DISSIPATION EFFECTS ON NATURAL CONVECTION FLOW OF A SECOND
GRADE FLUID ALONG A VERTICAL SURFACE WITH VARIABLE SURFACE
TEMPERATURE.
Naeem Mustafa, …
Principal Scientist, Theoretical Physics Division, PINSTECH, P. O. NILORE, Islamabad.
ABSTRATCT
A mathematical model is formulated to analyze the effects of dissipation term on
natural convective flow of second grade fluid along a vertical plate maintained at variable
surface temperature. It is pertinent to mention here that we have considered proper and full
dissipation term for second grade fluid instead of its simplified form. Results of the
governing equations are obtained using local non-Similarity method (LNS) as well as KellerBox method (KBM). The regular perturbation series method has also been employed to
obtain a generalized series of the governing equations and thus solution is obtained using this
generalized series for smaller values of the local Eckert number ξ together with the Padé
approximation. Values of local skin-friction and local heat transfer are tabulated and
compared for various values of physical parameters. Results obtained by LNS are found in
excellent agreement with that of KBM. Effects of local Eckert number ξ on the velocity and
temperature profiles for different values of Prandtl number Pr and Deborah number De is
shown graphically. Streamlines and isotherms are also shown for different values of pertinent
parameters.
KEYWORDS
Viscous dissipation, Local non-similarity method, Keller-Box method, Natural convection,
second grade fluid, Padé approximation, Eckert number.
INTRODUCTION
The natural buoyant flows of non-Newtonian fluids along cooled or heated surfaces is
one of the basic scenarios of heat transfer and are very important owing to their relevance in
engineering processes of heat removal, for example the cooling of used nuclear fuel pallets in
nuclear power applications and the solar collectors. The earliest research on natural buoyancy
along a vertical surface traced back to Beard and Walter [1]. They investigated twodimensional boundary layer flow of non-Newtonian elastico-viscous fluid near a stagnation
point. On the behalf of massive application of non-Newtonian fluids in industry, the Rivlin-
Erickson fluid flow past a stretching plate was analyzed by Siddappa and Khapate [2] for this
class of non-Newtonian fluids. Numerous studies of convection flows along a vertical wall
followed. Siddappa and Hiremath [3] extended the above study for the flow along permeable
surface with variable transpiration velocity. Similarly Rajagopal et al. [4] independently
examined the same problem investigated by Siddappa and Khapate in [2] numerically. Using
Karman-Pohlhausen approximate integral method, Bujurke et al. [5] treated the flow and heat
transfer of a second-order fluid along a stretched sheet. Recently, Heat transfer due to natural
convection is frequently encountered in our environment and in engineering devices; since,
natural convection arises from the buoyancy force induced by density difference in a fluid
and density difference is a consequence of temperature gradient within the fluid. Free
convection flow is a significant factor in several practical applications that include, for
example, cooling of electronic components. Laminar free convection along horizontal,
inclined and vertical plates with uniform surface temperature or uniform surface heat flux has
been extensively studied for viscous fluid numerically [7-13]. Numerous experimental
investigations are also available on natural convection flows from vertical, inclined and
horizontal surfaces, covering both laminar and turbulent regimes under either a constant wall
temperature or constant surface heat flux condition.
Natural convection in viscoelastic fluids has also been investigated because of the
applications in geophysics Dávalos-Orozco et al. [17]. The Oldroyd’s model was first
thoroughly investigated by Takashima [20] with fixed temperature at both walls and later by
Kolkka and Ierley [21] with fixed heat flux at the boundaries. This dissertation mainly
investigates the buoyant flow of non-Newtonian second grade viscoelastic fluid with variable
temperature distribution that has yet not been discussed in the literature.
Heat transfer because of natural convection play important role in several engineering
and practical applications that include, crystal growth, geothermal systems, heat exchangers,
nuclear reactors, metallurgical processes. Natural buoyancy along inclined, vertical and
horizontal surfaces having constant temperature of surface and/or at uniform heat flux of
surface has already been examined analytically [22-26] but only for viscous fluids.
Experimental investigations covering both laminar and turbulent buoyant flow from vertical,
inclined and horizontal plates, for constant temperature of the plate or for constant heat flux
of the wall e.g. Shaukatullah et. al. [27], Yousaf et. al. [28], Vliet [29], Fujii et. al. [30], Vliet
et. al. [31] and Siebers et. al. [32] also exist. Nevertheless, these experimental and analytical
investigations were carried for uniform thermal boundary conditions.
Boundary layer flow of viscoelastic fluid has drawn the attentions of many investigators as
one has to treat these fluids e.g. in polymer processing (Zakariya [33]). Second-grade fluid
flow past a stretched sheet has been examined (see Rajagopal et al. [4]). For second grade
fluid Bhattacharya et al. [34] investigated the steady flow past a stretched surface and
determined also the solution of temperature distribution. These fluids are called differential
type fluids or Rivlin Ericksen fluids (Rivlin et al. [35]). This fluid shows viscoelastic
behavior, meaning that very short portion of the deformation gradient history leaves an effect
on the stresses. In a differential type incompressible fluid stresses are dependent on velocity
gradient, on indeterminate pressure and some higher-time derivatives of velocity gradient.
This fluid does not reveal the “stress relaxation” that means the stress becomes pure pressure
when all local motion ceases. The status of second grade viscoelastic fluid and related
discussions are in Dunn et al. [36]. While on the other side, some fluids exhibit the “stress
relaxation” phenomena. Examples of such fluids according to Joseph [37] are polymer melts
e.g. high viscosity silicone oils, polymers mixed in Newtonian solvents and plastics in molten
form. But all these studies were either for constant temperature or for uniform heat flux
surface conditions.
Natural convection phenomenons in devices, which decelerate rapidly or rotate with high
speed, involve dissipative forces. Dissipation effects appear in situations where field of
gravitation is strong and/or arise in the large-scale processes. For example, it occurs in many
geological phenomena where fluids internals are contacting to the solid bodies, in a mass of
gas in space of relatively great size and on the large planets. In particular, processes of natural
convection in rotating cavities received great attention and similarly the cooling of turbine
blade by natural convection flow including dissipation effects are studied carefully by
Lighthill [38]. Ostrach [39] considered the effects of viscous dissipation in cavities with fully
developed velocity profiles and temperature distribution. Furthermore, he considered onedimensional steady flow and discussed the variations of buoyancy on the velocity fields. The
analysis of dissipation effects in the natural convection problems give rise to an independent
physical parameter known as the Eckert number which is not dependent on Prandtl number Pr
and Grashof number Gr.. The investigation of B. Gebhart [40] on the natural convection flow
with dissipation effects is of great interest. He considered two dimensional Navier Stocks
equations with boundary layer and Boussinesq approximations and also considered the
dissipation effects and reduced the governing relations to ODE’s employing proper similarity
technique and obtained the solution of the final equations employing perturbation technique.
Similarly, many researchers have studied steady state free convection flows. M. Subhas Abel
et. al. [41] analysed the transfer of heat for a viscoelastic fluid flow over a sheet which is
being stretched considering a variable source of heat and viscous dissipation. Energy
equation has been solved for second grade fluid flow over a surface being stretched for large
Prandtl numbers Pr by H. R. Nataraja [42] and he considered the proper dissipation term of
the second order fluid instead of viscous dissipation. According to our knowledge, for second
grade fluid the effect of dissipation term on buoyancy flow along a sheet considering second
grade dissipation term has not been studied yet. The defining equations for this problem are
derived from the Navier-Stokes eqs. and the energy eq. is taken with the tensor of second
grade fluid. These are converted to a system of parabolic PDE’s using proper similarity
variables. An asymptotic method and two numerical algorithms; the Keller Box technique
and the LNS are utilized to simulate the solution of the final equations.. Finally, the solutions
by all of these methods are exhibited for diverse range of the emerging parameters appearing
in the study. In this analysis, we have investigated the effects of various physical nondimensional numbers namely; Eckert number (  ), Grashof number (Gr), Deborah number
(De) and Prandtl number (Pr) on the flow fields, on the rates of heat transfer and on the
coefficients of skin friction.
Thermal
boundary layer
x
g
u
v
Tw(x)
Momentum
boundary layer
O
y
Fig. 1: The flow diagram and co-ordinate system.
GOVERNING EQUATIONS
u v

0
x y
u
u
u
 2 u 1     2 u   u  2 v
 3u 
v


u


v
 
  g  (T  T )

x
y
 y2   x   y2   y  y2
 y3 
2
 T
u 
T 
 2T
u   u
u 
cp  u
v

k

v

  1
u

2
y
y
 y y x
 y
 x
 y
The boundary conditions to be satisfied by the Eqs. are
u  v  0,
T  Tw ( x ), at y  0
u  0, T  T , as y  
Now we introduce the following transformation to non-dimensionalize the above equations:
y 1/ 4
Gr
x
T  T
 2Gr
g x


, 
, Tw  T  T0 x
2
c p Tx
cp
Tw  T
   Gr 1/ 4 f  ,   ,  
where
Gr 
g  Tx 3
2
f   ff   f  2    De  2 f f   f 2  ff iv 
 f 
 f 
f
f 
f 
f  
  f
 f 
 De  f 
 f 
 f 
 f iv




  
 
 
 


1
f
f
f    
   f    f     f 
 
 f 2  Def   ff   f f     f 
 f
 
 
Pr


   



The boundary conditions given in Eqs. thus take the form:
f  0,    0,
  0,    1,
5.1
f   0,    0,
f   ,    0
  ,    0
Mathematical formulation
Let us consider the flow induced by the buoyant forces of a heated vertical surface
within viscoelastic flow of ambient temperature T . Where surface is heated to a variable
temperature Tw ( x ) . For steady state flow conditions, under the Boussinesq approximation the
boundary layer equations for second grade fluid in the absence of heat generation including
dissipation term are given by [42]:
u  v  0 ,
x y
(5.1)
2
2
2
3 
 
u u  v u    u2  1   (u  u2 )  u  v2  v  u3   g  (T  T ) ,
x
y
y y
y
y 
 x y
(5.2)
2
2
 u   u
 u 
u 
u T  v T    T2 
 1
u
v



.
x
y
y
 c p  y   c p y y   x
y 
(5.3)
where  u , v  represent vertical & horizontal compositions of velocity,   k /  c p is the
thermal diffusivity and 1 is the material constant related to second-grade fluid, T is the
ambient temperature.
The BC’s for this study:
u  0, v  0, T  Tw  T  T0 x at y  0 ,
(5.4)
u  0, T  T as y   .
(5.5)
Here T0 represents reference temperature. Now we introduce the following transformation to
non-dimensionalize the above equations:
   Gr1/ 4 f  ,   ,  
y 1/ 4
T  T
 2 Gr
g x
Gr ,  

, 
,
2
x
c p Tx
cp
Tw  T
(5.6)
where
Grx 
g  Tx 3
2
(5.7)
.
represents the streamwise Grashof number. Using Eq. (5.6) into the Eqs. (5.1)-(5.5), equation
of continuity is identically satisfied and we develop the non-dimensionless equations as
given:
f   ff   f  2    De  2 f f   f 2  ff iv 
 f 
 f 
f
f 
f 
f   ,
 f
 f 
 De  f 
 f 
 f 
 f iv




  
 
 
1
   f    f 
Pr
,
 


f
f
f    
2
  f 
 
 f   Def   ff   f f     f 
 f
 
 


   



(5.8)
(5.9)
with boundary conditions


f 0,   0,


f  0,   0,


f  ,   0 ,
(5.10)
  0,    1,
  ,    0 .
(5.11)
In Eqs. (5.8) and (5.9), the parameters De (also known as viscoelastic parameter and is
the ratio of elastic to viscous stresses, Bird [16]) and Pr are, respectively, known as the
Deborah number and Prandtl number, which are already defined in Eq. (3.13).
5.2
Solution methodologies
The solution of Eq. (5.8)-(5.9) is again presented by three different methods: (1)
perturbation method for small values of local Eckert number  , (2) method initiated by
Sparrow and Yu [72] i.e. local non similarity method (LNS) because  is here the local
variable and is considered as non-similarity parameter, and (3) solutions for wide range of 
are then obtained by IFDM in conjunction with the Keller-box elimination technique [76] as
well. Detail of the above-mentioned methods is provided in the following sections.
5.2.1 Perturbation method for small 
As the local streamwise Eckert number,  , is small close to the principal edge, functions
f &  appearing in Eqs. (5.8) & (5.9) can be expanded in powers of  as given below:




  ,      k k   .
f  ,     k f k   ,
k 0
(5.12)
k 0
Expansions given in (5.12) are substituted in (5.8)-(5.9) & comparing different powers of ,
one can easily get the equations for various orders. The generalized kth order system for k  0
are obtained and are presented below:
k
f k  
i 0
  k  i  1  f f  f  f  De  f f  f  f     k  2  De f  f  ,
iv
k i
i
k i
k i
i
k i
i
k i
i
i
(5.13)
 k  0
k
k
i0
i 1

 k     k  i  1 f k i i  f i  k i     f k i f i 1




 De    k  i  1    f k i f i  j f j1  f ki f i  j f j1     0


i 1 
 j 1

k
i
Respective BC’s of the above problem for all values of k  0 are
.
(5.14)
f k  0   0,
f k   0   0,
 k  0   0,
k     0 .
fk     0 ,
(5.15)
(5.16)
Since Eqs. (5.13)-(5.14) are lnked and non-linear in nature, so solutions are
determined using iteration technique developed by Natscheim-Swigert. Writing these
equations in the generalized form it became possible to take the terms up to higher orders
then incorporated the Padé s approximation to correlate the results with those obtained by the
LNS as well as IFDM discussed below.
5.2.2 Local non-similarity method (LNS)
Governing system of equations of the current problem will now be rewritten to get the
solution with LNS as it was done in previous chapters by introducing the following new
functions.

 f ,   ,     .
g , 
(5.17).
Here, the Eqs. (5.8) and (5.9) for f &  are rewritten with the help of the newly defined
functions in (5.17) and equations for g and  are also written by differentiating Eqs. (5.8)(5.9) with respect to  .
f     ff   f 2  De  2 f  f   f 2  ff iv 



 f g   f g   De  f g   f g   f g   f iv g 
,
(5.18)
g     fg   2 f g  3 f g   De  3 f g   3 f g   3 f g   2 f iv g  fg iv 

  g g   g g  De  2 g g   g 2  g iv g 
1
   f    f     f    g  f 2   De   ff f   f f 2 
Pr
,
2
 De   f f g  f f g    0
1
    f    g     g   f     f    g  f 2   De  ff f   f f  2 
Pr
   g    g  2 f g    De  3 gf f   fg f   ff g   g f  2  4 f f g   .
 De  2  gg f   gf g   g f g   f g  2   0

,
(5.19)
(5.20)
(5.21)
Boundary conditions for the new set of ordinary differential equations are obtained by finding
the derivatives of Eqs. (5.10)-(5.11) with respect to  .




f 0,   0,
f  0,   0,
  0,    1,
  ,    0 ,
g 0,   0,


g  0,   0,

 (0,  )  0,
 (,  )  0 .


f  ,   0 ,
(5.22)
(5.23)



g  ,   0 ,
(5.24)
(5.25)
Now we are again at the position to get the exact solutions of Eqs. (5.18) to (5.21). As
before, here also, we have employed the Natcheim-Swigert iteration technique. The results
for local coefficient of skin-friction & local heat transfer rate are computed numerically.
5.2.3 Implicit finite difference method (IFDM)
To integrate set of Eqs. (5.8)-(5.11) exploiting IFDM together with the KBM, we
introduce the new variables as given U, V, G, P, W and Q below:
U  f , V  U , G  V , P  G , W   , Q  W  .
(5.26)
Use of Eq. (5.26) reduces the Eqs. (5.8) & (5.9) to the following first order system:
V   W  fV  U 2  De  2UG  V 2  fP 
 U
 G
f
U
V
f   ,
  U
V
 De  U
G
V
P




  
 
 
 W

 f
1
f
V   
Q  f Q  UW    U
Q
 V 2  DeV  fG  UV    G
U
  .
 
Pr

   
 


(5.27)
(5.28)
The respective BC’s are given below in Eq. (5.29).










f 0,   0, U 0,   0, U ,   0, W 0,   1, W ,   0 .
(5.29)
We now drop hat over  for convenience and put a grid of (, ) as:
0  0,  j   j 1  h j , j  1, 2,3,..., J ,
(5.30)
 0  0,  n   n1  k n , n  1, 2,3,... .
(5.31)
Then new variables f , U , V , G , P , W and Q are approximated at points  j ,  n  of the grid by
defining f jn , U nj , V jn , G nj , Pjn , W jn , Q nj . To represent all the functions or variables midway
between grid points, we employed the notation g nj .
 n 1 / 2 
1 n
1
   n 1  ,  j   j   j 1  ,

2
2
(5.32)
g nj 1 / 2 
1 n
1
g j  g nj 1  , g nj 1 / 2   g nj  g nj 1  .

2
2
(5.33)
We now develop finite difference approximations for variables and equations given through
(5.26)-(5.29) at middle points of grid  j 1/ 2 ,  n  .
h j 1 ( f jn  f jn1 )  U nj 1 / 2 ,
(5.34)
h j 1 (U nj  U nj 1 )  V jn1 / 2 ,
(5.35)
h j 1 (V jn  V jn1 )  G nj 1/ 2 ,
(5.36)
h j 1 (G nj  G nj 1 )  Pjn1/ 2 ,
(5.37)
h j 1 (W jn  W jn1 )  Q nj 1/ 2 ,
(5.38)
G nj1/ 2  1   n  fV  j 1/ 2  1   n  U 2 
n
n
j 1/ 2

  2   n  De UG  j 1/ 2  V 2 
n
n
j 1/ 2
  fP  j 1/ 2
n
 n V jn1/1 2 f jn1/ 2  f jn1/1 2V jn1/ 2  De  Pjn1/1 2 f jn1/ 2  f jn1/1 2 Pjn1/ 2   W jn1/ 2  R nj 1/1/22



n
1 1 n
n
n
n
h j  Q j  Q nj 1   1   n   fQ  j 1/ 2  UW  j 1/ 2  1   n  De  fVG  j 1/ 2  UV 2 
j 1/ 2
Pr
 n W jn1/1 2U nj 1/ 2  U nj 1/1 2W jn1/ 2  f jn1/1 2Q nj 1/ 2  Q nj 1/1 2 f jn1/ 2    n V 2 

,

n
j 1/ 2
(5.39)
.
(5.40)

 n n 1/ 2 De VG  j 1/ 2 f jn1/1 2  VG  j 1/ 2 f jn1/ 2  UV  j 1/ 2 V jn1/1 2  UV  j 1/ 2 V jn1/ 2  T jn1/1 2
n 1
n
n 1
n
where
 n   n 1/ 2 / k n ,
(5.41)


(5.42)
,
(5.43)
n 1
n 1
n 1
n 1
n 1
R nj 1/1 2   Lnj11/ 2   n  fV  j 1/ 2  U 2 
 De 2 UG  j 1/ 2  V 2 
  fP  j 1/ 2  ,
j 1/ 2
j 1/ 2


Lnj11/ 2  h j 1 V jn 1  V jn11    fV  j 1/ 2  U 2 
n 1
 De V 
2 n 1
j 1/ 2
W
n 1
j 1/ 2
 2De UG  j 1/ 2  De  fP  j 1/ 2
n 1
n 1
n 1
j 1/ 2


n 1
n 1
n 1
,
T jn1/1 2   M nj 1/1 2   n  fQ  j 1/ 2  WU  j 1/ 2    n n 1/ 2 De VGf  j 1/ 2  UV 2 
j 1/ 2


n
(5.44)
The boundary conditions are:
f 0n  0, U 0n  1, W0n  1, U Jn  0, WJn  0 .
(5.45)
Details of the IFDM is same as used in previous chapters hence it is skipped. for
numerous values of physical parameters Nusselt number and skin friction are calculated using
IFDM and results are displayed in graphs. An excellent agreement is noted while comparing
these numerical values with already mentioned methods.
5.3
Results and discussions
Here analysis for the dissipation effect on free convection flow considering the 2nd-
grade flow besides a level surface placed vertically having nonuniform temperature is made.
As the results obtained by all three methods agreed so for convenience, graphs are displayed
only for results by IFDM. Further for convenience in graphs we have dropped the hat on
 and used  for Eckert number.
In Figs. 5.1(a)-(b), the skin friction and heat transfer against local Eckert number 
and for various Pr are presented. It is noted that increase in  reduces the heat transfer and
enhances the skin friction. The temperature gradient of wall has a lower value for positive
Eckert number, as increasing the viscous dissipation the fluid gets heated which consequently
decreases rate of heat transfer and increases skin friction. Whereas, increase in Pr gives rise
to a larger temperature gradient and it give rise to increase in local Nusselt number. Figs.
5.2(a)-(b) reveal that skin-friction and Nusselt number both decrease owing to increase in
Deborah number, De as increase in De increases elasticity and decrease in viscosity.
We then depict the effect of various parameters on temperature and velocity in Figs.
(5.3)-(5.4). Both velocity and temperature decrease due to increase in Prandtl number Pr, as
shown in Fig. (5.3). Temperature and velocity both rise for larger values of dissipation
number i.e. local Eckert number. According to Fig. (5.4), the temperature increases but
velocity decreases with increase in Deborah number De. Figs. (5.3)-(5.4) support the physical
phenomena observed in Figs. (5.1)-(5.2).
The stream function lines and isotherms are plotted respectively in Figs. (5.5)-(5.6)
for different Pr from which we can again observe that velocity and temperature both decrease
at particular values of x and y due to increase in Pr as it was observed in Fig. (5.3).
Streamlines given in Fig. (5.7) show that increasing De velocity decreases as was noted in
Fig. (5.4)a. Similarly Fig. (5.8) displays isotherms for different De and supports Fig. (5.4)b
which tells us that due to increase in De temperature increases. We can also observe the
dissipation effects from Figs. (5.5) to (5.8) as local Eckert number is dependent on x and
again conclude that both temperature and velocity increase with increase in viscous
dissipation.
5.4
Conclusions
In this two-dimensional study, we have conducted numerical simulation to observe
the dissipation and free convection effects on visco-elastic second grade fluid flow along the
plate. Flow and thermal characteristics are studied over a range of Deborah number De,
Prandtl number Pr, and Eckert number  .

Both velocity and temperature increase with the increase in dissipation effect that is due
to increase in Eckert number  .

Whereas the coefficient of skin-friction is increased due to dissipation effect  & heattransfer rate reduces by increasing Eckert number  .

Both velocity and temperature decrease due to increase in Prandtl number Pr. Heat
transfer increases with increase in Pr while skin-friction is decreased due to rise in Pr.

Skin friction coefficient and heat transfer rate and velocity all decrease with increase in
Deborah number De whereas temperature increases with increase in De.
Fig. 5.1: (a) Local skin-friction (b) Nusselt number against  for different Pr, De =2.5.
Fig. 5.2: (a) Local skin-friction (b) Nusselt number against  for different De, Pr =10.
Fig. 5.3: (a) Velocity (b) Temperature against  for variation in Pr while De = 2.5 and  =
0.5.
Fig. 5.4: (a) Velocity (b) Temperature against  for different De and  having Pr = 50.
Fig. 5.5: Streamlines for (a) Pr=10 (b) Pr=50 & for De =2.5.
Fig. 5.6: Isotherms for (a) Pr=10 (b) Pr=50 & for De =2.5.
Fig. 5.7: Streamlines for Pr=10 & for (a) De = 1.0 (b) De = 5.0.
Fig. 5.8: Isotherms for Pr=10 & for (a) De = 1.0 (b) De = 5.0.
Data Files: oupkba 8,11,14, Layout Files: kbskn1.lay, kbhtc1.lay
Skin friction and heat transfer for different prandtl and for de 2.5
Data Files: oupkba7,8,9 , Layout Files: kbskn2.lay, kbhtc2.lay
Skin friction and heat transfer for different De and for pr=10
Data Files: oupkbvel 8,11,14, Layout Files: kbvel1.lay, kbtemp1.lay
Skin friction and heat transfer for different prandtl and for de 2.5, and xi =0.5
Data Files: oupkbvel 20,11,23 , Layout Files: kbvel3.lay, kbtemp3.lay
Skin friction and heat transfer for different Xi and for pr=50, and De =2.5
Data Files: oupkbvel 25,20,26 ,
Layout Files: kbvel3.lay, kbtemp3.lay Velocity and
temperature for different Xi and for pr=50, and De =2.5
Data Files: oupkbvel 19,20,21 , Layout Files: kbvel4.lay, kbtemp4.lay
Skin friction and heat transfer for different de and for pr=50, and xi =0.5
Data Files: Oupkbstm1,and Oupkbstm2 Layout Files: kbstrm1.lay, kbstrm2.lay
Stream lines for different Pr 10 & 50and de=2.5
Data Files: Oupkbstm1.dat and Oupkbstm2.dat, Layout Files: kbtherm1.lay, kbtherm2.lay
isotherms for Pr 10 & 50and de=2.5
Data Files: Oupkbstm3.dat and Oupkbstm4.dat, Layout Files: kbstrm3.lay, kbstrm4.lay
Stream lines for Pr =10 and de=1 and 5,
Data Files: Oupkbstm3.dat and Oupkbstm4.dat, Layout Files: kbtherm3.lay, kbtherm4.lay
Isotherms for Pr =10 and de=1 and 5,
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