Finite element computation of multi-physical micropolar transport phenomena from an inclined moving plate in porous media Shamshuddin, MD, Beg, OA, Ram, MS and Kadir, A http://dx.doi.org/10.1007/s12648-017-1095-y Title Finite element computation of multi-physical micropolar transport phenomena from an inclined moving plate in porous media Authors Shamshuddin, MD, Beg, OA, Ram, MS and Kadir, A Type Article URL This version is available at: http://usir.salford.ac.uk/id/eprint/42637/ Published Date 2017 USIR is a digital collection of the research output of the University of Salford. Where copyright permits, full text material held in the repository is made freely available online and can be read, downloaded and copied for non- commercial private study or research purposes. Please check the manuscript for any further copyright restrictions. For more information, including our policy and submission procedure, please contact the Repository Team at: [email protected].
28
Embed
Finite element computation of multi-physical micropolar ...usir.salford.ac.uk/42637/1/accepted INDIAN J... · Using a Galerkin formulation with a weighted residual scheme, finite
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Fini t e el e m e n t co m p u t a tion of m ul ti-p hysical mic ro pola r
t r a n s po r t p h e no m e n a fro m a n incline d m oving pl a t e in po ro us
m e diaS h a m s h u d din, MD, Be g, OA, Ra m, M S a n d Kadir, A
h t t p://dx.doi.o rg/1 0.10 0 7/s1 2 6 4 8-0 1 7-1 0 9 5-y
Tit l e Finit e el e m e n t co m p u t a tion of m ul ti-p hysical mic ropola r t r a n s po r t p h e no m e n a fro m a n incline d m oving pl a t e in po ro us m e dia
Aut h or s S h a m s h u d din, MD, Beg, OA, Ra m, M S a n d Kadir, A
Typ e Article
U RL This ve r sion is available a t : h t t p://usir.s alfor d. ac.uk/id/e p rin t/42 6 3 7/
P u bl i s h e d D a t e 2 0 1 7
U SIR is a digi t al collec tion of t h e r e s e a r c h ou t p u t of t h e U nive r si ty of S alford. Whe r e copyrigh t p e r mi t s, full t ex t m a t e ri al h eld in t h e r e posi to ry is m a d e fre ely availabl e online a n d c a n b e r e a d , dow nloa d e d a n d copied for no n-co m m e rcial p riva t e s t u dy o r r e s e a r c h p u r pos e s . Ple a s e c h e ck t h e m a n u sc rip t for a ny fu r t h e r copyrig h t r e s t ric tions.
For m o r e info r m a tion, including ou r policy a n d s u b mission p roc e d u r e , ple a s econ t ac t t h e Re posi to ry Tea m a t : u si r@s alford. ac.uk .
a significant modification in both linear and angular velocity (micro-rotation) distributions.
The effect of thermal radiation-conduction parameter ( R ) on linear velocity and temperature is
presented in Figs. 6-7. This parameter is defined as kk/TR 33
16 and features in the
17
augmented thermal diffusion term in eqn. (17) i.e. 2
21
y
. It defines the relative contribution of
thermal radiation heat transfer to thermal conduction heat transfer. When 1R , thermal
conduction dominates. When 1R , both thermal conduction and thermal radiation contributions
are equal. For 1R thermal radiation dominates over thermal conduction. In the present
simulations, we confine attention to the last of these three cases i.e. 1R wherein thermal
radiative flux is substantial. Fig. 6 clearly reveals that there is a strong deceleration in the linear
velocity with increasing R values. The energizing of the flow enhances thermal diffusion but
counteracts momentum diffusion. This leads to an increase in momentum boundary layer
thickness. A similar observation has been reported by Abo-Eldahab and Ghonaim [12] and
Olajuwon and Oahimire [14]. Increasing radiation-conduction parameter is also found to decrease
temperatures in the boundary layer (Fig. 7). Thermal boundary layer thickness is therefore also
reduced with greater values of R .
Figs. 8-9 shows the graphical representation of the non-dimensional velocity and temperature
profiles for some representative values of the temperature dependent and surface dependent heat
source (or sink) parameter 1012 ,,,H . It is to be noted that negative values of H indicate heat
source while positive values of H correspond to heat sink. From Fig.8, it is observed that due to
heat source 0H the buoyancy force increases which in turn manifests in higher velocities in the
boundary layer i.e. flow acceleration. On the other hand, when heat sink 0H is present, the
buoyancy force decreases inducing flow deceleration. For both heat source and sink, the peak
velocities occur near the surface of the plate. Fig. 9 depicts heat source/sink effect on temperature
and indicates that as H increases from negative to positive values, the temperature as well as
thermal boundary layer thickness increases. This is due to fact that the heat source introduces
thermal energy to the plate which increases temperature, energizes the boundary layer and
elevates thermal boundary layer thickness. For the case of heat sink more heat is removed from
the plate which decreases temperature and allows effective cooling of the boundary layer. These
thermal effects may therefore be exploited to advantage in materials processing systems to control
temperatures in manufactured materials which in turn influence other characteristics.
18
Figs. 10-11 represents the influence of chemical reaction parameter ( ) on the velocity and
concentration profiles. The reaction parameter is based on a first-order irreversible chemical
reaction which takes place both in the bulk of the fluid (homogeneous) as well as at the plate which
is assumed to be catalytic to chemical reaction. Although chemical reactions generally fall into one
of two categories i.e. homogenous or heterogenous, the former is of interest in the present study.
Homogenous chemical reactions take place uniformly throughout a given phase and exert a similar
influence to an internal source of heat generation. We consider the destructive type of
homogenous chemical reaction. Increasing the chemical reaction parameter produces a
decrease in velocity. The momentum boundary layer thickness is therefore increased substantially
with greater chemical reaction effect. It is noticed that concentration distributions decrease when
the chemical reaction increases. Physically, for a destructive case, chemical reaction takes place
and progressively destroys the original species. This, in turn, suppresses molecular diffusion of the
remaining species which leads to a fall in concentration magnitudes and a decrease in
concentration boundary layer thickness.
The profiles of the velocity and microrotation in the boundary layer for various values of the plate
moving velocity pU are shown in Figures 12 -13 in the direction of the fluid flow. It is noticed that
the peak value of velocity across the boundary layer increases near the porous plate as the plate
velocity increases. The results also show that the magnitude of microrotation on porous plate
decreases as pU increases. The linear flow is therefore accelerated with greater plate velocity
whereas the micro-rotation (angular flow) of micro-elements is inhibited i.e. decelerated.
Figs. 14-15 represents the influence of Grashof number Gr and modified Grashof number Gc on
velocity and microrotation profiles. The thermal Grashof number, Gr , quantifies the relative
magnitude of the buoyancy force and the opposing frictional (viscous) forces acting on the
micropolar fluid. Physically the positive, negative and zero )GrandGr,Gr.,e.i( 000 values of
the Grashof number correspond to cooling, heating of the boundary surface and absence of free
convection currents, respectively. The species (solutal) Grashof number i.e. Gc embodies the
relative contribution of species buoyancy force to viscous hydrodynamic force. It is observed that
the velocity increases as Gr or Gc increases. Furthermore, the peak value of velocity increases
19
rapidly near the wall of the porous plate. However, the converse behavior is computed in the case
of micro-rotation profiles. Thermal and species buoyancy therefore modify linear and angular
velocity fields in a different fashion with different implications for boundary layer thicknesses.
Figs. 16-17 show the pattern of the velocity and microrotation for different values of magnetic field
parameter M . It is observed that the amplitude of the velocity as well as the boundary layer
thickness decreases when M is increased. Physically, in magnetohydrodynamic materials
processing, the applied magnetic field exerts a retarding effect on the free convective flow,
transverse to the direction of imposition of the magnetic field. With increasing the values of M , this
type of resisting force slows down the fluid i.e. with stronger magnetic field strength the flow is
decelerated and this is confirmed with the decreasing velocity distribution across the boundary
layer. In case of Fig. 17 an increase in magnetic parameter is observed to significantly accelerate
the angular velocity i.e. enhance the magnitude of micro-rotation, although the effect is more
localized at the plate surface and progressively grows further from the plate. In both Figs. 16 and
17 asymptotically smooth solutions are obtained indicating that a sufficiently large infinity boundary
condition is prescribed in the free stream. Linear momentum boundary layer thickness is therefore
increased with greater magnetic parameter whereas angular momentum boundary layer thickness
is reduced.
Figs. 18-19 visualize the effect of the porous medium permeability parameter (K) on both velocity
and microrotation fields. This parameter characterizes the hydraulic transmissivity of the porous
medium. It arises in the Darcian drag force term in the composite linear momentum eqn. (16), viz
uK/1 .With increasing permeability the regime, the quantity of solid fibers progressively
decreases. The Darcian bulk impedance to flow is therefore also decreased. This results in
acceleration in the velocity u , as observed in Fig. 18. This behaviour is sustained across the
boundary layer i.e. for all values of transverse co-ordinate, y . It is also apparent that micro-rotation
i.e. angular velocity is enhanced with greater permeability parameter although the effect is more
prominent near the plate surface and is weakened with further distance into the boundary layer.
Since the permeability parameter does not arise in the angular momentum conservation (boundary
layer) eqn. (17) the accelerating effect on micro-rotation is sustained via the boost in linear
momentum experienced through the coupling terms which link both linear and angular momentum
20
fields. The increase in permeability implies greater void space in the porous medium. This allows
an enhancement in gyratory motions as the micro-elements are afforded greater space in which to
spin. Similar observations have been reported by Zueco et al. [48] and Mohammadein et al. [49].
Figs. 20-21 present the effects of the viscous dissipation parameter i.e., the Eckert number Ec on
the velocity and temperature fields. Eckert number signifies the quantity of mechanical energy
converted via internal friction to thermal energy i.e. heat dissipation. Increasing Ec values will
therefore cause an increase in thermal energy contributing to the flow and will heat the regime.
Positive Eckert number implies cooling of the wall and therefore a transfer of heat to the micropolar
fluid. Convection is enhanced and we observe that in consistency with this, the micropolar fluid is
accelerated i.e. linear velocity is elevated (Fig. 20). Temperatures are markedly increased with
greater Eckert number (Fig. 21). For all non-zero values of Ec the temperature overshoot near the
wall is distinct; this overshoot migrates marginally further into the boundary layer with an increase
in Ec. Very smooth decays in temperature profiles are observed for all values of Eckert number
and the convergence of profiles in the free stream indicates that an adequately large infinity
boundary condition has been imposed in the finite element model.
The velocity and concentration profiles for different values of Schmidt number, Sc are illustrated in
Figs. 20-21. The Schmidt number embodies the ratio of the momentum to the mass diffusivity i.e.
DvSc / . The Schmidt number therefore quantifies the relative effectiveness of momentum and
mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary
layers. For 1Sc momentum diffusion rate exceeds the species diffusion rate. The opposite
applies for 1Sc . For 1Sc both momentum and concentration (species) boundary layers will
have the same thickness and diffusivity rates will be equal. It is observed that as the Schmidt
number increases velocity decreases. The momentum boundary layer thickness is also reduced
with greater Schmidt number. However, it is apparent that species (concentration) profiles
gradually increase with higher Schmidt number. Smaller values of Sc are equivalent to increasing
the chemical molecular diffusivity and vice versa for larger values of Sc.
21
Fig. 2: Velocity profiles for various values
of
Fig. 3: Micro-rotation profiles for various
values of
Fig. 4 : Velocity profiles for various values
of
Fig. 5 : Microrotation profiles for various
values of
Fig. 6: Velocity profiles for varoius values of
R
Fig. 7 : Temperature profiles for various
values of R
22
Fig. 8: Velocity profiles for various values of
H
Fig. 9 : Temperature profiles for various
values of H
Fig. 10: Velocity profiles for various values of
Fig. 11 : Concentration profiles for various
values of
Fig. 12 : Velocity profiles for various values
Up
Fig. 13 : Microrotation profiles for various
valuesUp
23
Fig. 14 : Velocity profiles for various values
of Gr & Gc
Fig. 15 : Microrotation profiles for various
values of Gr & Gc
Fig. 16 : Velocity profiles for various values
of M
Fig. 17 : Microrotation profiles for various
values of M
Fig. 18 : Velocity profiles for various values
of K
Fig. 19 : Microrotation profiles for various
values of K
24
Fig. 20: Velocity profiles for various values
of Ec
Fig. 21 : Temperature profiles for various
values of Ec
Fig. 22 : Velocity profiles for various values
of Sc
Fig. 23 : Concentration profiles for various
values of Sc
25
5. Conclusions
A mathematical model has been presented for radiative magnetic free convection heat and
mass transfer in transient flow of an incompressible, micropolar fluid from an inclined plate in
porous media. Heat source/sink and homogeneous chemical reaction effects have been included
in the formulation. The conservation equations for momentum, angular momentum (micro-rotation
component), energy and concentration have been non-dimensionlized with appropriate variables.
The resulting non-linear, transient, coupled system of partial differential equations and set of initial
and boundary conditions has been solved numerically, using the variational finite element method
with a Galerkin weighted residual scheme. Validation of the finite element solutions for selected
cases has been conducted with previous published works i.e. Roja et al. [19] and excellent
correlation achieved. The computations have been executed in MATLAB software, and have
shown that the flow is accelerated and momentum boundary layer thickness decreased with
increasing values of Up, Gr, Gc, H, K and Ec but in case of , , M, R, Sc and the flow is
decelerated and momentum boundary layer thickness increased. Angular velocity (Microrotation)
is suppressed as , Up, Gr, Gc, and K increases, conversely angular velocity is elevated with
and M increases. Increasing heat source/sink H parameter and Eckert number Ec elevates
temperature and enhances thickness of thermal boundary layer. Increasing Schmidt number
elevates concentration and enhances the thickness of the species boundary layer. Increasing
homogeneous chemical reaction parameter decreases concentration and reduces concentration
boundary layer thickness. Sherwood number (wall mass transfer rate) is enhanced with increasing
permeability and homogeneous chemical reaction but reduced with increasing angle of inclination
of the plate. Wall heat transfer rate is decreased with an increase in heat source/sink (H)
parameter and increased with an increase in angle of inclination of the plate. With an increase in
heat source/sink H parameter there is initially a significant rise in both wall skin friction (flow
acceleration) and wall couple stress coefficient (flow acceleration), however with further increase
in , there is a subsequent declaration in the flow.
The finite element code developed has resolved efficiently the nonlinear micropolar transport
phenomena in inclined plate magnetohydrodynamic heat and mass transfer. Future studies will
consider magnetic induction effects and will be reported soon.
26
Acknowledgement
The Authors wish to express their cordial thanks to Reviewers for their valuable suggestions and constructive comments which have served to improve the quality of this paper.
References
[1] A C Eringen J. Applied Math. Mech 16 1 (1996).
[2] A C Eringen J. Math. Analys. Appl 38 480 (1972).
[3] A C Eringen Micro-continuum field theories II Fluent media, New York: Springer (2001).
[4] G Lukaszewicz Micropolar Fluids, Modelling and Simulation, Boston: Birkhauser Boston (1999).
[5] T Ariman, M A Turk and N D Sylvester Int. J. Eng. Sci 11 905 (1973).
[6] T Ariman, M A Turk and N D Sylvester Int. J. Eng. Sci. 12 273 (1974).
[7] G Swapna, L. Kumar, O. Anwar Bég and Bani Singh Heat Transfer- Asian Research 1 (2014).
DOI 10.1002/htj.21134.
[8] S Jangili and J.V. Murthy Front. Heat Mass Transfer 6(1) 1 (2015).
[9] S Rawat, R. Bhargava, R. Bhargava and O. Anwar Bég Proc.IMechE Part C- J. Mechanical Engineering Science
223 2341 (2009).
[10] O. Anwar Bég, J. Zueco and T.B. Chang Chemical Engineering Communications 198(3) 312 (2010).
[11] O Anwar Bég, J Zueco, M Norouzi, M Davoodi, A A Joneidi and A F Elsayed Computers in Biology and Medicine
44 44 (2014).
[12] F M Abo-Eldahab and A F Ghonaim App. Math. Comput. 169(1) 500 (2005).
[13] M Ferdows, P Nag, A Postelnicu and K Vajravelu J. App.Fluid. Mech 6(2) 285 (2013).
[14] B I Olajuwon and J I Oahimire Int. J. Pure and Appl. Math. 84 015 (2013).
[15] P K Kundu, K Das and S Jana Bull. Malays. Math. Sci. Soc 38 1185 (2015).
[16] M M Rahman and Y Sultana Nonlinear Analysis, Modelling and Control 13 71 (2008).
[17] P Cheng Int. J. Heat Mass Transfer 20 807 (1977).
[18] P K Singh Int. J. Scientific. Eng. Research 3 2229 (2012).
[19] M. Sudheer Babu, J Girish Kumar and T Shankar reddy Int. J. Appl. Math. Mech 9(6) 48 (2013).
[20] P Roja, T Shankar Reddy and N Bhaskar Reddy Int. J. Scientific and Research Publications 3 (2013) Issue 6.
[21] C H Chen Acta Mechanica 172 219 (2004).
[22] Aurangzaib, A R M Kasim, N F Mohammad and S Shafie Heat Transfer Asian Research 42(2) 89 (2013). DOI:
10.1002/htj.21034.
[23] J Srinivas, J V Ramana Murthy and A J Chamkha Int.J. Numerical Methods for Heat and Fluid Flow 26(3) 1027
(2016). DOI: 10.1108/HFF-09-2015-0354.
[24] S K Bhaumik and R Behera ICCHMT, Procedia Engineering 127 155 (2015). DOI: 10.1016/j.proeng.201.11.318
[25] M K Nayak and G C Dash Modelling, Measurement and Control B 84(2) 1 (2015).
[26] M E M Khedr, A J Chamkha and M Bayomi Nonlinear Analysis Modelling and Control 14 27 (2009).
[27] E Magyari and A J Chamkha Int. J. Thermal Sci 49 1821 (2010).
[28] A J Chamkha and A R A Khaled Heat and Mass Transfer 37 117 (2001).
[29] M M Rahman, M J Uddin and A Aziz Int. J. Thermal. Sci 48(3) 2331 (2009).
[30] D Srinivasacharya and M Upender Turk. J. Eng. Environmental. Sci 38 184 (2015).
[31] S Siva Reddy and MD Shamshuddin ICCHMT, Procedia Engineering 127 885 (2015).
[32] S Siva Reddy and MD Shamshuddin Theoretical and Applied Mechanics 43 117 (2016).
[33] S Rawat, S Kapoor, R Bhargava and O Anwar Bég Int.J. Computer Applications 44 40 (2012).
[34] K Das Int. J. Numerical Methods in Fluids 70(1) 96 (2012).
[35] D Pal and B Talukdar Central European J. Physics 10 1150 (2012).
[36] D Srinivasacharya and M Upender Chem. Ind. Chem. Eng. Q 20 (2) 183 (2014).
[37] T G Cowling, Magnetohydrodynamics, New York: Wiley inter Science (1957).
[38] O Anwar Bég New developments in Hydrodynamics Reaserch, Maximiano J. Ibragimov and A. Anisimov, Eds.,
Ch1. 1, New York: Nova Science (2012).
[39] T Adunson and B Gebhart J. Fluid Mechanics 52 57 (1972).
27
[40] A Rapits and C Perdikis ZAMP 78, 277 (1998).
[41] R Cortell Chin Physics Let: 25 1340 (2008).
[42] S R Rao The Finite Element Method in Engineering, 2nd Edition, BPCC Wheatons Ltd., Exeter USA (1989).
[43] J N Reddy An Introduction to the Finite Element Method, New York: McGraw-Hill (1985).
[44] O Anwar Bég, M M Rashidi and R Bhargava Numerical Simulation in Micropolar Fluid Dynamics Lambert: 288
pp, Germany: Sarbrucken (2011).
[45] D Gupta, L Kumar, O Anwar Bég and B Singh Comp.Thermal. Sci. 6 (2) 155 (2014).
[46] O Anwar Bég, S Rawat, J Zueco, L Osmond and R S R Gorla Theoret. Appl. Mech 41 (1) 1 (2014).
[47] R Bhargava, S Sharma, P Bhargava, O Anwar Bég and A Kadir Int. J. Applied Computational Mathematics (2016).
DOI: 10.1007/s40819-106-0180-9 (13 pages).
[48] J Zueco, O. Anwar Bég and H S Takhar Computational Materials Science 46(4) 1028 (2009).
[49] A Mohammadein, M A El-Hakiem, S M M El-Kabeir and M A Mansour Int. J. Appl. Mechanics Engineering. 2 187