An important class of two dimensional time dependent flow problem dealing with the response of boundary layer to external unsteady fluctuations of the free stream velocity about a mean value attracted the attention of many researchers. Besides that convective flow through porous medium has applications in geothermal energy recovery, thermal energy storage, oil extraction, and flow through filtering devices. Nowadays Magneto hydrodynamics is very much attracting the attention of the many authors due to its applications in geophysics and engineering. MHD flow with heat and mass transfer has been a subject of interest of many researchers because of its varied application in science and technology. Such phenomena are observed buoyancy induced motions in the atmosphere, in water bodies, quasi solid bodies such as earth, etc.
Hayat et al. [1] have investigated oscillatory rotating flows of a fractional Jeffrey fluid filling a porous space. Hayat et al. [2] have studied the effect of thermal radiation on the unsteady mixed convection flow of a Jeffrey fluid past a porous vertical stretching surface using homotopy analysis method. Hayat et al. [3] have investigated radiative flow of Jeffery fluid in a porous medium with power law heat flux and heat source. Hussain et al. [4] have examined radiative hydro magnetic flow of Jeffrey nanofluid by an exponentially stretching sheet. Lakshiminarayana et al. [5] studied effect of slip and heat transfer on peristaltic transport of a Jeffrey fluid in a vertical asymmetric porous channel. Santhosh et al. [6] noticed Jeffrey fluid flow through porous medium in the presence of magnetic field in narrow tubes. Shehzad et al. [7] have studied influence of thermophoretic and joule heating on the radiative flow of Jeffrey fluid with mixed convection. Shehzad et al. [8] have noticed MHD three-dimensional flow of Jeffrey fluid with Newtonian heating.Sreenath et al. [9] has examined oscillatory flow of a conducting Jeffrey fluid in a composite porous medium channel. Sreenadh et al. [10] have investigated free convection flow of a Jeffrey fluid through a vertical deformable porous stratum. Kavitha et al. [11] investigated Influence of heat transfer on MHD oscillatory flow of Jeffrey fluid in a channel. Batti et al. [12] studied simultaneous effects of slip and MHD on peristaltic blood flow of Jeffrey fluid model through a porous medium. Krishna Murthy et al. [13] investigated flow of Jeffrey fluid in a porous channel with heat source and chemical reaction. Akbar et al. [14] studied Influence of magnetic field and slip on Jeffrey fluid in a ciliated symmetric channel with metachronal wave pattern. Sandeep et al. [15-16] investigated momentum and heat transfer behavior of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface with non-uniform heat source/sink.
2. Formulation of the problem:
The unsteady free convective flow of a radiative, chemically reactive, heat absorbing, Jeffry fluid past an infinite vertical porous plate in a porous medium with time dependent oscillatory suction as well as permeability in presence of radiation absorption and a transverse magnetic field is considered. Let x*-axis be along the plate in the direction of the flow and y*-axis normal to it. Let us consider the magnetic Reynolds number is much less than unity so that induced magnetic field is neglected in comparison with the applied transverse magnetic field. The basic flow in the medium is, therefore, entirely due to the buoyancy force caused by the temperature difference between the wall and the medium. Ii is assumed that initially, at t* <0, the plate as well as fluids are at the same temperature and also concentration of the species is very low so that the Soret and Dofour effects are neglected. When t*, the temperature of the plate is instantaneously raised to Tw* and the concentration of the species is set to Cw* .Let the permeability of the porous medium and the suction velocity be considered in the following forms respectively.
(1)
Where
v0>0 and
are
positive constants. Under the above assumption with usual Boussinesq’s
approximation, the governing equations and boundary conditions are given by
(2)
(3)
(4)
(5)
Introducing the non-dimensional quantities,
(6)
The equations, (2)-(5) reduce to following non-dimensional form
(7)
(8)
(9)
(10)
3. Method of Solution:
In view of periodic suction, temperature and concentration at the plate let us assume the velocity, temperature, concentration the neighborhood of the plate be
(11)
Substituting equations (11) into (7)-(9) and comparing the no harmonic and harmonic terms we get
(12)
(13)
(14)
(15)
(16)
(17)
The boundary conditions now reduce to
(18)
Solving these differential equations (12-18) with the help of boundary conditions we get,
(19)
(20)
(21)
(22)
(23)
The skin friction at the plate in terms of amplitude and phase angle is given by
, at y = 0
(24)
The
rate of heat transfer, i.e. heat flux at the 
in
terms of amplitude and phase is given by,
at y = 0
(25)
The
mass transfer coefficient, i.e., the Sherwood number 
at
the plate in terms of amplitude and phase is given by
at y = 0
(26)
4. RESULTS AND DISCUSSION:
In order to assess the effects of the dimensionless thermo physical parameters on the regime calculations have been carried out on velocity field, temperature field, and concentration field for various physical parameters like magnetic parameter, Prandtl parameter, Grashof number, modified Grashof number, chemical reaction parameter etc. The results are represented through graphs in figures 1 to 14. Figure 1, displays the velocity profiles for various values of magnetic parameter M. It is observed that the velocity decreases with an increase in M. This is due to fact that the applied magnetic field which acts as retarding force that condenses the momentum boundary layer. From figure 2, it is displays that the velocity increases with an increases in Gc number. A similar effect is noticed from figure 3, in the presence of Schmidt number where velocity decreases. Figure 4, depicts the effects of Grashof number on velocity, from this figure it is noticed that the velocity increases with an increase in Gr. Influence of the m on velocity is presented in figure 5, from this figure it is noticed that the velocity decreases with an increase in m. From figure 6, it is seen that the velocity increases with an increase in Kp. From figure 7 shows velocity increases for the increasing values of Prandtl number. Effect of radiation absorption is presented in figure 8, from this figure, it is noticed that the velocity decreases with an increase in R. Form figure 9 noticed velocity decreases for increases in chemical reaction parameter. This figure 10 witnesses that velocity increases with an increase in H. Effect of Schmidt number on temperature is shown in figure 11, which concludes that that temperature decreases as the values of Sc increase. In figure 12, effect of chemical reaction parameter on temperature is shown, it is seen that temperature decreases as the values of Kc increase. From figure 13, it is concluded that the concentration decreases as Kc increases. Effect of Schmidt number parameter on concentration is presented in figure 14, which witnesses that concentration decreases as the values of Sc increase.
Effects of various parameters on skin friction, the rate of heat transfer and also the rate of mass transfer are presented in tables 1-3. From table 1 it is noted that skin friction increase due to an increase in Grashof number Gr. But modified Grashof number has a different effect on skin friction. Skin friction decreases due to an increase in M. From this table it is also observed that the skin friction increases due to an increase in porosity parameter. From table 2 it is observed that skin friction increases for increasing values of R where as Nusselt number decreases with the increasing values of R. of course skin friction, as well as Nusselt number increase for increasing values of Pr and also heat source parameter H. From table 3, it is found that skin friction and Sherwood number both increase for increasing values of Sc. whereas skin friction decreases with increase values of Kc, but a reverse effect is noticed in the case of Sherwood number.
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Fig. 1: Effect of M on Velocity |
Fig. 2: Effect of Gc of number on Velocity |
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Fig. 3: Effect of Sc of number on Velocity |
Fig. 4: Effect of Grashof number on Velocity |
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Fig. 5: Effect of m on Velocity |
Fig. 6: Effect of Kp on Velocity |
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Fig. 7: Effect of Pr on Velocity |
Fig. 8: Effect of R on Velocity |
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Fig. 9: Effect of Kc on Velocity |
Fig. 10: Effect of H on Velocity |
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Fig. 11: Effect of Sc on Temperature |
Fig. 12: Effect of Kc on Temperature |
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Fig. 13: Effect of Kc on Concentration |
Fig. 14: Effect of Schmidt number Sc on Concentration |
Table 1: Effects of Gr, Gc, M and Kp on skin friction coefficient
|
Gr |
Gc |
M |
Kp |
Τ |
|
11 |
4 |
0.9 |
1.5 |
10.1811 |
|
12 |
4 |
0.9 |
1.5 |
11.1621 |
|
13 |
4 |
0.9 |
1.5 |
12.1734 |
|
14 |
4 |
0.9 |
1.5 |
13.2867 |
|
9 |
5 |
0.9 |
1.5 |
12.3545 |
|
9 |
6 |
0.9 |
1.5 |
11.4456 |
|
9 |
7 |
0.9 |
1.5 |
10.5332 |
|
9 |
8 |
0.9 |
1.5 |
10.1321 |
|
9 |
4 |
1.5 |
1.5 |
12.4615 |
|
9 |
4 |
2.0 |
1.5 |
12.3477 |
|
9 |
4 |
2.5 |
1.5 |
11.2173 |
|
9 |
4 |
3.0 |
1.5 |
11.1195 |
|
9 |
4 |
0.9 |
0.3 |
8.6748 |
|
9 |
4 |
0.9 |
0.6 |
9.4895 |
|
9 |
4 |
0.9 |
0.8 |
9.8940 |
|
9 |
4 |
0.9 |
1.0 |
9.9784 |
Table 2: Effect of R and H on skin friction coefficient and Nusselt number
|
R |
Pr |
H |
Τ |
Nu |
|
1 |
0.71 |
1 |
98.5648 |
0.6784 |
|
2 |
0.71 |
1 |
127.9863 |
0.4532 |
|
3 |
0.71 |
1 |
180.5674 |
0.3286 |
|
4 |
0.71 |
1 |
230.8521 |
0.2341 |
|
1.5 |
1.71 |
1 |
14.8954 |
2.4724 |
|
1.5 |
2.71 |
1 |
21.8532 |
2.5342 |
|
1.5 |
3.71 |
1 |
32.9845 |
2.6282 |
|
1.5 |
4.71 |
1 |
37.6283 |
2.7845 |
|
1.5 |
0.71 |
2 |
7.6382 |
1.6784 |
|
1.5 |
0.71 |
3 |
8.8352 |
1.8745 |
|
1.5 |
0.71 |
4 |
9.7642 |
1.9564 |
|
1.5 |
0.71 |
5 |
10.7342 |
2.6745 |
Table 3: Effect of Sc and Kc on skin friction coefficient and Sherwood number
|
Sc |
Kc |
Τ |
Sh |
|
1.33 |
2 |
11.6754 |
3.6753 |
|
2.33 |
2 |
12.8643 |
4.6734 |
|
3.33 |
2 |
13.9743 |
5.6342 |
|
4.33 |
2 |
14.9345 |
6.5463 |
|
0.33 |
5 |
73.8453 |
1.3456 |
|
0.33 |
6 |
54.8542 |
1.4674 |
|
0.33 |
7 |
43.8645 |
1.6435 |
|
0.33 |
8 |
23.8956 |
1.8456 |
CONCLUSIONS:
We have considered an unsteady MHD free convection flow of a viscoelastic, incompressible, electrically conducting fluid past a vertical porous plate through a porous medium with time dependent oscillatory permeability and suction in presence of a uniform transverse magnetic field. Some of the notable conclusions are given below.
a. Application of magnetic field decelerates the fluid flow.
b. The heavier species with low conductivity reduces the flow within the boundary layer.
c. An increase in elasticity of the fluid leads to decrease the velocity which is an established result.
d. Impact of Casson parameter leads to decrease the fluid velocity.
APPENDIX:
REFERENCES:
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2. T. Hayat, M. Mustafa, and Z. Naturforsch, “The Effect of Thermal Radiation on the Unsteady Mixed Convection Flow of a Jeffrey Fluid Past a Porous Vertical Stretching Surface Using Homotopy Analysis Method (HAM),” 65a, (2010)711-719.
3. T. Hayat, S.A. Shehzad, M. Qasim, and S. Obaidat, “Radiative Flow of Jeffery Fluid in a Porous Medium with Power Law Heat Flux and Heat Source,” Nucl. Eng, 243, (2012) 15–19.
4. T. Hussain, S.A. Shehzad, T. Hayat, A. Alsaedi, F. Al-Solamy, and M. Ramzan, “Radiative Hydromagnetic Flow of Jeffrey Nanofluid by an Exponentially Stretching Sheet,” Plos One, 9(8), (2014)1-9.
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7. S.A. Shehzad, A. Alsaedi, and T. Hayat, “Influence of Thermophoresis and Joule Heating on the Radiative Flow of Jeffrey Fluid with Mixed Convection”, Braz. J. Chem. Eng., 30(4), (2013) 897-908.
8. S.A. Shehzad, T. Hayat, M.S. Alhuthali, and S. Asghar, “MHD Three-Dimensional Flow of Jeffrey Fluid with Newtonian Heating,” J. Cent. South Univ., 21, (2014) 1428-1433.
9. S. Sreenadh, J. Prakash, A. Parandhama, and Y.V.K. Ravi Kumar, “Oscillatory Flow of a Conducting Jeffrey Fluid in a Composite Porous Medium Channel,” 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, 14-16 July2014, Florida. 2350-2358.
10. S. Sreenadh, M.M. Rashidi, K. Kumara Swamy Naidu, and A. Parandhama, “Free Convection Flow of a Jeffrey Fluid through a Vertical Deformable Porous Stratum,” Journal of Applied Fluid Mechanics, 9(5), (2016) 2391-2401.
11. K. Kavita, K. Ramakrishna Prasad, B. Aruna Kumari, Influence of heat transfer on MHD oscillatory flow of Jeffrey fluid in a channel Adv Appl Sci Res, 3 (2012), pp. 2312-2325
12. M.M. Bhatti, M. Ali Abbas, Simultaneous effects of slip and MHD on peristaltic blood flow of Jeffrey fluid model through a porous medium Alexandr Eng J, 55 (2016), pp. 1017-1023
13. M. Krishna Murthy, M.H.D. Couette, Flow of Jeffrey fluid in a porous channel with heat source and chemical reaction Middle-East J Scient Res, 24 (2016), pp. 585-592
14. N.S. Akbar, Z.H. Khan, S. Nadeem, Influence of magnetic field and slip on Jeffrey fluid in a ciliated symmetric channel with metachronal wave pattern J Appl Fluid Mech, 9 (2016), pp. 565-572
15. N. Sandeep, C. Sulochana, Momentum and heat transfer behaviour of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface with non-uniform heat source/sink A in Shams Eng J, 9 (4) (2018), pp. 517-524
16. N. Sandeep, C. Sulochana, I.L. Animasaun, Stagnation-point flow of a Jeffrey nano fluid over a stretching surface with induced magnetic field and chemical reaction Int J Eng Research in Afrika, 20 (2016), pp. 93-111
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Received on 05.11.2020 Modified on 17.11.2020 Accepted on 26.11.2020 ©A and V Publications All right reserved Research J. Science and Tech. 2021; 13(1):33-40. DOI: 10.5958/2349-2988.2021.00006.1 |
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