Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Finite Difference Technique for Unsteady MHD Periodic Flow of Viscous Fluid through a Planer Channel

Pawan Preet Kaur, SP Agrawal, Anil Kumar*

Department of Applied Mathematics, World Institute of Technology Sohna, Gurgaon, India

Abstract: In the present paper we investigate an unsteady periodic flow of a viscous incompressible fluid through a porous planer channel in the presence of transverse magnetic field. The governing equations have been solved by finite difference method. The analytical expressions for the velocity profiles of the fluid have been obtained and numerical computations involved in the solution have been shown by tables. The effects of various parameters on the flow field have been discussed with the help of graphs and tables.

Keywords: magnetohydrodynamics, planer channel, periodic flow, viscous fluid, Hartmann number, Finite difference technique

Received October 24, 2013; Revised November 13, 2013; Accepted November 15, 2013

Citation: Pawan Preet Kaur, SP Agrawal, Anil Kumar, Finite Difference Technique for Unsteady MHD Periodic Flow of Viscous Fluid through a Planer Channel. American Journal of Modeling and Optimization, 1 (3). 47-55. January 2013. doi:10.12691/ajmo-1-3-5

1. Introduction

The fluid flow through porous medium has been of great interest for a long time. In the history of fluid flow through channels, Navier investigated a boundary condition of fluid slip at solid surface such that, where h is the slip coefficient and u the velocity along x axis. In a case =0 indicates that there is no slip at the boundary. Sparrow and Cess (1961) investigated the effect of a magnetic field on a free convective heat transfer. Raptis et al. (1982) have been discussed hydro magnetic free convection flow through a porous medium between two parallel plates. Pal et al. (1984) investigated a longitudinal dispersion of tracer particles in a channel bounded by porous media using slip condition. Singh and Kumar (1993) investigated a free convective fluctuating flow through a porous medium with variable permeability. Aldoss et al. (1995) depicted a mixed convection flow from a vertical plate embedded in porous medium in the presence of magnetic field. Makinde (1995) investigated a laminar flow in a channel of varying width with permeable boundaries. Soltani and Yilmazar (1998) studied a slip velocity and slip layer thickness in flow of concentrated suspensions. Jha (1998) discussed an effect of applied magnetic field on transient free convective flow in a vertical channel.

Helmy (1998) investigated MHD unsteady free convective flow past a vertical porous plate.Hossain et al. (1998) investigated heat transfer response of MHD free convective flow along a vertical plate to surface temperature oscillation. Kim (2000) discussed an unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. Yu and Ameel (2002) investigated a slip low heat transfer in rectangular micro channels. Derek et al. (2002) established an apparent fluid slip at hydrophibic microchannel walls. Sharma et al. (2002) investigated the flow between annular spaces surrounded by coaxial cylindrical porous medium. Steinruck (2003) investigated about the physical relevance of similarity solution of the boundary layer flow equation describing mixed convection flow along a vertical plate. Khaled and Vafai (2004) discussed the effect of slip condition on Stokes and Coutte flows due to an oscillating wall. Makinde and Mhone (2005) investigated a heat transfer to MHD Oscillatory flow in a channel filled with porous medium. Makinde and Osalusi (2006) discussed a MHD steady flow in a channel with slip at permeable boundaries. Kumar et al. (2007) investigated a viscous flow coaxial cylinder in the presence of magnetic field. Mahmood and Ali (2007) established the effect of slip condition on unsteady MHD oscillatory flow of a viscous fluid in a planer channel. Mishra et al. (2008) investigated a flow and heat transfer of a MHD viscoelastic fluid in a channel with stretching walls. Kumar et al. (2010) investigated Finite difference technique for reliable MHD steady flow through channels permeable boundaries. Kumar et al. (2013) investigated a Finite difference technique for reliable MHD steady flow through channels permeable boundaries. Kumar et al. (2013) made a Perturbation Technique of MHD Free Convection Flow through Infinite Vertical Porous Plate with Constant Heat Flux. Kumar and Agrawal (2013) profounded Mathematical and Simulation of lid driven cavity flow at different aspect ratios using single relaxation time lattice Boltzmann technique. Kumar et al. (2013) investigated Finite Element Galerkin’s Approach for Viscous Incompressible Fluid Flow through a Porous Medium in Coaxial Cylinders.

2. Formulation of the Problem

We are considering an unsteady periodic flow of viscous incompressible fluid through a planer channel with saturated porous medium in the presence of transverse magnetic field. The governing equations of the motion are given below:

[picture] | (1)

[picture] | (2)

Boundary conditions are:

[picture] | (3)

[picture] | (4)

where is the velocity, the fluid density, the pressure, the time, the kinematics viscosity coefficient, the porous medium permeability coefficient, the conductivity of the fluid, the electromagnetic induction, g the acceleration due to gravity, the coefficient of volume expansion due to temperature, the fluid temperature, the thermal conductivity, the specific heat at constant pressure and q is the radiative heat flux. Consider the fluid is optically thin with a relatively low density and the radiative heat flux is given by

[picture] | (5)

Where is the mean radiation absorption coefficient,

where V is the mean of flow velocity

Using the above dimensionless quantities into equations (1) to (4), we have

[picture] | (6)

[picture] | (7)

Subject to condition

[picture] | (8)

[picture] | (9)

where

[picture] | (10)

[picture] | (11)

The corresponding boundary conditions are

[picture] | (12)

[picture] | (13)

The system of algebraic equations can be solved by Gauss-Siedal method or any other similar method. We use the Thomas algorithm.

The shearing stress is given by

3. Computational Algorithm

We solve these coupled non linear partial differential equations by standard Finite difference technique [Kumar et al. (2013)]. The spetially discrete form of the governing equations can be obtained by Finite difference technique. A computational iteration procedure for the equations (10)-(11) with the boundary conditions (12) and (13) has been followed which takes care of highly non linear coupled set of algebraic equations. The computational iteration process may be divided into the following steps:

Step 1. Simplify equation (10) for j = 2, 3,…n-1 and i = 2,3,….m-1. Calculate the velocity profile using boundary conditions (12) and (13).

Step 2. The algebraic equations are solved by using Guass-Seidal method

Step 3. Update from the previous step.

Step 4. Repeate steps (1)-(3) until convergence are reached.

4. Result and Discussion

The effects of various parameters like Hartmann number M, Grashoff number , Peclet number, Reynolds number , permeability of porous medium k, Radiation parameter N, time t and wall slip parameter h on velocity and temperature profiles have been shown in tables and graphs. The brief description is as given below:

Table 1. Velocity profiles for different values of parameter

Figure 1. Velocity profiles for various values of wall slip parameter h

Table 2. Shear stress for different values of parameter

Figure 1 shows that the velocity is everywhere increases corresponding to every increase in the wall slip parameter h in the region, which clearly tallies the natural phenomenon.

Figure 2 depicts the tendency of shear stress distribution corresponding to different values of wall slip parameter h. Hence for every smaller values of h the value of shear stress is seen much more than that of larger one.

Figure 2. Shear stress for various values of wall slips parameter h

Table 3. Velocity profiles for different values of parameter

Figure 3. Velocity profile for various values of time t

Figure 3 depicts that the velocity profile of the fluid is periodic with period.

Figure 4 depicts velocity profile for various values of magnetic parameter M. It shows that the flow decelerates on the imposition of magnetic field i.e. on the increase of magnetic parameter M.

Table4. Velocity profiles for different values of parameter

Figure 4. Velocity profile for various values Hartmann number M

Table 5. Velocity profiles for different values of parameter

Figure 5 depicts that velocity profile for different values of porosity parameter k.

The effect of Peclet number on the velocity profiles are shown in Table 6 and Figure 6. It is found that the fluid velocity decreases as the values of Peclet number increases corresponding to the same value of y.

Table 7 & Figure 7 also show similar effects of Reynolds number as the Peclet number has the flow.

Figure 5. Velocity profile for various values permeability parameter k

Table 6. Velocity profiles for different values of parameter

Figure 6. Velocity profile for various values Peclet number

Table 7. Velocity profiles for different values of parameter

Figure 7. Velocity profile for various values Reynolds number

Figure 8. Velocity profile for various values radiation parameter N

Table 8. Velocity profiles for different values of parameter

Table 8 & Figure 8 depict the velocity profiles for various values of radiation parameter N. From Table 8 or Figure 8, it is observed that the increasing values of radiation parameter N have the similar effects on the velocity distribution as parameter and number have.

Nomenclature:

: Velocity along x axis

: Fluid pressure

: Fluid density

: Kinematic viscosity

: Electromagnetic induction

: Fluid conductivity

: Intensity of magnetic field

: Coefficient of volume expansion due to temperature

: Magnetic permeability

: A constant

: Specific heat at constant pressure

: Acceleration due to gravity

M: Hartmann number

: Thermal conductivity

K: Porous medium permeability

N: Radiation parameter

k: Porous medium shape factor

T: Fluid temperature

: Temperature at the boundary y = 0

: Temperature at the boundary y = a

: Frequency of the periodic flow

: Non dimensional temperature

: Reynolds number

: Grashoff number

: Darcy number

References

[1]   | Sparrow, E.M. and Cess, R.D. (1961): The effect of a magnetic field on a free convective heat transfer, International Journal of Heat and Mass Transfer, 3, 267-274.
[2]   | Raptis, A.; Massias, C. and Tzivanidis, G. (1982): Hydromagnetic free convection flow through a porous medium between two parallel plates, Phys. Lett., 90(A), 288-289.
[3]   | Pal, D.; Veerbhadraiah, R.; Shivakumar, P.N. and Rudraiah, N. (1984): Longitudinal dispersion of tracer particles in a channel bounded by porous media using slip condition, International Journal of Mathematical science, 7,755-764.
[4]   | Singh, K.D. and Kumar, Suresh (1993): Free convective fluctuating flow through a porous medium with variable permeability, Journal of Mathematical and Physical Sciences, 27(2), 141-148.
[5]   | Aldoss, T.K.; Al-Nimr, M.A.; Jarrah, M.A. and Al-Shaer, B.(1995): Mixed convection flow from a vertical plate embedded in porous medium in the presence of magnetic field, Heat Transfer, 28(A), 635-645.
[6]   | Malcinde, O.D. (1995): Laminar flow in a channel of varying width with permeable boundaries, Romanian Journal of Physics, 40(4-5), 403.
[7]   | Soltani, F. and Yilmazar, U. (1998): Slip velocity and slip layer thickness in flow of concentrated suspensions, Journal of Applied Polymer Science, 70, 515-522.
[8]   | Jha, B.K. (1998): Effect of applied magnetic field on transient free convective flow in a vertical channel, Indian Journal of Pure and Applied Mathematics, 29(4), 441-445.
[9]   | Helmy, K.A. (1998): MHD unsteady free convective flow past a vertical porous plate, Z AMP, Z.Angew Math. Mech., 78(4), 255-270.
[10]   | Hossain, M.A.; Das, S.K. and Pop, I.(1998): Heat transfer response of MHD free convective flow along a vertical plate to surface temperature oscillation, International Journal of Non-Linear Mechanics, 33(3), 541-553.
[11]   | Kim, Y.J. (2000): Unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction, International Journal of Engineering Science, 38,833-842.
[12]   | Yu, S. and Ameel, T.A. (2002): Slip low heat transfer in rectangular micro channels, International Journal of Heat and Mass Transfer, 44, 4225-4234.
[13]   | Derek, C.; Tretheway, D.C. and Mainhart, C.D. (2002): Apparent fluid slip at hydrophibic micro channel walls, Physics of Fluids, 14, L9-L12.
[14]   | Sharma, G.C.; Jain, M. and Kumar, Anil (2002): The flow between annular space surrounded by coaxial cylindrical porous medium, Published in National Conference on 67th Indian Mathematical Society, AMU, Aligarh, Jan. 27-30, 2002.
[15]   | Steinruck, H. (2003): About the physical relevance of similarity solution of the boundary layer flow equation describing mixed convection flow along a vertical plate, Fluid Dynamics Research, 32, 1-13.
[16]   | Khaled, A.R.A and Vafai, K. (2004): The effect of slip condition on stokes and coutte flows due to an oscillating wall, International Journal of Nonlinear Mechanics, 39, 795-809.
[17]   | Makinde, O.D. and Mhone, P.Y. (2005): Heat transfer to MHD Oscillatory flow in a channel filled with porous medium, Romanian Journal of physics, 50(9-10), 931-938.
[18]   | Makinde, O.D. and Osalusi, E. (2006): MHD steady flow in a channel with slip at permeable boundaries, Romanian Journal of Physics, 51(3-4), 319-328.
[19]   | Kumar, Anil; Varshney, C.L. and Sajjan Lal (2007): Viscous flow coaxial cylinder in the presence of magnetic field, Published in the 15th International conference on Interdisciplinary Mathematical and Statistical Technique, Shanghai, China, May20-23.
[20]   | Mahmood, A. and Ali, A. (2007): The effect of slip condition on unsteady MHD Oscillatory flow of a viscous fluid in a planer channel, Romanian Journal of Physics, 52, Nos. 1-2, 85-91.
[21]   | Mishra, J.P.; Shit, G.C. and Rath, H.J. (2008): Flow and heat transfer of a MHD visco-elastic fluid in a channel with stretching walls, Computers and Fluids, 37(1), 1-11.
[22]   | Anil Kumar, CL Varshney and Sajjan Lal (2013): Performance Modeling and analysis of Non-Viscous Incompressible Fluid Flow and Convective Temperature Flow in the Magnetic effects, International Journal of Mathematical Modeling and Physical Sciences Vol. 01, ( 01) pp 10-15, March 2013.
[23]   | Anil Kumar and S P Agrawal (2013): Mathematical and Simulation of lid driven cavity flow at different aspect ratios using single relaxation time lattice Boltzmann technique, American Journal of theoretical and Applied Statistics, vol 2 (3), pp 87-93.
[24]   | Anil Kumar , SP Agrawal and Pawan Preet Kaur (2013)Finite Element Galerkin’s Approach for Viscous Incompressible Fluid Flow through a Porous Medium in Coaxial Cylinders, Journal of Mathematical Sciences and Applications, 2013, Vol. 1, No. 3, 39-42.
[25]   | Anil Kumar, CL Varshney and Sajjan Lal (2013): Perturbation Technique of MHD Free Convection Flow through Infinite Vertical Porous Plate with Constant Heat Flux, International Journal of Mathematical Modeling and Physical Sciences Vol. 01, (02) pp 1-5, 2013.
[26]   | Anil Kumar, R. K. Saket, C L Varshney and Sajjan Lal : Finite difference technique for reliable MHD steady flow through channels permeable boundaries, International Journal of Biomedical Engineering and Technology (IJBET) UK, Vol. 4(2) pp 101-110, 2010.