and downstream walls and the step itself were heated to a uniform and constant temperature. The experiment was carried out for the ranges of step heights 3.5 mm ≤ s ≤ 9 mm, temperature differences of 5.8°C ≤ ΔT ≤ 23°C between the heated wall and the free stream (corresponding to 2.238 × 107 < Grxi < 8.877 × 107), and reference velocities of 0.24 m/s ≤ u* ≤ 0.47 m/s. Laser-Doppler velocimeter and cold-wire anemometer were utilized to measure, respectively, the velocity and the temperature distributions simultaneously. Flow visualization was also performed to determine the reattachment length. Measurements compare favorably with predictions. These results reveal that the step height significantly affects the velocity and temperature distributions, the friction coefficient, and the rate of heat transfer downstream of the backward-facing step.

Measurements and predictions are reported for buoyancy-assisting, laminar mixed convection boundarylayer flow of air along a two-dimensional, vertical backward-facing step, in which the upstream wall and the step are heated to a uniform temperature while the downstream wall is heated to the same or different uniform temperature. The experiments were performed on a backward-facing step geometry, with a step height of 0.8 cm, over a range of freestream velocities 0.26 < u^ < 0.41 m/s and a range of temperature differences 0 ^ Ar < 34°C between the heated walls and the freestream. Laser Doppler velocimeter and cold-wire anemometer were utilized to measure, respectively, the air velocity and the temperature simultaneously. Flow visualizations were also performed to determine the flow reattachment length behind the backward-facing step. Reported measurements are limited to the case where the upstream wall and the step are heated to the same uniform temperature as the downstream wall, but numerical results are presented for the cases where these two temperatures are different and also for the situation when the upstream wall and the step are maintained as adiabatic surfaces. These results reveal that heating of the upstream wall and the step significantly affects the reattachment length, the velocity and temperature distributions, and the rate of heat transfer downstream of the backward-facing step.

Flow and heat transfer characteristics of mixed convection from horizontal surfaces in a saturated porous medium are investigated. Two conditions of surface heating were considered, a variable wall temperature (VWT) and a variable surface heat flux (VHF). Two different nonsimilarity parameters for VWT and VHF cases, due to the nonuniform heating conditions, are found by nondimensionalizing the governing equations. The nonsimilarity parameters cover the entire regime of mixed convection from the pure forced convection limit at [chi] = 1 (or [chi]* = 1) to the pure free convection limit at [chi] = 0 (or [chi]* = 0). The resulting transformed governing equations are solved by a finite difference scheme. Numerical results for both VWT and VHT boundary conditions, including velocity and temperature profiles and local Nusselt numbers, are presented for selected values of the exponents n and m. Simple and accurate correlation equations valid for the entire mixed convection regime are also presented for the local and average Nusselt numbers.

The problem of mixed convection along an isothermal vertical plate in porous media with uniform surface injection or suction is studied. The analysis deals with the entire regime of mixed convection by introducing a single parameter X= [1+ (Rax / Pex )1/2]-1 where X= 1 corresponds to pure forced convection and x = 0 to pure free convection. This x parameter describes both buoyancy and forced flow effects. The nonsimilar variable ξ=(ϑ0x /α) (pex 1/2 + Rax 1/2)-1 represents the effect of injection (ϑ0> 0 or ξ> 0) or suction (ϑ< 0 or xi < 0) at the wall. The transformed nonlinear system of equations involving these parameters is solved using a finite difference method. Results are presented for temperature and velocity profiles, local wall shear stress, and local Nusselt number for the buoyancy assisting flow condition. It is found that suction increases, whereas injection decreases, the rate of heat transfer at the wall. Correlation equations are given for the local and average Nusselt numbers.

Mixed convection adjacent to a vertical cylinder in porous media is studied. The problem is solved using the nonsimilarity solution method for the case of uniform wall temperature (UWT). In the first approach, mixed convection is studied by examining separately the buoyancy effect on forced convection and the forced flow effect on free convection. These effects are represented, respectively, by RaJPex and PexlRax. Curvature is represented by gf = (xlr^Pe~* /2 in the forced-convection-dominated regime and by £„ = (x/r0)Raxl/2 in the free-convection-dominated regime. The second approach examines the entire regime of mixed convection by introducing the single parameter \ = [1 + (Rax/Pex)1/2]~l, where \ = I corresponds to pure forced convection and x = 0 to pure free convection. In this regime, the curvature is represented by £ = (xlr^[Pe*x2 + Ralx2]~*. In the analysis, the non-Darcian and thermal dispersion effects are neglected. The resulting nonsimilar systems of equations from the different approaches are solved using a finite difference method. Results are presented for temperature and velocity profiles, and local Nusselt number. Correlation equations are given for local and average Nusselt numbers in all regimes of mixed convection.

Measurements and predictions of buoyancy-assisting laminar mixed convection flow over a horizontal, two-dimensional forward-facing step are reported. Laser-Doppler velocimeter (LDV) and cold wire anemometer were used to simultaneously measure the velocity and the temperature distributions, respectively. Flow visualizations were conducted to determine the reattachment lengths for different inlet velocities (u(0) between 0.255 m/s and 0.50 m/s), wall freestream temperature differences (Delta-T between 0 C and 37 C), and step heights (s between 0.79 cm and 1.75 cm). The results reveal that the buoyancy force due to wall heating has a negligible effect on the velocity and temperature distributions and the reattachment lengths, as long as the flow remains stable and two-dimensional. The inlet velocity and the step height, on the other hand, significantly affect the flow and thermal fields. The local heat transfer coefficient is found to increase as the inlet velocity increases and the step height decreases. On the other hand, the length of the recirculation regions upstream and downstream of the step are found to increase as the inlet velocity and the step height increase. Correlation equations are developed to predict the reattachment lengths that appear upstream and downstream of the step. The measured results agree well withmore » numerical predictions. 10 refs.« less

Mixed convective heat transfer results for two-dimensional laminar flow in an inclined duct with a backward-facing step are presented for both the buoyancy assisting and the buoyancy opposing flow conditions. The wall downstream of the step is maintained at a uniform heat flux, while the straight wall that forms the other side of the duct is maintained at a constant temperature equivalent to the inlet fluid temperature. The wall upstream of the step and the backward-facing step are considered as adiabatic surfaces. The inlet flow is fully developed and is at a uniform temperature. The effects of the inclination angle and Prandtl number on the velocity and temperature distributions are reported.

Measurements and predictions of velocity and temperature distributions in buoyancy-assisting, laminar, mixed convection boundary layer flow over a vertical two-dimensional backward-facing step are presented. The wall downstream of the step was subjected to a uniform wall heat flux, and the wall upstream of the step and the step itself were kept adiabatic. Velocity and temperature distributions were measured simultaneously by using a laser-Doppler velocimeter (LDV) and a cold-wire anemometer. Flow visualization was also carried out to determine the reattachment length. The experiment covered a range of levels of wall heat flux (0 ⩽ q w ⩽ 290 W/m 2 ), free-stream velocities (0.25 ⩽ u ∞ ⩽ 0.7 m/s), and step heights (0.35 ⩽ s ⩽ 0.8 cm). The results indicate that as the wall heat flux increase the Nusselt number increases but the reattachment length decreases. The measured results reveal that axial conduction along the constant heat flux surface is very significant in the recirculation region. This finding necessitates the coupling of the convective heat transfer from the wall to the fluid with the axial conduction along the constant heat flux wall in order to accurately predict the measured data by the numerical solution.

Measurements of velocity and temperature distributions in buoyancy-assisting laminar mixed convection boundary-layer flow over a vertical, two-dimensional backward-facing step are reported. The leading surface upstream of the step and the step itself were adiabatic, and the surface downstream of the step was heated and maintained at a uniform temperature. A laser-Doppler velocimeter and a cold-wire anemometer were utilized to measure simultaneously the velocity and the temperature distributions in the recirculation and the reattached region downstream of the step. Flow visualization was used to study the flow and to measure the reattachment length for different free-stream velocities (0.37 m/s ≤ u0 ≤ 0.72 m/s), wall temperature differences (10°C ≤ ΔT ≤ 30°C), and step heights (0.38 cm ≤ s ≤ 1 cm). Results show that for a given step height the reattachment length decreases as the buoyancy force parameter, Grs/Res2, increases. The Nusselt number at the heated wall downstream of the step increases and the location of its maximum value moves closer to the step as the buoyancy force parameter increases. For the present experimental range, it is found that the location of the maximum Nusselt number occurs downstream of the reattachment point and the distance between the reattachment point and the location of the maximum Nusselt number increases as the buoyancy force parameter increases. Predicted behavior agrees favorably with the measured results.

The problem of mixed convection about a vertical flat plate embedded in a porous medium is analyzed. Nonsimilarity solutions are obtained for the cases of variable wall temperature (VWT) in the form T w (x) = T ∞ + ax n and variable surface heat flux (VHF) in the form q w (x) = bx m . The entire mixed convection regime is covered by two different nonsimilarity parameters χ = [1 + ( Ra x / Pe x ) 1/2 ] − and χ * = [1 + ( Ra x * / Pe x 3/2 ) 1/3 ] −1 , respectively, for VWT and VHF cases, from pure forced convection (χ = 1 or χ * = 1) to pure free convection (χ = 0 or χ * = 0). A finite-difference scheme was used to solve the system of transformed governing equations. Velocity and temperature profiles, and local Nusselt numbers are presented. It is found that as χ or χ * decreases from 1 to 0, the thermal boundary layer thickness increases first and then decreases, but the local Nusselt number in the form Nu x ( Pe x 1/2 + Ra x 1/2 ) −1 or Nu x ( Pe x 1/2 + Ra x *1/3 ) −1 decreases first and then increases. The correlation equations for the local and average Nusselt numbers are also obtained for the two surface heating conditions.

An analysis is performed to study the heat transfer characteristics of laminar mixed convective boundary-layer flow over a semi-infinite horizontal flat plate with non-uniform surface temperatures. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A nonsimilar mixed convection parameter χ and a pseudo-similarity variable η are introduced to cast the governing boundary layer equations and their boundary conditions into a system of dimensionless equations which are solved numerically by a weighted finite-difference method. The mixed convection parameter χ is chosen so that χ = 0 corresponds to pure free convection and χ = 1 corresponds to pure forced convection. Numerical results are presented for Prandtl numbers of 0.1, 0.7, 7 and 100 and representative values of the exponent n for the power-law variation in wall temperature. The heat transfer results are compared with existing correlations for the uniform wall temperature case and new correlations are derived for the general case of power-law wall temperature variations. It is found that an increase in the Prandtl number and exponent value n increases the local heat transfer rate.

Measurements and predictions of velocity and temperature distributions are presented for buoyancy-assisting mixed convection laminar boundary-layer flow over horizontal and inclined, twodimensional backward-facing steps. Laser-Doppler velocimeter and cold wire anemometer are used to measure simultaneously the velocity and the temperature distributions, respectively. Flow visualizations are carried out to determine the reattachment length and the onset/start of vortex instability for different wall temperatures (0°C ⩽ ΔT⩽ 30°C), free stream velocities (0.285 m s−1 ⩽ u∞ ⩽ 0.7 m s−1), step heights (0.35 cm ⩽ s ⩽ 0.8 cm), and inclination angles (30° ⩽ φ ⩽ 90°) as measured from the vertical direction. For the horizontal case, the buoyancy force has a negligible effect on the velocity and temperature distributions, but it influences significantly the onset/start of instability. The inclination angle, on the other hand, affects significantly the temperature and velocity distributions. As the inclination angle from the vertical increases, the local Nusselt number decreases, while the reattachment length and the location of the maximum Nusselt number behind the backward-facing step increase.

Mixed convection in a porous medium from horizontal surfaces with variable wall temperature distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξΓ=Rax/Pex3/2 is found to characterize the effect of buoyancy forces on the forced convection. The second region covers the free convection dominated regime where the dimensionless parameter ξn=Pex/Rax2/3 is found to characterize the effect of the forced flow on the free convection. To obtain the solution that covers the entire mixed convection regime, the solution of the first region is carried out for ξf=0. the pure forced convection limit, to ξf=1 and the solution of the second region is carried out for ξn=0, the pure free convection limit, to ξn=1. The two solutions meet and match at ξΓ=ξn=1. Numerical results for different wall temperature variations are presented. In addition, correlation equations for the local and average Nusselt numbers are obtained.

Nonsimilarity solutions for mixed convection from a vertical flat plate embedded in a porous medium are reported for two surface heating conditions: variable wall temperature (VWT) and variable surface heat flux (VHF) of the power-law form. The entire mixed convection regime is divided into two regions. One region covers the forced convection dominated regime and the other one covers the free convection dominated regime. The governing equations are first transformed into a dimensionless form by the nonsimilar transformation and then solved by a finite-difference scheme. Four nonsimilarity parameters are introduced. The parameters Rax/Pex and Rax*/Pex3/2 characterize the effect of buoyancy forces on the forced convection for the VWT and VHF cases, respectively; while the parameters Pex/Rax and Pex/Rax*2/3 characterize the effect of forced flow on the free convection for VWT and VHF cases, respectively. Numerical results for both heating conditions are presented. Correlation equations for the local and average Nusselt numbers are also presented.

An analysis is performed to study the flow and heat transfer characteristics of laminar free convection on isothermal vertical and horizontal flat plates with uniform vectored surface mass transfer. The governing equations are first cast into a dimensionless form by a nonsimilar transformation and the resulting equations are then solved by a finite difference method. Numerical results for gases with a Prandtl number of 0.7 are presented for representative values of the normal and streamwise components of the uniform surface mass transfer. For both vertical and horizontal plates, it has been found that an increase in the normal injection ( v w >0) results in a decrease in the local surface heat transfer rate for both downstream vectoring ( u w >0) and upstream vectoring ( u w >0). On the other hand, an increase in the normal suction ( v w >0) results in an increase in the local surface heat transfer rate. For a given value of the normal injection or suction parameter, an increase in the downstream vectoring increases the local surface heat transfer rate and decreases the local wall shear stress. The opposite is true for the case of increasing upstream vectoring. No experimental data are available for comparison.

A nonsimilarity solution for mixed convection from impermeable horizontal surfaces in a saturated porous medium is obtained for the case of variable heat flux. Solutions that cover the entire regime of mixed convection, including the two limits of pure forced convection and pure free convection, are made possible through using two different transformations to the governing equations. The nonsimilarity parameter ξ f = Ra x /Pe x 2 results from transformation of the governing equations for the forced flow dominated regime and the nonsimilarity parameter ξ f = Pe x /Ra x 12 arises from transformation of the governing equations for the buoyancy dominated regime. The two solutions provide results that cover the entire mixed convection regime from pure forced to pure free convection limit. Numerical results for different values of surface heat variation are presented. Correlation equations for the local and average Nusselt numbers, valid for the entire mixed convection regime, are also presented.

A linear theory based on the nonparallel flow model is employed to study the onset of longitudinal vortex instability of laminar mixed convection flow over horizontal and inclined flat plates with variable surface temperature, Tw(x) − T∞ = Axn. In the analysis, the streamwise dependence of the disturbance amplitude functions is taken into account. Neutral stability curves as well as the critical values for the parameter G ∗ = Gr ∗ x /Re x ∗ 3 2 and wave number α ∗ are presented for Prandtl numbers Pr = 0.7, 7, 100, and 1000 over a range of the exponent values −0.5 ⩽ n ⩽ 1.0 and inclination angles 0° ⩽ φ ⩽ 85°. For a given Prandtl number and inclination angle, thermal instability is found to decrease as the value of the exponent n increases. Also, for given values of the exponent n and Prandtl number Pr, the critical value of Gr x ∗ /Re x ∗ 3 2 increases with increasing inclination angle from the horizontal. However, the critical wave number α ∗ appears to be unaffected by the inclination angle. The results from the present nonparallel flow analysis are compared with available analytical and experimental results from previous studies. The nonparallel flow analysis that accounts for the streamwise dependence of the amplitude functions is found to have a stabilizing effect as compared with the parallel flow analysis in which the streamwise dependence of the disturbance is neglected.

The vortex instability characteristics of laminar boundary-layer flow in natural convection on inclined flat plates heated from below, under the variable surface temperature Tw(x) − T∞ = Axn, are studied analytically by the linear theory. The analysis is performed by using the non-parallel flow model in which the steady main flow is treated as two-dimensional and account is taken of the streamwise dependence of the disturbance amplitude functions. Neutral stability curves as well as critical Grashof numbers and the corresponding critical wave numbers are presented for fluids having Pr = 0.7 and 7 over the range of inclination angles, 0° ⩽ φ ⩽ 70° from the horizontal, for a range of the exponent values n from − 1 3 to 1. For a given Prandtl number and a given exponent value n, the flow is found to become more stable to the vortex mode of instability as the inclination angle increases from the horizontal. In addition, the local non-similarity non-parallel flow model provides a larger critical Grashof number than that of the local similarity non-parallel flow model. Results from the present non-parallel flow analysis are compared with previous results from the parallel flow analyses and with available experimental data. The streamwise dependence of the disturbances leads to a stabilization of the main flow, which brings the present predictions to a close and qualitative agreement with available experimental data.

An analysis is performed to study the vortex instability of laminar boundary-layer flow in natural convection over a horizontal flat plate with variable surface temperature, T w ( x ) − T ∞ = Ax n . The analysis is based on the linear non-parallel flow model in which the steady main flow is treated as two-dimensional and account is taken of the streamwise dependence of the disturbance amplitude functions. Neutral stability curves as well as critical Grashof numbers and critical wave numbers are presented for Prandtl numbers of 0.7 ⩽ Pr ⩽ 100, over a range of the exponent values n from- sol1 3 to 1.0. For a given Prandtl number, the flow is found to become more stable to the vortex mode of instability as the value of the exponent n increases. However, fluids with a larger Prandtl number are found to be more susceptible to the instability than fluids with a lower Prandtl number. Results from the present non-parallel flow analysis are compared with available results from the parallel flow analyses and experiments. The streamwise variation of the disturbances is found to stabilize the flow.

Mixed convective heat transfer results for laminar, buoyancy-assisting, two-dimensional flow in a vertical duct with a backward-facing step are reported. The present numerical study examines a wide range of inlet flow and wall temperature conditions to cover the domain from pure forced convective flow, where the buoyancy force effects are not present, to the inlet starved convective flow where the buoyancy force effects are significant and where the average inlet velocity is smaller than the corresponding natural convective value. The results compare very favorably with existing, but limited, experimental and numerical data. This study focuses on a backward-facing step geometry with an expansion ratio of 2, but the general observed behaviors are applicable to similar geometries with different expansion ratios. The buoyancy-induced flow decreases the reattachment length and pushes the recirculating region away from the heated wall. Velocity and temperature distributions along with Nusselt numbers and wall friction coefficients are presented for wide ranges of flow and temperature parameters.