EDO UNIVERSITY IYAMHO FACULTY OF SCIENCE SEMINAR SERIES - PowerPoint Presentation

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EDO UNIVERSITY IYAMHOFACULTY OF SCIENCE SEMINAR SERIES

Analysis of a Chemically Reactive MHD Flow of a Radiating Third Grade Fluid in a Cylindrical Pipe with Variable Viscosity and Hall Effect

Prof.

Aiyesimi

Y. M.Department of Mathematics/ICTFaculty of ScienceEdo University Iyamhoyomi.aiyesimi@edouniversity.edu.ng

March, 2018

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ABSTRACT

The combined effects of thermal radiation, temperature dependent viscosity (Reynolds model), Hall effects and magnetic field on a steady chemically reacting third grade fluid flow through uniformly circular pipe is studied. The governing equations are presented and the resulting nonlinear equation are solved analytically using

Adomian Decomposition Method. A parametric study of important parameters involved is conducted with graphical representation of the results. Our investigation thus reveals that the viscosity, chemical and to a small extent the Hall parameter enhance the flow fields but the magnetic parameter has an inhibiting effect on the flow fields.Keywords

: Magnetohydrodynamics (MHD), Third grade fluid, Hall effects, variable viscosity

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INTRODUCTIONMagnetohydrodynamics

(MHD) flows in rectangular and cylindrical system continue to stimulate significant interest in the field of Engineering Science and Applied Mathematics. This interest is owned to the numerous important applications in engineering and industry such as reactive polymer flows, extraction of crude oil, synthetic fibers and paper production (Chinyoka & Makinde

2010).The dynamics of reactive fluids through pipe at low Reynolds numbers has long been an important subject in the area of chemical, biomedical, and environmental engineering and science.

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In 2013, Aiyesimi

et al considered a mathematical model for a dusty viscoelastic fluid flow in a circular channel, observing that an increase in the value of magnetic field and viscoelastic parameter reduces the horizontal velocity of the fluid and particles, and thereby reducing the boundary layer thickness, hence inducing an increase in the absolute value of the velocity gradient at the surface. The effect of radiation on unsteady MHD flow of a chemically reacting fluid past a hot vertical porous plate using finite difference approach was examined by

Srihari & Avinash (2015).

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Hall current and chemical reaction effects on a hydromagnetic flow of a stretching vertical surface with internal heat generation/absorption was studied by Salem

et al (2008). A finite element solution of heat and mass transfer flow with Hall current, heat source and viscous dissipation were presented by Siaiah & Srinivasa

(2013).

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MOTIVATIONThe

motivation comes from a desire to gain more understanding into the combined effect of variable viscosity, magnetic field and Hall current on the flow of chemically reactive third grade fluid using analytical method. The relevant governing equations have been solved analytically by Adomian Decomposition Method [Adomian

(1994), Wazwaz (1998)]. The effects of the various parameters of interest on the velocity, temperature and concentration are presented.

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EQUATION FORMULATION

Consider the MHD steady flow of a third grade fluid in a horizontal circular insulated pipe of radius of infinite length. The fluid is assumed incompressible and electrically conducting in the presence of transversely applied uniform magnetic field

The velocity field is given by

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A temperature dependent variable viscosity (Reynold's type) is assumed of the form

where is an ambient fluid dynamic viscosity at temperature

1

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Under these conditions the continuity, momentum, energy and concentration equations for the problem may be written as follows:

2

3

4

5

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with the boundary conditions

Where

are

fluid velocity, fluid temperature, applied magnetic strength, modified pressure, electrical conductivity, Hall parameter, thermal conductivity, thermal radiation, molecular diffusivity, thermal diffusivity, specific heat capacity, chemical reaction rate constant, reference temperature, wall temperature, reference concentration and wall

concentration.

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Introducing the following non-dimensional quantities (Ellahi, 2013) into (3) to (5) and the boundary

conditions.

6

7

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and using Rosselands approximation

represents third grade parameter, magnetic parameter, pressure drop, Prandtl number, Eckert number, heat source/sink parameter, material constant parameter,

Dufour number, radiation parameter, Schmidt number, chemical reaction parameter, Stefan-Boltzmann constant and mean absorption coefficient.

8

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Following Ellahi (2010), can

be expressed as where , we obtain

With boundary conditions

9

10

11

12

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Method of Solution (Adomian Decomposition Method)

Equations (9)-(11) together with boundary conditions (12) are coupled nonlinear boundary value problem which are solved semi-analytically using Adomian decomposition method as in Wazwaz (2001, 2009) and Chen & Lu (2004). The nonlinear and dependent variables are represented by introducing infinite series solutions of the forms:

13

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The Adomian polynomials are computed following Choi & Shin (2003) and Peon-on &

Viriyapong (2013) and the recursive relations are obtained with initial guess.

14

15

16

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For where

The computations are done using the symbolic and computational mathematical tool MAPLE.

The gradient of the velocity at the pipe wall “Skin friction coefficient” and the heat transfer at the pipe wall “

Nusselt

number” in dimensionless forms are given by

17

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Results and Discussion

Figure 1 velocity profile for various values of Hall parameter and Viscosity

parameter

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Figure 2 Temperature profile for various values of Hall and Viscosity parameter

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Figure 3 Concentration profile for various values of Hall and Viscosity parameter

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Figure 4 Velocity flow for various values of Chemical reaction and Third grade parameter

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Figure 5 Temperature profile for various values of Chemical reaction and Third grade parameter

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Figure 6 Concentration profile for various values of Chemical reaction Third grade

parameter

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Table

1 Skin

Friction and Nusselt number for

Pr = 0.72, Ec

= 0.1 and M = 0.1 0.10.10.1

1

0.943500352

-1.103087404

0.5

0.1

0.1

1

1.126929845

-1.077579209

1

0.1

0.1

1

1.344332949

-1.049681564

1

0.1

0.1

1

0.940038705

-0.197824133

1

0.5

0.1

1

0.975471816

-0.184089821

1

1

0.1

1

1.052508911

-0.153938225

1

0.1

0.1

1

1.144744380

-1.112624658

1

0.1

0.1

1.5

1.123225530

-1.262469521

1

0.1

0.1

2

1.096176071

-1.442332020

1

0.1

0.1

1

1.227979066

-1.074045285

1

0.1

0.5

1

2.176114319

-1.084296264

1

0.1

1

1

9.846556201

-0.899155727

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Conclusion

In this investigation, the influence of Hall parameter and variable viscosity on a steady flow of a reactive third grade magnetohydrodynamic fluid in a circular pipe is presented using the semi-analytic Adomian

Decomposition method. Expression for the velocity, temperature and concentration were obtained. Special emphasis has been focused here to the variations of , and on the velocity, temperature and concentration.

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The main findings of the present analysis are:The

velocity increases largely values of . and have the similar effects on the velocity and temperature.

have considerable effects on the wall skin friction as it increases.

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Thank You

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References

Adomian G. (1994). Solving Frontiers Problems of Physics: The Decomposition Method,

SpringerScience+Business media, USA.Aiyesimi, Y.M., Okedayo, G.T. & Lawal O. W (2013). Analysis of

magnetohydro-dynamics flow of adusty viscoelastic fluid through a horizontal circular channel. Academia Journal of

Scientific Research 1(3) Pp 056-062.Aiyesimi, Y.M., Yusuf, A., & Jiya, M., (2015) Hydrmagnetic boundary-layer flow of a Nanofluid past astretching sheet embedded in a Darcian porous medium with radiation, Nigerian Journal of Mathematical Applications, 24, pp.13-29.Ajadi, S.O. (2009). A note on the thermal stability of a reactive non-Newtonian flow in a cylindrical pipe. International Communications in Heat and Mass Transfer, 36, 63-68.Chen, W. & Lu, Z. (2004). An Algorithm for Adomian Decomposition Method, Applied Mathematics and Computation, 159, pp.221-235.Chinyoka, T. & Makinde, O.D. (2010). Computational dynamics of unsteady flow of a variable viscosity reactive fluid in a porous pipe, Mechanics Research Communications, 37, pp.347-353.

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Choi, H.-W. & Shin, J.-G., (2003). Symbolic implementation of the algorithm for calculating Adomian

polynomials, Applied Mathematics and Computation, 146, pp.257-271.Ellahi

, R. & Riaz, A., (2010). Analytical solutions for MHD flow in a third grade with variable viscosity, Mathematical and Computer Modelling, 52, pp.1783-1793.

Ellahi, R., (2013).The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid pipe: Analytical solution, Applied Mathematical

Modelling, 37, pp.1451-1467.Makinde, O.D., (2007). On steady flow of a reactive variable fluid in a cylindrical pipe with an isothermal wall. International Journal of Numerical Methods for Heat & Fluid Flow, 17(2), pp.187-194.Makinde, O.D., Olajuwon, B.I. & Gbolagade, A.W., (2007). Adomian Decomposition approach to a boundary layer flow with thermal radiation past a moving vertical porous plate. Internal Journal of Applied Mathematics and Mechanics, 3(3), pp.62-70.}Makinde, O.D., Chinyoka, T. & Eegunjobi, A.S. (2013). Numerical investigation of entropy generation in an unsteady flow through a porous pipe with suction. Int. J. Exergy, 12(3), pp.279-297.Peu-on, P. & Viriyapong, N., (2012). Modified Adomian Decomposition for solving particular third order ordinary differntial equation. Applied Mathematical Science, 6(30), pp.1463-1469.Ramachandra Prasad, V., Abdul Gaffar, S., Keshava Reddy, E. & Anwar Bég, O. Numerical study of non‑Newtonian Jeffreys fluid from a permeable horizontal isothermal cylinder in non‑Darcy porous medium, Journal of the Brazilian Society of Mechanical Sciences and Engineering doi: 10.1007/s40430-014-0301-5.

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, S. &

Marewo

, G. (2014). Hall Effect on MHD flow and heat transfer over an

unsteady

stretching

permeable surface in the presence of thermal radiation and heat Source/Sink.

Journal of Applied Mathematics

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