EFFECT OF SLIP CONDITIONS AND
HALL CURRENT ON UNSTEADY MHD
FLOW OF A VISCOELASTIC FLUID
PAST AN INFINITE VERTICAL
POROUS PLATE THROUGH POROUS
MEDIUM
RAKESH KUMAR*
*
Department of Mathematics, Govt. College for Girls, (R.K.M.V.), Shimla, India 171 001
KHEM CHAND#
#
Department of Mathematics and Statistics, H.P. University Shimla, India 171 005
Abstract
The purpose of this paper is to present a theoretical analysis of an unsteady hydromagnetic free convection flow of viscoelastic fluid (Walter’s B’) past an infinite vertical porous flat plate through porous medium. The temperature is assumed to be oscillating with time and the effect of the Hall current is taken into account. Assuming constant suction at the plate, closed form solutions have been obtained for velocity and temperature profiles. The effect of the various parameters, entering into the problem, on the primary, secondary velocity profiles, the axial and transverse components of skin-friction are shown graphically followed by quantitative discussion.
Keywords: Hydromagnetic, Viscoelastic, Hall effect, Porous medium, Slip-flow regime.
1. Introduction
The phenomenon of slip-flow regime has attracted the attention of a large number of scholars due to its wide ranging application The problem of the slip flow regime is very important in this era of modern science, technology and vast ranging industrialization. In many practical applications, the particle adjacent to a solid surface no longer takes the velocity of the surface. The particle at the surface has a finite tangential velocity; it slips along the surface. The flow regime is called the slip flow regime and its effect cannot be neglected. The fluid slippage phenomenon at the solid boundaries appear in many applications such as micro channels or Nano channels and in applications where a thin film of light oils is attached to the moving plates or when the surface is coated with special coating such as thick monolayer of hydrophobic octadecyltrichlorosilane (Derek; 2002) i e. lubrication of mechanical device where a thin film of lubricant is attached to the surface slipping over one another or when the surfaces are coated with special coating to minimize the friction between them. The slip-flow regime can also occur in the working fluid containing concentrated suspensions (Soltani and Yilmazer;1998 ) Recently , several scholars have suggested that the no-slip boundary conditions may not be suitable for hydrophilic flows over hydrophobic boundaries at both the micro and nano scale (Watanebe et al;
(1998) ,Watanebe et al; (1999), Ruckenstein and Rahora; (1983). A detailed account of the developments on the
fluid in slip flow regime through porous medium with periodic temperature and concentration, Sharma (2005) discussed the unsteady free convective viscous incompressible flow past an infinite vertical plate with periodic heat and mass transfer in slip flow regime.
Many common liquids such as oils, certain paints, polymer solution, some organic liquids and many new material of industrial importance exhibit both viscous and elastic properties. Therefore, the above fluid, called viscoelastic fluids, is being studied extensively. Many researchers have shown their interest in the fluctuating flow of a viscous incompressible fluid past an infinite or semi-infinite flat plate. Viscoelastic fluid flow through porous media has attracted the attention of scientists and engineers because of its importance notably in the flow of the oil through porous rocks, the extraction of energy from geothermal region, the filtration of solids from liquids and drug permeation through human skin. The knowledge of flow through porous media is useful in the recovery of crude oil efficiently from the pores of reservoir rocks by displacement with immiscible water. The flow through porous media occurs in the ground water hydrology, irrigation, and drainage problems and also in absorption and filtration processes in chemical engineering, the scientific treatment of the problem of irrigation, soil erosion and tile drainage are the present developments of porous media.
Rajagopal (1983) studied the heat transfer in the forced convection flow of a visco-elastic fluid of Walter’s model. Rahmann and Sarkar (2004) investigated the unsteady MHD flow of a viscoelastic Oldroyd fluid under time varying body forces through a rectangular channel. Singh and Singh (1996) analyzed MHD flow of a dusty visco-elastic (Oldroyd B-liquid) through a porous medium between two parallel plates inclined to the horizon. Ibrahim et. al. (2004) discussed the flow of a viscoelastic fluid between coaxial rotating porous disks with uniform suction or injection .Oscillatory motion of an electrically conducting visco-elastic fluid over a stretching sheet in a saturated porous medium was studied by Rajagopal (2006). Prasuna et al. (2010) examined
an unsteady flow of a visco-elastic fluid through a porous media between two impermeable parallel plates. When the strength of the magnetic field is strong one cannot neglect the effects of the Hall currents. It is of considerable importance and interest to study how the results of the hydro dynamical problems are modified by the effects of the Hall currents. Singh and Kumar (2010) examined the exact solution of an oscillatory MHD flow through a porous medium bounded by rotating porous channel in the presence of Hall current. Chaudhary and Jain (2006) investigated the effects of Hall current and radiation on MHD mixed convection flow of a visco-elastic fluid past an infinite vertical plate. Biswal and Sahoo (1999) also studied Hall current effects on free convective hydromagnetic flow of visco-elastic fluid past an infinite vertical plate.
Hence, the objective of the present paper is to investigate the influence of the slip conditions on unsteady hydromagnetic flow of a visco-elastic fluid past an infinite vertical porous plate through porous medium taking into consideration the effect of the Hall current.
2. Mathematical Formulation
We consider the unsteady flow of a viscous incompressible and electrically conducting viscoelastic fluid with oscillating temperature. The flow occurs over an infinite vertical porous plate. The ∗-axis is assumed to be oriented vertically upwards along the plate and ∗-axis is taken normal to the plane of the plate. It is assumed that the plate is electrically non-conducting and a uniform magnetic field of strength is applied normal to the plate. The induced magnetic field is assumed to be negligible so that , , . The plate is subjected to a constant suction velocity .
The constitutive equations for the rheological equation of the state for the viscoelastic fluid (Walter’s liquid B’) are:
= − + ∗ (1)
∗ = − ∗
∞ ∗ ∗ (2)
In which − ∗ = ∞ ∗/
is the distribution function of relaxation times . In the above equation pik is the stress tensor, p is an
arbitrary isotropic pressure, is the metric tensor of a fixed coordinate system xi, and is the rate of strain
tensor. It was shown by Walters (1964) that equation (2) can be put in the following generalized form which is valid for all types of motion and stress
∗ , = − ′
∗ ∗
where x*i is the position at times t*of the element which is instantaneously at the point xi at the time t. The fluid with equation of the state (1) to (3a) has been designated as the liquid B’. In the case of the liquid with short memories i.e. short relaxation times, the above equation can be written in the following simplified form:
∗ , = − (3b)
In which = ∞ is limiting viscosity at the small rates of shear,
= ∞ and denotes the convected time derivative.
The equation of conservation of electric charge is ∇. = which gives ∗=constant, where =
∗, ∗, ∗ . Since the plate is electrically non-conducting, ∗= and is zero everywhere in the flow.
Considering the magnetic field strength to be very large the generalized Ohm’s laws including Hall current, in the absence of electric field neglecting the ion-slip and thermo electric effect takes the following form
+ × = × , (4)
Where is the velocity vector, is the electron frequency, is electrical-conductivity and is the electron collision time and
∗= ∗− ∗ ,
∗= ∗+ ∗ , where = is the Hall current parameter.
Since the plate is infinite in extent all physical quantities are the function of y* and t* only. Thus the governing equations of flow under the usual Boussinesq approximation are:
∗
∗= ,⟹ ∗= − , (5) ∗
∗+ ∗
∗
∗=
∗
∗ −
∗
∗ ∗ − ∗+ ∗ + ∗− ∞∗ −
∗
∗ , (6) ∗
∗+ ∗
∗
∗=
∗
∗ −
∗
∗ ∗ − ∗− ∗ −
∗
∗ , (7)
∗
∗+ ∗
∗
∗=
∗
∗ . (8)
The boundary conditions are
∗= ∗ ∗
∗ , ∗= ∗
∗
∗ , ∗= ∞∗+ ∗− ∞∗
∗ ∗
, ∗= ,
∗→ , ∗→ , ∗→
∞∗, ∗→∞ . (9)
Now we introduce the following non–dimensional parameters as follows:
= ∗, = ∗ ∗, = ∗, = ∗, = ∗ ∞∗
∗ ∞∗, =
∗ ∞∗
,I
= , = , = , = ∗ , = ∗, ℎ = ∗ , (10)
where , is the dimensional less temperature, , is the Grashoff number, ,is the Hartmann number, , is the Prandtl number, , is the viscoelastic parameter, , is the frequency of the oscillations, ℎ ,is the slip parameter. ∞∗ ,denotes the temperature of the fluid far away from the plate, ∗ ,denotes the temperature of the fluid at the plate ,K is the thermal conductivity, , is the specific heat at constant pressure, , is the density of the fluid, , is the volumetric coefficient of thermal expansion, , is the acceleration due to gravity , , is the molecular diffusivity and ∗ is the characteristics length of the plate.
− = − − + + − (11)
− = − − − − (12)
− = (13)
The corresponding boundary conditions become
= ℎ , = ℎ , = , =
→ , ⟶ , ⟶ , ⟶∞ (14)
3. Method of Solution
Introducing = , + , , = √− , the equations (11) and (12) transform to
− = − − − − + , (15)
The corresponding boundary conditions become
= ℎ , = , = ,
→ , → , →∞. (16) In order to solve the equations (13) and (15) under the boundary conditions (16), we assume
, = , = (17) Substituting (17) into equations (13), (15) and (16), we obtain
− ′′ + ′ − + = − , (18) ′′ + ′ − = . (19)
The corresponding boundary condition reduce to
= ℎ , = , = , = = →∞ (20)
Solving equations (18) and (19) under the boundary conditions (20) and using (17), we have
, = − − − , (21)
, = + + . (22)
Since , = , + , , therefore from equation (21) we get
, = cosω − + sin − − cos ω − + sin ω −
(23) and
, = sin ω − − cosω − − − − − . (24)
Separating (22) into real and imaginary parts, the real part is given by
, = cos − sin (25)
Skin-Friction
The axial component of the skin friction at plate for primary velocity is:
= = − – + cosω + − + − sinω (26)
The transverse component of the shearing stress at plate for secondary velocity is:
= = − – + sinω + − − + cosω (27)
4. Results and Discussions
In order to illustrate the influence of the various parameters on the velocity profile, temperature profile, axial shearing stress and the transverse shearing stress, numerical calculations of the solutions, obtained in the preceding section, have been carried out. The value of the Prandtl number, , ,is taken 3, 5, and10. The value of Prandtl number 3 corresponds to Freon. Freon represents several different chlorofluorocarbons or CFC’s which are used in commerce and industries. The value of the Prandtl number equal to 10, represent Gasoline at 1-atm. Pressure and at ′C. The values of other parameters are chosen arbitrarily.
Figs.-1 & 2 represent the variations of primary and secondary velocity profiles u and w, respectively with the Grashoff number, , the Prandtl number, , the Hartmann number, M , and Hall current parameter, m. It is observed from these two figures that the primary velocity decreases with the increase of Grashoff number and the Hall current parameter and increases with the increase of the Prandtl number, but the opposite trend is noted for the secondary velocity with the increase of these parameters.
Figs.-3 & 4 illustrate the effects of viscoelastic parameter, , the slip parameter h, and the Hartmann number m, on the primary and secondary velocity profiles. It is noted from Fig.-3 that the influence of increasing viscoelastic parameter and the Hartmann number is to reduce the primary velocity, but on moving away from the plate ≥ . . this behavior is reversed. It is also clear from the Figs.-3 & 4 that the primary velocity decreases and the secondary velocity increases with the increase of the slip parameter.Fig.-4 explicitly emphasizes that the secondary velocity decreases with the increase of the viscoelastic and the slip parameters. The effects of the porous medium parameter K, and the frequency of the oscillations, , are represented in Figs.-5 & 6. It is noted from these figure that the primary velocity decrease with the increase of porous medium permeability K, and increase with the increase of the frequency of oscillations, , but the pattern is reversed for the secondary velocity with the increase of these parameters.
Fig.-7 displays the temperature profiles variations with the Prandtl number , and the frequency of the oscillations, , it clearly depicts that the increasing Prandtl number has reducing effects on the temperature. The temperature profile increases with the increase of the frequency of oscillations, but on moving away from the plate, the opposite trend is observed. Also it is observed that the temperature attains the maximum values at
≥ . . and then it start declining.
The variations of the axial shearing stress , and the transverse shearing stress , with the Grashoff number, Prandtl number, Hall current parameter and the Hartmann number are plotted against frequency of the oscillations in Figs.-8 & 9 respectively. It can be interpreted from these that axial shearing stress initially increases with the increase of the Prandtl number and thereafter moving away from the plate it decreases. The opposite trend is observed for the other parameters. Fig.-9 clearly emulates that the transverse component of the shearing stress decreases with the increase of Prandtl number and the Hartmann number, but Grashoff number and the Hall current have opposite effect on the transverse shearing stress.
5. Concluding Remarks:
In the present paper, the effects of the slip conditions on the unsteady, MHD, flow of viscoelastic fluid past an infinite vertical plate through the porous medium is studied. The effect of the Hall current is also taken into the consideration. The closed form solution of the governing equation under the prescribed boundary conditions is obtained. The conclusion of the study is as follows:
1. Primary velocity decreases significantly with the increase of the Hall current parameter, but the increasing Grashoff number have reverse effect on the secondary velocity profiles.
2. It is relevant to note that for the Newtonian fluids = as compared to the Non-Newtonian fluids initially the primary velocity increases and then on moving away from the plate ≥ . . , it decreases whereas the secondary velocity increases.
3. The primary velocity initially increases and thereafter for ≥ . . , it decreases for h=0 i.e. no-slip conditions, in comparison to flow in slip-flow regime.
4. The secondary velocity profiles decrease for no-slip conditions, in comparison to flow in slip-flow regime.
5. The temperature profile attains its maximum values at ( = . .) and then start decreasing sharply.
6. For the Newtonian fluids as compared to the Non-Newtonian fluids the axial component of the shearing stress is increased.
7. The axial component of the shearing stress decreases whereas the transverse component of the shearing stress increases when no-slip velocity is applied at the plate.
Fig. 1: Variation of primary velocityu withGr,Pr and mat
2
π
ω =t .
Fig. 2: Variation of secondary velocity w
withGr,Pr and
m
at ω =t π2. rG Pr
m
Curve2 3 0.5 I 4 3 0.5 II 2 3 1 III 2 3 1.5 IV 2 5 0.5 V 2 10 0.5 VI
,
r
G Pr m Curve
2 3 0.5 I 4 3 0.5 II 2 3 1 III 2 3 1.5 IV 2 5 0.5 V 2 10 0.5 VI
Fig. 3: Variation of primary velocityu withα, h and
M at
2
π
ω =t .
Fig. 4: Variation of secondary velocity w
with α , h and M at
2
π
ω =t .
Fig. 5: Variation of primary velocity u with K and ω
at
2
π
ω =t .
Fig. 6: Variation of secondary velocity w with
K and ω at
2
π
ω =t .
α h M Curve
0.05 0.2 5 I 0 0.2 5 II 0.2 0.2 5 III 0.05 0.4 5 IV 0.05 0 5 V 0.05 0.2 10 VI
,
α h M Curve 0.05 0.2 5 I 0 0.2 5 II 0.2 0.2 5 III 0.05 0.4 5 IV 0.05 0 5 V 0.05 0.2 10 VI
,
K ω Curve
1 1 I 3 1 II 1 3 III 1 5 IV 1 10 V
,
K ω Curve
1 1 I 3 1 II 1 3 III 1 5 IV 1 10 V
Fig. 7: Variation of temperature θr with Pr and ωatω =t π2.
Fig. 8: Axial shearing stressτ1 forGr, Pr, m
and M at
2
π
ω =t .
Fig. 9: Transverse shearing stress forτ2Gr, Pr, m and
Mat
2
π
ω =t .
r
G Pr m M Curve
2 3 0.5 5 I 4 3 0.5 5 II 2 10 0.5 5 III 2 3 1 5 IV 2 3 1.5 5 V 2 3 0.5 10 VI
r
P ω Curve
3 1 I 5 1 II 3 5 III 3 10 IV 3 15 V 3 20 VI
r
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Fig. 11: Transverse shearing stress τ2 forα, h and K at
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α h K Curve 0.05 0.2 1 I 0 0.2 1 II 0.2 0.2 1 III 0.05 0 1 IV 0.05 0.4 1 V 0.05 0.2 3 VI
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Appendix
= + , = − , = , = , = + cos ,
= sin , = − + − , = − − − + , =
, = , = + , = − , = + ℎ ,
= − ℎ , = , = , = + ,
= tan , = ,
