Asymptotic Model of Free Convection Flow on a Vertical Surface in
Porous Media with Newtonian Heating
Anna A.Bocharova a, Irina V. Plaksina b and Andrey A. Obushnyy c
1
Far Eastern Federal University, School of Engineering, 8, Suhanova str., Vladivostok,
690950, Russia
a
b
[email protected], [email protected], [email protected]
Keywords: Free convection, asymptotic analysis, porous media.
Abstract. Free convection on a vertical surface with Newtonian heating of the
form proposed by Merkin [1994]
in the fluid-filled porous medium
is
considered on the basis of the full equations of a viscous liquid. With the
help
of
dimensional
analysis
a
set
of
criteria
that
define
the
characteristics of flow and heat transfer was derived. Asymptotic analysis of the full
equations allowed us to determine the region of applicability of the boundary
layer approximation, which was used in the previous studies of this problem. Darcy parameter
influence was studied, the composite numerical and analytical solution for stream function and
temperature was derived.
Introduction
A mathematical model free-convective boundary layer occurring near a vertical surface used
in most scientific studies to research the characteristics of heat transfer in porous media. A broad
overview of the solutions obtained for different boundary conditions and geometric configurations
presented in reviews Ingham, Pop (1998) [1] and Nield, Bejan (1999). [2] Typical boundary
conditions are the conditions of constant temperature and constant heat flux at the surface. The
problem of free convection, which arises under the condition that the rate of heat transfer from the
boundary surface thermally coupled with the convective flow is proportional to the local surface
temperature has applications in various engineering problems than the problem with a
predetermined surface temperature and a predetermined heat flux.
Free convection boundary layer in porous media with Newtonian heating on a vertical surface
considered in [1], is based on the Darcy model, which does not make it possible to obtain a solution
satisfying homogeneous boundary conditions for the longitudinal velocity on the surface. In this
paper, free convection on a vertical surface with Newtonian heating of the form proposed by
Merkin [2] in the fluid-filled porous medium is considered on the basis of the full equations of a
viscous liquid.
Мathematical model
We will consider the problem of free convection near a hot vertical plane bounding a semiinfinite porous medium filled with a fluid and having a temperature T . It is assumed that on the
surface the heat flux is proportional to the local surface temperature. It is assumed that the fluid and
the porous matrix are in thermodynamic equilibrium, and that the properties of the fluid and matrix
are isotropic and constant. Neglecting the effect of viscous dissipation, we have the following
system of mass, momentum and energy conservation equations [6]:
V  0,
We consider the problem of free convection near a hot vertical plate adjacent to a semiinfinite porous medium filled with a fluid and having a temperature T . It is assumed that on the
surface the heat flux is proportional to the local surface temperature. It is also assumed that the
fluid and the porous matrix are in thermodynamic equilibrium, and that the properties of the fluid
and matrix are isotropic and constant.
the
The choice of dimensionless variables [2]: dimensionless coordinates x  x l and y  y l ,
gKT
stream function, the   0  , scale velocity 0  U 0 l , length l 
, and
a f hs 2
temperature   T  T  T , the system of the equations and boundary conditions is based on the
viscous, convective and inertia forces (Brinkman’s quadratic correction) terms.
 Pr *
  2
  2
m Pr *  
2m Pr * Gr    2    2  

,

  
  
  2    2  4  

2
y x
x y
Da  y
Da
x x 2 
 Ra
 y y
   
1
(1)


 2 ,
2
y x x y Ra




 



 0,
 0 ,   0 as y  , x  0,
  Ra 1   at y  0, x  0,
x
y
y
y
 

 0 ,   0 at x  0 .
y
x
Here, Pr 
 f
a
, Da  Khs , Gr 
2
(2)
CgT K 2
2f
Gr = , and Ra  hsl
are the dimensionless
parameters governing the process, parameters the designations are as follows: ε is porosity, νf is
kinematic viscosity, a is the coefficient of temperature conductivity of the porous medium, K is the
porous medium permeability, hs is a constant coefficient of heat transfer from the surface, C is the
theoretically determined inertia coefficient [2], g is the gravity acceleration, and β is the thermal
expansion coefficient
Asymptotic analysis of the full equations allowed us to determine the region of applicability
of the boundary layer approximation as a region of order Ra 1 ; within which the previous studies
of this problem were held [1,2]. Taking into account the viscous forces we will draw an asymptotic
analysis of the system of equations (1), (2), as Ra   . It is assumed that the thickness of the
thermal and dynamic boundary layers are of the same order, i.e. Pr *  О1 .
We will construct an external solution for the boundary layer in the following form:
 x, y, Ra   Ra 1  0 x, Y   Ra 2  1 x, Y   ..., x, y, Ra    0 ( x, Y )  Ra 1 1 x, Y   ....,
where y  Ra 1Y .
For the unknown functions Ф0 ,  0 we obtain a system of differential equations:
4
 0  3  0  0  3  0 m Pr *   0  2  0 
2mGr Pr *  0  2  0
*  0






Pr

,
Y xY 2
x Y 3
Da  Y
Da
Y Y 2
Y 2 
Y 4
 0  0  0  0  2 0


,
Y x
x Y
Y 2
(3)
with boundary conditions
at Y  0 ,
 0  0

 0,
x
Y
 0
 0 , 0  0 at Y   , which was solved numerically [3].
Y
4
 0  3 1  1  3 0  1  3 0  0  3 1  Pr *  1  2 1 
*  1








Pr

Y xY 2 Y xY 2
x Y 3
x Y 3
Da  Y
Y 2 
Y 4
  0  2 1  1  2 0 

,

2
Y Y 2 
 Y Y
 1  0  0 1  0 1  1  0  21




,
Y x
Y x
x Y
x Y
Y 2

2 xGr Pr *
Da
1
1 1
 1 at Y  0 , 1  0 at

 0,
Y
x
Y
Y   , and when conditions obtained from the matching of the outer and inner expansions for the
 1
longitudinal velocity
x ,   1 x ,0 .
Y
y
Appling the transformation 1 ( x,Y )  F1 ( Y ), 1 ( x,Y )  H1 ( Y ) we reduce system to system of
ordinary differential equations
with the boundary conditions on the surface
1
F0 F1 F0F1   H1  F1  2 xGr F0F   0,
*
Da
Da
Pr
H1  F0 H 1  2 F0H 1  F1 H 0  2 F1 H 0 ,
F1 0  F10  0, H1( 0 )  H1 ( 0 )  1, F1   F0  , H1   0 .
F1
(8)
(9)
Numerical solution of equations (8), (9) is similar to the solution of (6), (7). Inhomogeneous
boundary conditions at infinity, take into account the effect of the displacement of the outer flow on
the boundary layer near the surface.
Results
The results of numerical solutions of equations (3) are presented in Fig. 1 (without
Brinkman's term) F0Y  H 0 Y 
F0Y 
1
H 0 Y 
Da=0.1
Da=0.01
Da=0.001
0.8
4
3.2
0.6
2.4
0.4
1.6
а)
0.2
0
0
2
4
6
8
Da=0.1
Da=0.01
Da=0.001
Da=0
10
0.8
б)
0
0
2
4
6
8
10
Fig. 1. Plots of the solutions of system of equations (6) for different values of the parameter
Darcy, Pr*=1 and m  0.5 .
Numerical solution for the longitudinal flow velocity and temperature is represented in Fig 1,
a and b, respectively, for different values of the governing parameter Da . The obtained results are
in a good agreement with the previous studies. As it is shown in Fig. 1b, the temperature flow
decreases with decrease in Da , and for Da  0 the solution corresponds to the temperature
profile obtained by Lesnic [1]. The results of the solution for the stream function for different
values of parameter Pr

show that the characteristics of the process are weakly dependent on this
parameter.
To investigate the influence of parameter Darcy let us consider the asymptotic analysis of the
system of equations in the approximation of the boundary layer under Da  0 , external expansion
has the following form:
x, Y , Da   0 x, Y   Da 2 1 x, Y   ... , x, Y , Da   T0 x, Y   Da 2T1 x, Y   .... ;
1
1
internal expansion:
1
1
x, Y , Da   Da 2  0 x, ~
y   Da1 x, ~
y   ..., x, Y , Da    0 ( x, ~
y )  Da 21 x, ~
y   ....
For the functions of zero-order of the external and internal expansions the system of
differential equations that was solved analytically was obtained. The composite solution for the
longitudinal velocity and temperature suitable for the entire range of applicability of the boundary
layer equations was constructed:


Y
 с
x, Y   h0 0e  h0 0 Y  h0 0e Da ,  с x, Y   h0 0e  h0 0 Y .
(4)
Y
The results of comparing the analytical solution of (4) with the corresponding numerical
solution [3] of the external problem show their good agreement.
Summary
The model of free convection flow near an impermeable vertical surface bounded by a semiinfinite porous medium with the given heat transfer from the surface is described. The thickness of
the viscous boundary layer is determined as the quantity of the order Ra 1 on the basis of on
asymptotic analysis of the momentum and energy conservation equations. Аn analytical solution for
the longitudinal velocity and the temperature was obtained; the solution is uniformly applicable
throughout the entire boundary layer region and satisfies the no-slip condition on the surface.
Acknowledgment
This work was supported by the Ministry of Education and Sciences of the Russian Federation,
Agreement № 14.578.21.0024, RFMEFI57814X0024.
Reference
[1]
Lesnic, D., D.B. Ingham, I. Pop, Free convection boundary-layer flow along a vertical surface in a
porous medium with Newtonian heating // Int. J. Heat Mass Transfer, 1999. – 42. – p. 2621-2627.
[2]
Merkin J.H., Natural-convection boundary-layer flow on a vertical surface with Newtonian
heating // Int. J. Heat and Fluid Flow, 1994. – 15(5). – p. 392-398.
[3]
Bocharova A.A., Plaksina I.V., Boundary Effect on free convection flow in a porous
medium at given heat transfer from a vertical surface // Fluid Dynamics, 2011. 46(6). – p. 984-991.