Hydromagnetic Stability of Al2O3-Water
and CuO-Water Nanofluids:
Effect on Critical Rayleigh Number
Jyoti Ahuja1, Urvashi Gupta2,
R.K. Wanchoo2
1Energy Research Centre, Panjab University, Chandigarh-160014, India
2Dr. S.S. Bhatnagar
University Institute of Chemical Engineering & Technology, Panjab University, Chandigarh-160014, India
*Corresponding Author:
dr_urvashi_gupta@yahoo.com, jyotiahuja1985@gmail.com
ABSTRACT:
In the present paper, hydromagnetic
stability of alumina-water and copper oxide-water nanofluids
is investigated for top heavy configuration of nanoparticles.
Brownian motion and thermophoretic forces are
introduced due to the presence of nanoparticles and
Lorentz force term is added in the momentum equation due to the presence of
magnetic field. The problem has application in geophysics due to earth’s
magnetic field and enhanced heat transfer characteristics of nanofluids.
Computations are carried out within the frame work of normal mode
technique and single term Galerkin approximation. An eigen value equation representing the relation between
various nanofluid parameters, magnetic field, growth
rate of disturbance and wave number is obtained. It is observed that the
thermal Rayleigh number increases with the increase in Chandrasekhar number and
decreases with the increase in volume fraction of nanoparticles.
Nanofluids with Al2O3 nanoparticles
exhibit higher stability than the nanofluids
containing CuO nanoparticles.
The mode of instability is found to be through stationary convection. It is
expected since for oscillatory motions to exist two of the buoyancy forces must
act in the opposite directions.
KEY WORDS: Nanofluids; Brownian motion; Hydromagnetics;
Critical Rayleigh number.
INTRODUCTION:
In nanofluids
small sized particles (nanoparticles, 1-100nm) are suspended in
base fluids using various techniques and this name was coined by Choi [1]. With
the invention of nanofluids a keen interest in the development
of energy efficient heat transfer equipments has been observed because of its
high thermal conductivity. A comprehensive model for
the enhanced thermal conductivity of the nanofluid
has been given by Wang et al. [2]. The problem of
thermal convection for regular fluids has been initiated by Chandrasekhar [3]. Afterwards,
the problem of thermal convection for nanofluids
based on Buongiorno’s model [4] caught the eyes of
many scientists. Buongiorno’s model demonstrated the
conservation equations of nanofluids in convective
transport where the Brownian motion and thermophoretic
forces of nanoparticles are introduced. These
conservation equations have been utilized by Tzou [5] to study the problem of
thermal instability for nanofluids whereas in porous
medium investigation has been carried out by Nield
and Kuznetsov [6]. The growing volume of work on the
area of thermal convection of nanofluids by
considering different aspects of hydrodynamics and hydromagnetics
is devoted by Bhadauria et al. [7] and Yadav et al. [8]. Recently, Gupta
et al. [9] & Yadav et al. [10] considered thermal convection problem with
magnetic field for bottom heavy and top heavy configuration of nanoparticles,
respectively and used arbitrary nanofluid parameters.
Keeping in
mind the importance of magnetic field in geophysics, astrophysics and heat
transfer enhancement/reduction characteristics of nanofluids;
our interest is to bring out the effect
of magnetic field on the thermal instability of a nanofluid
layer. Our objective is to study top heavy configuration of nanoparticles by
taking Al2O3-water and CuO-water
nanofluids. The presence
of magnetic field in fluid introduces Lorentz force term in addition to the
body and buoyancy forces. In the present model normal mode technique and
weighted residuals method has been employed to obtain the stability analysis. Magnetic
field is found to delay the onset of convection and Al2O3
–water nanofluid is found to be more stable
than CuO-water nanofluid. Further, it is found
that oscillatory motions do not exist in the top heavy distribution of
nanoparticles. Since for oscillatory motions to exist density gradient caused
by a distribution of nanoparticles at the bottom must compete with the density
variation caused by heating from the bottom.
1. Mathematical Formulation of
the Problem
Equations
(11) and (12) are integrated to find 
and 
for free-free boundary
conditions and using the fact that the parametric value of 
is large and that of 
is small. One gets the best approximated solution as
4. Perturbation Equations and Normal Mode Analysis
Let
us apply small disturbance to the system by heating it from the bottom. The imposition
of disturbance on the initial solution is mathematically expressed as
5. RESULTS AND DISCUSSION:
5.1 Oscillatory Convection
For oscillatory
convection 
. Therefore, the state at
which the instability sets as an oscillatory motions is determined by separating
real and imaginary parts of the eigen value equation
(22) after putting 
. Solution of the eigen value problem for which 
is real gives the
critical values of thermal Rayleigh number in case of overstability.
But in the present case all the values of are imaginary for the parametric values of Al2O3-water
nanofluid and CuO-water nanofluid. Therefore, oscillatory motions do not exist for the
system with distribution of nanoparticles at the upper layer of the fluid. This
result is expected because for the existence of overstability
two of the buoyancy forces must act in opposite directions.
5.2 Stationary
Convection
For stationary convection
. Therefore, eigen value Eq. (22)
reduces to
It is noteworthy that the expression for thermal Rayleigh number is
independent of both the Prandtl numbers. Also, Brownian motion and thermophoretic
forces produce their effect through conservation equation for nanoparticles. For 
;
reduces to the one given by Chandrasekhar [3] and for 
it agrees with the result of Nield and
Kuznetsov [6]. For
sufficiently large
, the thermal Rayleigh
number 
takes negative
value. This means the instability of top heavy arrangement is so high that the
applied magnetic field must be increased or temperature at the lower boundary
must be decreased in comparison to the upper boundary in order to attain
neutral stability.
The critical values are obtained by putting 
. As a function of 
given by Eq. (25) attains its minimum when
Thus the critical wave number 
shows a
substantial rise with the rise in Chandrasekhar number Q and is independent
of nanoparticles. For 
the minimum is
attained at 
and the corresponding critical Rayleigh number comes out to
be 657.5 which is identical with the result of Chandrasekhar [3]. Further, it
is noted that 
is positive which
means that with the increase in Chandrasekhar number thermal Rayleigh number
increases. Further 
is negative which shows that the increase in nanoparticles
for top heavy distribution destabilizes the system. According to Buongiorno [4] and Nield and Kuznetsov [6], the
thermal characteristics values for alumina-water nanofluid
and CuO-water nanofluid are:
We fix. Figures 2-3 depict the impact of
magnetic field, on thermal Rayleigh number and critical Rayleigh number,
respectively for Al2O3 –water and CuO
–water nanofluids while Figs. 3-4 show the influence
of volume fraction of nanoparticles on 
and 
, respectively. These figures confirm the analytical results
by showing that as well as increases with the
rise in Chandrasekhar number whereas these values decrease with the rise in volume
fraction of nanoparticles for both types of nanofluids.
Thus, magnetic field shows a stabilizing character and its effect is to advance
the onset of convection whereas volume fraction of nanoparticles shows a
destabilizing character with the effect of quickening the onset of instability.
The curves showing the effect of magnetic field and nanoparticles for Al2O3
–water nanofluid lie above the CuO-water
nanofluid. Thus, Al2O3 –water nanofluid exhibit higher stability than CuO-water
nanofluid in the presence of magnetic field. The mode
of heat transfer is through stationary motions as overstability
does not occur for the present configuration.
To differentiate our work from the previous works on nanofluids
it is necessary to mention that Gupta et al. [9] & Yadav et al. [10]
considered thermal convection problems with magnetic field for bottom heavy and
top heavy configuration of nanoparticles, respectively, and used arbitrary nanofluid parameter values without taking the values for Al2O3-water
& CuO-water nanofluids.
In the present paper a comparative study of thermal instability in the presence
of magnetic field has been done for these two nanofluids.
6.
CONCLUSIONS:
Present paper investigates
the effect of magnetic field on the Rayleigh-Bénard
convection for top heavy configuration of nanoparticles for Al2O3–water
and CuO-water nanofluids.
Using normal mode technique and Galerkin single term
approximation it has been found that the mode of heat transfer is through stationary
motions. The Prandtl number and the magnetic Prandtl number do not contribute towards the thermal
instability for stationary convection as the expression of thermal Rayleigh
number for stationary convection is independent of both the numbers. Critical
value of thermal Rayleigh number increases with the increase in magnetic field
and decreases with the increase in volume fraction of nanoparticles
for both types of fluids. Thus, Magnetic field delays the onset of convection
to a large extent and volume fraction of nanoparticles quickens the onset of
convection. Further, Al2O3–water nanofluid
exhibit higher stability as compared to CuO-water nanofluid in the presence/absence of magnetic field.
