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1.
The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water
at 4°C (maximum density) when the surface heat flux varies as x
m
and the velocity outside the boundary layer varies as x
(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions
of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions
when the mixed convection parameter λ is greater than some critical value λ
c
(dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ
c
is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and
the limits as m → ∞ and as m → −1. 相似文献
2.
Steady mixed convection boundary layer flow from an isothermal horizontal circular cylinder embedded in a porous medium filled
with a nanofluid has been studied for both cases of a heated and cooled cylinder. The resulting system of nonlinear partial
differential equations is solved numerically using an implicit finite-difference scheme. The solutions for the flow and heat
transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle
volume fraction φ and the mixed convection parameter λ. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2. It is found that for each particular nanoparticle, as the nanoparticle volume fraction φ increases, the magnitude of the skin friction coefficient decreases, and this leads to an increase in the value of the mixed
convection parameter λ which first produces no separation. On the other hand, it is also found that of all the three types
of nanoparticles considered, for any fixed values of φ and λ, the nanoparticle Cu gives the largest values of the skin friction coefficient followed by TiO2 and Al2O3. Finally, it is worth mentioning that heating the cylinder (λ > 0) delays separation of the boundary layer and if the cylinder
is hot enough (large values of λ > 0), then it is suppressed completely. On the other hand, cooling the cylinder (λ < 0) brings
the boundary layer separation point nearer to the lower stagnation point and for a sufficiently cold cylinder (large values
of λ < 0) there will not be a boundary layer on the cylinder. 相似文献
3.
Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law 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 ζ
f
=Ra*
x
/Pe2
x
is found to characterize the effect of buoyancy forces on the forced convection with K
′
U
∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where
the dimensionless parameter ζ
n
=Pe
x
/Ra*1/2
x
is found to characterize the effect of the forced flow on the natural convection, with (K
′
U
∞/ν)Ra*1/2
x
/Pe
x
characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the
solution of the first regime is carried out for ζ
f
=0, the pure forced convection limit, to ζ
f
=1 and the solution of the second is carried out for ζ
n
=0, the pure natural convection limit, to ζ
n
=1. The two solutions meet and match at ζ
f
=ζ
n
=1, and R
*
h
=G
*
h
.
Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall
temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers
the forced convection dominated regime where the dimensionless parameter ξ
f
=Ra
x
/Pe
x
3/2 is found to measure the buoyancy effects on mixed convection with Da
x
Pe
x
/ɛ as the wall effects. The second region covers the natural convection dominated region where ξ
n
=Pe
x
/Ra
x
2/3 is found to measure the force effects on mixed convection with Da
x
Ra
x
2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature
exponents are presented.
Received on 8 July 1996 相似文献
4.
The effect of lateral mass flux on mixed convection heat and mass transfer in a saturated porous medium adjacent to an inclined
permeable surface is analyzed. A similarity solution is obtained when surface temperature and concentration, free stream velocity
and injection/suction velocity of fluid are prescribed as power functions of distance from the leading edge. The cases when
the flow and buoyancy forces are in the same and opposite directions are discussed both for aiding and opposing buoyancy effects.
The governing parameters are the mixed convection parameter Gr, the Lewis number Le, the buoyancy ratio N, the lateral mass flux parameter f
w, representing the effects of injection or withdrawal of fluid at the wall, and λ which specifies three cases of the inclined
plate. The interactive effect of these parameters on heat and mass transfer rates are presented. It is observed that the diffusion
ratio (Le) has a more pronounced effect on concentration field than on flow and temperature fields. It is found that the rates
of heat and mass transfer increase with suction and decrease with injection of the fluid.
Received on 31 August 2000 / Published online: 29 November 2001 相似文献
5.
M. Thiele 《Heat and Mass Transfer》1997,33(1-2):7-16
An analysis has been performed to study the influence of velocity dependent dispersion on transverse heat transfer in mixed
convection flow above a horizontal wall of prescribed temperature in a saturated porous medium. The Boussinesq approximation
and boundary layer analysis were used to numerically obtain gravity affected temperature and velocity distributions within
the frames of Darcy's law and a total thermal diffusivity tensor comprising both of constant coefficient heat conduction and
velocity proportional mechanical heat dispersion. Dependending on Pe∞, the molecular Peclét number basing on the effective thermal diffusivity and the velocity of the oncoming flow, density coupling
has distinct influences on heat transfer rates between the wall surface and the porous medium flow region. For small Peclét
numbers, when heat conduction is the prevailing mechanism, wall heat fluxes are the higher the larger the density difference
between the oncoming and the near wall fluid is. The opposite is true for larger Peclét numbers, when mechanical heat dispersion
is the main cause of heat spreading. For Pe∞ tending to infinity these wall heat fluxes approach finite maximum values in the total heat diffusivity model, they grow
beyond any limit if only constant coefficient heat conduction is considered. Thus, the inclusion of mechanical heat dispersion
effects yields physically more realistic predictions.
Received on 18 September 1996 相似文献
6.
Nonsimilarity solutions for non-Darcy mixed convection from a vertical impermeable surface embedded in a saturated porous
medium are presented for variable surface heat flux (VHF) of the power-law form. The entire mixed convection region is divided
into two regimes. One region covers the forced convection dominated regime and the other one covers the natural 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. Computations are based on Keller Box method and a tolerance of iteration of
10−5 as a criterion for convergence.
Three physical aspects are introduced. One measures the strength of mixed convection where the dimensionless parameter Ra*
x
/Pe3/2
x
characterizes the effect of buoyancy forces on the forced convection; while the parameter Pe
x
/Ra*2/3
x
characterizes the effect of forced flow on the natural convection. The second aspect represents the effect of the inertial
resistance where the parameter K′U
∞/ν is found to characterize the effect of inertial force in the forced convection dominated regime, while the parameter (K′U
∞/ν)(Ra*2/3
x
/Pe
x
) characterizes the effect of inertial force in the natural convection dominated regime. The third aspect is the effect of
the heating condition at the wall on the mixed convection, which is presented by m, the power index of the power-law form heating condition.
Numerical results for both heating conditions are carried out. Distributions of dimensionless temperature and velocity profiles
for both Darcy and non-Darcy models are presented.
Received on 26 May 1997 相似文献
7.
In this study, the mixed convection of water at 4°C along a wedge in a porous medium is investigated numerically using finite
difference method. In order to explore the effect of mixed convection, both forced and free convection-dominated regimes are
considered. Non-similarity solutions are obtained for the variable wall flux boundary condition. Velocity and temperature
profiles as well as local dimensionless skin friction and Nusselt number are obtained and compared with the available numerical
results for various values of different parameters. The wedge angle geometry parameter m and mixed convection parameter ξ are ranged from 0 to 1 in both regimes whereas different values of λ are considered for the purpose of comparison of heat transfer results. 相似文献
8.
The steady mixed convection boundary-layer flow on an upward facing horizontal surface heated convectively is considered. The problem is reduced to similarity form, a necessary requirement for which is that the outer flow and surface heat transfer coefficient are spatially dependent. The resulting similarity equations involve, apart from the Prandtl number, two dimensionless parameters, a measure of the relative strength of the outer flow M and a heat transfer coefficient γ. The free convection, M=0, case is considered with the asymptotic limits of large and small γ being derived. Results for the general, M>0, case are presented and the asymptotic limit of large M being treated. 相似文献
9.
Laminar mixed convection from a continuously moving vertical surface with suction or injection 总被引:1,自引:0,他引:1
The boundary layer flow over a uniformly moving vertical surface with suction or injection is studied when the buoyancy forces
assist or oppose the flow. Similarity solutions are obtained for the boundary layer equations subject to power law temperature
and velocity boundary conditions. The effect is of various governing parameters, such as Prandtl number Pr, temperature exponent
n, injection parameter d, and the mixed convection parameter λ=Gr/Re2, which determine the velocity and temperature distributions and the heat transfer coefficient, are studied. The heat transfer
coefficient increases as λ assisting the flow for all d at Pr=0.72 however, for n=−1 it decreases sharply with λ. On the other hand, increasing λ has no effect on heat transfer coefficient for Pr=10 at n=0, and 1 for almost all values of d studied. However, for n=−1 it has similar effect as for Pr=0.72. It is also found that Nusselt number increases as n increases for fixed λ and d.
Received on 26 March 1997 相似文献
10.
The steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in
a fluid-saturated porous medium is investigated. Using appropriate similarity variables, the governing system of partial differential
equations is transformed into a system of ordinary differential equations. This system is then solved numerically. The features
of the flow and the heat transfer characteristics for different values of the governing parameters, namely, the Darcy–Brinkman,
Γ, mixed convection, λ, and slip, γ, parameters, are analysed and discussed in detail for the cases of assisting and opposing
flows. It is found that dual solutions exist for assisting flows, as well as those usually reported in the literature for
opposing flows. A stability analysis of the steady flow solutions encountered for different values of the mixed convection
parameter λ is performed using a linear temporal stability analysis. This analysis reveals that for γ = 0 (slip absent)
and Γ = 1 the lower solution branch is unstable while the upper solution branch is stable. 相似文献
11.
Robert McKibbin 《Transport in Porous Media》1986,1(3):271-292
The theory describing the onset of convection in a homogeneous porous layer bounded above and below by isothermal surfaces
is extended to consider an upper boundary which is partly permeable. The general boundary condition p + λ ∂p/∂n = constant is applied at the top surface and the flow is investigated for various λ in the range 0 ⩽ λ < ∞. Estimates of the magnitude and horizontal distribution of the vertical mass and heat fluxes at the surface, the horizontally-averaged
heat flux (Nusselt number) and the fraction of the fluid which recirculates within the layer are found for slightly supercritical
conditions. Comparisons are made with the two limiting cases λ → ∞, where the surface is completely impermeable, and λ = 0, where the surface is at constant pressure. Also studied are the effects of anisotropy in permeability, ξ = K
H
/K
V
, and anisotropy is thermal conductivity, η = k
H
/k
V
, both parameters being ratios of horizontal to vertical quantities. Quantitative results are given for a wide variety of
the parameters λ, ξ and η. In the limit ξ/η → 0 there is no recirculation, all fluid being converted out of the top surface, while in the limit ξ/η → ∞ there is full recirculation. 相似文献
12.
The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated
with water at 4°C (maximum density) when the temperature of the plate varies as x
m
and the velocity outside boundary layer varies as x
2 m
, where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel
to a free stream velocity U
∞ oriented in the upward or downward direction, while the ambient temperature is T
∞ = T
m (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further
transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential
equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system
of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme.
Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles
are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible
for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ
and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at
4°C (maximum density) compared to water at normal temperature. 相似文献
13.
A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal
vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical
method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest.
The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be
uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters
(λ
1, λ
2) in the range of −2.0≤λ
1≤−0.5 and −1.0≤λ
2≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow
of water. 相似文献
14.
An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe x =Re x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases. 相似文献
15.
Ali Nouri-Borujerdi Amin R. Noghrehabadi D. Andrew S. Rees 《Transport in Porous Media》2007,69(3):343-357
This paper considers the onset of free convection in a horizontal fluid-saturated porous layer with uniform heat generation.
Attention is focused on cases where the fluid and solid phases are not in local thermal equilibrium, and where two energy
equations describe the evolution of the temperature of each phase. Standard linearized stability theory is used to determine
how the criterion for the onset of convection varies with the inter-phase heat transfer coefficient, H, and the porosity-modified thermal conductivity ratio, γ. We also present asymptotic solutions for small values of H. Excellent agreement is obtained between the asymptotic and the numerical results. 相似文献
16.
A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between
two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux
and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The
Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes
equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow
and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio,
K
r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity
ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than
the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K
r ≈ 1) and for all values of λ the porous material acts as thermal insulation. 相似文献
17.
H. C. Agrawal 《Applied Scientific Research》1960,9(1):177-189
Summary The problem of heat transfer for laminar flow between two infinite parallel plates, y=±l, x≤0, kept at a constant temperature T
0, and y=±l, x≥0, kept at a different constant temperature T
s
is formulated to take into account the effect of heat diffusion on the incident fluid. This has been achieved by obtaining
solutions of the energy equation for the regions x≤0 and x≥0 and by imposing continuity conditions on the temperature and its derivative at the junction x=0. It is found that at small Péclét numbers the incident temperature is affected by the diffusion of heat from the right
(x>0) to the left (x<0). This effect is negligible for large Péclét numbers (Pe ∼ O(1000)). Further the temperature of the incident fluid at x=0 cannot be taken as constant (=T
0) if the heat generated by viscous dissipation is taken into consideration. Detailed solutions are given for Pe=1. Mean-mixed temperatures and local Nusselt numbers for x>0 and x<0 are tabulated and shown graphically. 相似文献
18.
In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a
vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to
a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations
and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation,
(2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values
of ξ ( = Gr
x
/Re
x
2, defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned
methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients
have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of
the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied.
We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity
gradient is higher than that of temperature.
On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh. 相似文献
19.
The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal
and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are
included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u
w/(u
w + u
∞), where u
w and u
∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1
it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have
been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation
technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution.
For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase
with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and
the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction
decreases as the relative velocity between the surface and free stream decreases.
Received on 17 May 1999 相似文献
20.
The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving
thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper.
The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated
into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed
into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference
scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include
the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall)
temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented
in graphical form and are discussed in detail. 相似文献