Unsteady Boundary Layer Flow Induced by a ... - Science Direct

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ScienceDirect Procedia Engineering 127 (2015) 678 – 685

International Conference on Computational Heat and Mass Transfer-2015

Unsteady Boundary Layer Flow Induced by a Stretching Sheet in a Rotating Fluid with Thermal Radiation Sreelakshmi Ka, Nagendramma Vb, Sarojamma c,* Dept. of Applid Mathematics, SPMVV, Tirupati, 517502, India

Abstract The MHD unsteady flow heat and mass transfer of a viscous incompressible fluid induced by a stretching surface in a rotating fluid taking the effects of thermal radiation, heat absorption and first order chemical reaction into account with convective boundary condition. The rotation parameter is found to reduce the velocities and the magnetic parameter prevents the flow reversal in the x – direction. The increasing values of Biot number generate thicker thermal boundary layers resulting in the rise of temperature. The Schmidt number and chemical reaction parameter are found to have a strong influence on the species concentration resulting in small values. The rate of heat transfer is enhanced by magnetic field, thermal radiation parameter and rotation parameter. The rotation parameter and time are observed to enhance the rate of mass transfer. here insert your abstract . Click © 2015 Theand Authors. Published by text Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). under responsibility responsibility of of the the organizing organizing committee committee of of ICCHMT ICCHMT –– 2015 2015. Peer-review under Peer-review Keywords: MHD; Unsteady boundary layers; Rotating flows; Convective boundary condition

1. Introduction The study of fluid flow over a stretching surface in a rotating fluid finds application to study the geological stretching of a techtonic surface in a rotating ocean (Wang [1]). Gorla et al. [2] discussed the steady flow of an incompressible power law fluid past a horizontal stretching plate that rotates around vertical axes. Takhar and Nath [3] analysed the effect of the magnetic field on the unsteady flow and heat transfer of a viscous incompressible electrically conducting fluid due to a stretching surface in a rotating fluid. Takhar et al. [4] analysed the steady flow

* Corresponding author. E-mail address:[email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015

doi:10.1016/j.proeng.2015.11.365

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and heat transfer on a stretching surface in a rotating fluid subject to a magnetic field. Nazar et al. [5] studied the flow due to a suddenly stretching surface in a rotating fluid.

Thermal radiation plays a significant role in manufacturing process in industry. For example, in casting and levitation, metallic rolling, design of furnace, fins. In engineering, many processes involve very high temperatures and the application of radiative heat transfer is essentially required to design the specific equipment. Nuclear power plants, gas turbines, satellites and space vehicles are some of the examples (Seddeek [6]) which involve radiative heat transfer. In this paper an effort is made to investigate the effect of thermal radiation on the unsteady flow of a viscous, incompressible, electrically-conducting fluid caused by the stretching of a surface in a rotating fluid to know the influence of thermal radiation, magnetic field, chemical reaction and heat sink. 2. Mathematical Formulation Let us consider the unsteady motion of a viscous incompressible electrically conducting fluid induced by the stretching of a surface in the x – direction in a rotating fluid. The rotation of the fluid makes the flow threedimensional.

Fig. 1 Physical model and coordinate system Fig. 1 shows the coordinate system, where u, v and w be the velocity components in the x, y and z respectively. A uniform magnetic field B is applied in the z – direction. The velocity components u, v and w, temperature T and concentration C depend only on x and z as the flow is induced by stretching the surface in the x – direction only. The induced magnetic field is neglected by assuming that the magnetic Reynolds number is small. The surface temperature and the fluid temperature at the edge of the boundary layer are all assumed to be constant. Initially (i.e. at – ‫ כ‬ൌ Ͳ) the stretching surface varies linearly at a distance from leading edge (i.e. —୵ ൌ ƒšǡ ƒ ൐ Ͳ) and the fluid is rotating with an angular velocity ȳ଴ . At – ‫ כ‬൐ Ͳ, the velocity of stretching surface is taken as—୵ ൌ ƒš ሺͳ െ Ƚ– ‫ כ‬ሻିଵ and the fluid is rotating with an angular velocity ȳ ൌ ȳ଴ ሺͳ െ Ƚ– ‫ כ‬ሻିଵ about z – axis. Under these assumptions, the equations of continuity, motion, heat and mass transfer can be written as —୶ ൅ ˜୷ ൅ ™୸ ൌ Ͳ —୲ ൅ ——୶ ൅ ˜—୷ ൅ ™—୸ െ ʹπ˜ ൌ  െɏିଵ ’୶ ൅ ɋ‫׏‬ଶ — െ ɐɏିଵ  ଶ — ˜୲ ൅ —˜୶ ൅ ˜˜୷ ൅ ™˜୸ െ ʹπ— ൌ  െɏିଵ ’୷ ൅ ɋ‫׏‬ଶ ˜ െ ɐɏିଵ  ଶ ˜ ™୲ ൅ —™୶ ൅ ˜™୷ ൅ ™™୸ ൌ  െɏିଵ ’୸ ൅ ɋ‫׏‬ଶ ™ ୲ ൅ —୶ ൅ ˜୷ ൅ ™୸ ൌ 

ଵ

஡ୡ౦

ቂ‫׏‬ଶ  െ ଶ

ப୯౨ ப୸

െ ‫ כ‬ሺ െ ஶ ሻቃ

୲ ൅ —୶ ൅ ˜୷ ൅ ™୸ ൌ ‫  ׏‬െ ሺ െ ஶ ሻ ଶ

Where ‫ ׏‬ൌ ቀ

பమ

ப୶మ

൅

பమ

ப୷మ

൅

பమ

ப୸మ

ቁ.

The boundary conditions are ப୘ u(x, y, 0, – ‫ ୵—= ) כ‬, v(x, y, 0, – ‫ = ) כ‬w(x, y, 0, – ‫ = ) כ‬0,െ ୤ ൌ Š୤ ሺ୵ െ ሻ, C(x, y, 0, – ‫ = ) כ‬Cw, ப୸ p(x, y, 0, – ‫ = ) כ‬pw

(1) (2) (3) (4) (5) (6)

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u(x, y, λ, – ‫ = ) כ‬v(x, y, λ, – ‫ = ) כ‬0, T(x, y, λ, – ‫ = ) כ‬ஶ , C(x, y, λ, – ‫ = ) כ‬ஶ (7) where ɋ is the kinematic viscosity, B is the strength of the magnetic field, p is the pressure, t is the dimensional time, – ‫ כ‬ൌ ƒ– is the dimensionless time, ƒ ൌ ሺ†—୵ Τ†šሻ୲‫כ‬ୀ଴ is the gradient of the velocity of the stretching surface at time – ‫ כ‬ൌ Ͳ . The parameter Ƚ denotes unsteadiness in the flow field Ƚ ‫ Ͳ س‬corresponds to whether the flow is accelerating or decelerating, the subscripts t, x, y, z denote partial derivatives with respect to t, x, y, z respectively, and the subscripts w, λ denote condition at the surface and condition far away from the surface respectively, …୮ is the specific heat at constant pressure, ɏ is the density of the fluid,Š୤ is the convective heat transfer coefficient, ୵ is the convective fluid temperature, π is the angular velocity of the fluid, T is the fluid temperature, C is the fluid concentration, ɐ is the electrical conductivity, K is the thermal conductivity of the medium, ‫ כ‬is the uniform volumetric heat absorption, “ ୰ is the radiation heat flux, D is the mass diffusivity and k is the chemical reaction parameter. The radiation heat flux by using Rosseland approximation and expanding  ସ linearly using Taylor’s series can be ଷΤ written as “ ୰ = െ ሺͳ͸ɐୱ ஶ ͵ ‫ כ‬ሻሺμΤμœሻ (8) where ɐୱ is the Stefen-Boltzman constant and  ‫ כ‬is the absorption coefficient. The equations (2) – (6) can be transformed into a set of ordinary differential equations on introducing the following similarity variables:

Ʉ ൌ ඥƒΤɋሺͳ െ Ƚ– ‫ כ‬ሻ œ, – ‫ כ‬ൌ ƒ–, — ൌ ƒšሺͳ െ Ƚ– ‫ כ‬ሻିଵ ˆ ᇱ ሺɄሻ, ˜ ൌ ƒšሺͳ െ Ƚ– ‫ כ‬ሻିଵ ‰ሺɄሻ, ™ ൌ െඥƒɋΤሺͳ െ Ƚ– ‫ כ‬ሻ ˆሺɄሻ,  ൌ ஶ ൅ ሺ୵ െ ஶ ሻɅሺɄሻ,  ൌ ஶ ൅ ሺ୵ െ ஶ ሻԄሺɄሻ, ’ ൌ ’୵ െ ʹିଵ ɏƒሺͳ െ Ƚ– ‫ כ‬ሻିଵ ሺɄሻ, —୵ ൌ ƒšሺͳ െ Ƚ– ‫ כ‬ሻିଵ , ˜୵ ൌ ™୵ ൌ Ͳ, ȳ ൌ ȳ଴ ሺͳ െ Ƚ– ‫ כ‬ሻିଵ ,  ൌ ଴ ሺͳ െ Ƚ– ‫ כ‬ሻିଵȀଶ , ‫ כ‬ൌ  ଴ ሺͳ െ Ƚ– ‫ כ‬ሻିଵ (9) where ȳ଴ is the angular velocity of the fluid at – ‫ כ‬ൌ Ͳ, ଴ is the magnetic field at – ‫ כ‬ൌ Ͳ. From the above transformations, the equations (2) – (6) reduce to the non – dimensional, nonlinear ordinary differential equations: ஗ ˆ ᇱᇱᇱ ൅ ˆˆ ᇱᇱ െ ˆԢଶ െ ˆ ᇱ െ Ƚሺˆ ᇱ ൅ ˆԢԢሻ ൅ ʹɉ‰ ൌ Ͳ (10) ஗

ଶ

‰ ᇱᇱ ൅ ˆ‰ ᇱ െ ˆ ᇱ ‰ െ ‰ െ Ƚሺ‰ ൅ ‰Ԣሻ െ ʹɉˆԢ ൌ Ͳ ସ

ᇱᇱ

ᇱ

ଶ

(11)

஗

ቀͳ ൅ ”ቁ Ʌ ൅ ” ቀˆɅ െ Ʌ െ ȽɅԢቁ ൌ Ͳ ଷ

ᇱᇱ

ᇱ

஗

(12)

ଶ

Ԅ ൅ … ቀˆԄ െ ȽԄԢ െ ɀԄቁ ൌ Ͳ (13) ଶ (14) ‫ ܨ‬ᇱ ൌ ʹ݂ ᇱᇱ ൅ ʹ݂݂ ᇱ െ ߙሺ݂ ൅ ߟ݂Ԣሻ The associated boundary conditions are: (15) ߟ ൌ Ͳ : ݂ ൌ Ͳǡ ݂ ᇱ ൌ ͳǡ ݃ ൌ Ͳǡ ߠ ᇱ ൌ െ‹ሺͳ െ Ʌሻǡ ߶ ൌ ͳǡ ‫ ܨ‬ൌ Ͳ, (16) ߟ ՜ λ : ݂ ᇱ ՜ Ͳǡ ݃ ՜ Ͳǡ ߠ ՜ Ͳǡ ߶ ՜ Ͳ, Where the primes denote the differentiation with respect to ߟ, ߣ ൌ ߗ଴ Ȁܽ is the rotation parameter, ‫ ܯ‬ൌ ߪ‫ܤ‬଴ଶ Τߩܽ is the magnetic parameter, ܲ‫ ݎ‬ൌ ߩܿ௣ ߥΤ݇ is the Prandtl number, ܰ‫ ݎ‬ൌ ͳ͸ߪ௦ ܶஶଷ Τ͵݇ ‫ ܭ כ‬is the thermal radiation parameter, ܳ ൌ ܳ଴ Τܽߩܿ௣  is the heat source/sink, ܵܿ ൌ ߥ Τ‫ ܦ‬is the Schmidt number, ߛ ൌ ݇ Τܽ is the chemical reaction parameter and ‫ ݅ܤ‬ൌ ܿ Τ݇௙ ඥߥ Τ‫ݑ‬௪ is the Biot number. Equations (10) – (13) have been solved numerically, but the solution of equation (14) governing pressure F under condition (15) and (16) can be expressed as: (17) ‫ ܨ‬ൌ ʹ݂ ᇱ ൅ ݂ ଶ െ ߙߟ݂ െ ʹ The physical quantities of engineering interest are the skin friction, heat transfer and mass transfer coefficients. The local skin friction coefficients in the x and y directions, the local Nusselt number and Sherwood number are expressed as: ିଵΤଶ

‫ܥ‬௙௬ ൌ െʹ൫ܴ݁௬ ൯ ݃ᇱ ሺͲሻ, (18) ‫ܥ‬௙௫ ൌ െʹሺܴ݁௫ ሻିଵΤଶ ݂ ᇱᇱ ሺͲሻ, ܰ‫ݑ‬௫ ൌ െሺܴ݁௫ ሻିଵΤଶ ߠ ᇱ ሺͲሻ, S݄௫ ൌ െሺܴ݁௫ ሻିଵΤଶ ߶ ᇱ ሺͲሻ, (19) where ܴ݁௫ ൌ ‫ݑ‬௪ ‫ݔ‬Τߥ , ‡୷ ൌ —୵ ›Τɋ are the local Reynolds numbers, ɋ is the coefficient of kinematic viscosity.

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3. Results and Discussion In this analysis the unsteady flow of a viscous rotating fluid under the influence of a magnetic field, thermal radiation and heat absorption in the presence of convective boundary condition is analysed. The similarity equations governing the flow are numerically solved by Runge-Kutta method along with shooting technique. To validate the numerical scheme the skin friction coefficient and Nusselt number are compared with those of Takhar and Nath [3] for the case of steady flow in the absence magnetic field, thermal radiation, heat absorption and concentration for different values of Prandtl number and rotation parameter. Also the Nusselt number is compared with those of Grubka and Bobba [7], Takhar and Nath [3] for steady flow in the absence of thermal radiation, heat absorption and concentration. From Tables 1 and 2, the compared values of ݂ƍƍ, ݃ƍ, ߠƍ at ߟ ൌ Ͳ are found to be in excellent agreement. Table 1. Comparison of surface shear stress and heat transfer results (െ݂ ƍƍ ሺͲሻǡ െ݃ƍ ሺͲሻǡ െߠ ƍ ሺͲሻ) with those Takhar and Nath [3] for ߙ ൌ ‫ ܯ‬ൌ ܰ‫ ݎ‬ൌ ܳ ൌ ܵܿ ൌ ‫ ݅ܤ‬ൌ ߛ ൌ Ͳ.

0.0000 ሺͲǤͲͲͲͲሻற ሺͲǤͲͲͲͲሻ‫כ‬

Pr=0.7 0.4550 ሺͲǤͶͷͷሻற ሺͲǤͶͷͷͲሻ‫כ‬

െߠ ᇱ ሺͲሻ Pr=0.2 0.9114 ሺͲǤͻͳͳሻற ሺͲǤͻͳͳͶሻ‫כ‬

Pr=7.0 1.8954 ሺͳǤͺͻͶሻற ሺͳǤͺͻͷͻሻ‫כ‬

1.1384 ሺͳǤͳ͵ͺͶሻற ሺͳǤͳ͵ͺ͵ሻ‫כ‬

0.5128 ሺͲǤͷͳʹͺሻற ሺͲǤͷͳʹ͹ሻ‫כ‬

0.3904 ሺͲǤ͵ͻͲሻற ሺͲǤ͵ͻͲͶሻ‫כ‬

0.8525 ሺͲǤͺͷ͵ሻற ሺͲǤͺͷʹͷሻ‫כ‬

1.8511 ሺͳǤͺͷͲሻற ሺͳǤͺͷͳ͹ሻ‫כ‬

1

1.3250 ሺͳǤ͵ʹͷͲሻற ሺͳǤ͵ʹͷͲሻ‫כ‬

0.8371 ሺͲǤͺ͵͹ͳሻற ሺͲǤͺ͵͹Ͳሻ‫כ‬

0.3216 ሺͲǤ͵ʹͳሻற ሺͲǤ͵ʹͳ͹ሻ‫כ‬

0.7703 ሺͲǤ͹͹Ͳሻற ሺͲǤ͹͹Ͳ͸ሻ‫כ‬

1.788 ሺͳǤ͹ͺͺሻற ሺͳǤ͹ͺͺͳሻ‫כ‬

2

1.6524  ሺͳǤ͸ͷʹ͵ሻற ሺͳǤ͸ͷʹͶሻ‫כ‬

1.2873 ሺͳǤʹͺ͹͵ሻற ሺͳǤʹͺ͹Ͳሻ‫כ‬

0.2429 ሺͲǤʹͶʹሻற ሺͲǤʹͶʹͻሻ‫כ‬

0.6382 ሺͲǤ͸͵ͺሻற ሺͲǤ͸͵ͻͺሻ‫כ‬

1.6643 ሺͳǤ͸͸Ͷሻற ሺͳǤ͸͸Ͷͺሻ‫כ‬

ߣ

െ݂ԢԢሺͲሻ

െ݃ԢሺͲሻ

0

1.0000 ሺͳǤͲͲͲͲሻற ሺͳǤͲͲͲͲሻ‫כ‬

0.5

‫ כ‬Results obtained by Takhar and Nath [3] Table 2. Comparison of surface heat transfer Ȃ Ʌƍ ሺͲሻ with that of Grubka and Bobba [8] and Takhar and Nath [3] for Ƚ ൌ  ൌ ɉ ൌ ” ൌ ‹ ൌ  ൌ … ൌ ɀ ൌ Ͳ. Pr 0.72 1.0 3.0 10.0 100.0

Grubka and Bobba [8] 0.4631 0.5820 1.1652 2.3080 7.7657

Takhar and Nath [3] 0.4651 0.5823 1.1654 2.3090 7.7657

Present Results 0.4631 0.5820 1.1652 2.3080 7.7657

Fig. 1 reveals that the primary velocity is found to decelerate with increasing rotation parameter as rotation lessens the` fluid entrained. Also for smaller values of ߣ a monotonic and exponential decay in the velocity is noted whereas for higher values of the rotation parameter it exhibits an oscillatory behavior with a depreciation in the velocity. From Fig. 2 it is evident that the rotation of the fluid generates secondary velocity and is absent when ߣ ൌ Ͳ and the secondary velocity takes on its peak value -0.2754 at ߟ ൌ ͲǤ͵͹ͺ͸ for higher values of ߣ. From Fig. 3 it is observed that a back flow in the primary velocity exists when there is no magnetic field and in the presence of magnetic field this flow is absent. We may conclude that the flow field can be made free from back flow by using magnetic field. The primary velocity depreciates for increasing strength of magnetic field which is in conformity with the fact that the Lorentz force that arises due to the magnetic field decelerates the velocity. From

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Fig.4 the secondary velocity is significantly influenced by the magnetic field throughout the boundary layer. In the vicinity of the boundary, the secondary velocity profile (-g) attains a maximum value and increasing values of M reduce this peak value. The primary velocity (Fig.5) along the sheet falls initially for increasing unsteady parameter ߙ and hence the boundary layer shrinks in the vicinity of the wall while it increases the velocity away from the wall. However, the effect of ߙ on primary velocity is not appreciable whereas its effect is more pronounced on the secondary velocity (Fig.6). The secondary velocity enhances in the vicinity of the boundary while it decreases away from the boundary eventually satisfying the free stream condition. Fig. 7 shows that as Biot number increases convection becomes stronger resulting in higher surface temperatures and the thermal effect permeates deeper into the quiescent fluid resulting in thicker thermal boundary layers. Fig. 8 depicts that the temperature reduces with increasing in Prandtl number. Since higher Prandtl number fluid has low thermal conductivity and opposes conduction, the temperature of higher Prandtl number fluids falls rapidly compared to lower Prandtl number fluids. The thickness of the thermal boundary layers for lower Prandtl fluids enlarges. From Fig. 9 it is observed that presence of the thermal radiation Nr enhances the temperature significantly throughout the region. The temperature is further increased for increasing thermal radiation parameter. From Fig. 10 it is evident that the concentration reduces with increased values of Schmidt number which is associated with the reduction in the thickness of the solutal boundary layer. This may physically be explained that enhancement of the Schmidt number implies reduction in the molecular diffusivity and thus the concentration of the species is more for smaller values of Schmidt number than for higher values of Schmidt number. Also the gradient of concentration is always negative and hence the mass transfer always takes place from boundary to the ambient fluid for all values of Schmidt’s number. From Fig. 11 the wall shear stress is seen that the dependence on the magnetic field and rotation is quite appreciable. For a fixed value of the rotation parameter an increase in magnetic field parameter decreases the wall shear stress. This is because; an increase in the magnetic field induces a reduction in the velocity (Fig. 3) which in turn reduces the wall shear stress. The wall shear stress shows a similar trend for an increase in the rotation parameter due to the reduction in the velocity (Fig. 1). Fig. 12 shows that Nusselt number is an increasing function of Prandtl number and the rate of heat transfer is observed to increase with elapsed time. Fig.13 reveals that the Nusselt number increases with heat absorption parameter and it is reduced by the radiation parameter. We may conclude that the thermal radiation enhances the rate of heat transfer. It is noticed from Fig. 14 that the local Sherwood number significantly increases with chemical reaction parameter and Schmidt number. 0.05

1.2 λ = 1.0 λ = 2.0 λ = 3.0

1

0

λ = 4.0

λ = 1.0

-0.05

0.8

λ = 2.0

-0.15

0.2

-0.2

0

-0.25

0

1

2

3

4

5

η

Fig. 1. Primary velocity profiles for different values of Ȝ

λ = 4.0

g(η )

f ' (η ) 0.4

-0.2

λ = 3.0

-0.1

M=1;α =-0.5;Pr=0.7;Nr=1;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

0.6

6

-0.3

M=1;α =-0.5;Pr=0.7;Nr=1;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

0

1

2

3

4

5

6

η

Fig. 2. Secondary velocity profiles for different values of Ȝ

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Sreelakshmi K et al. / Procedia Engineering 127 (2015) 678 – 685 1.2

0.05

M=0.0 M=1.0 M=2.0 M=4.0

1

0 M=0.0 M=1.0 M=2.0 M=4.0

-0.05

0.8

-0.1

λ=1.0;α =-0.5;Pr=0.7;Nr=1.0;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

g(η )

f ' (η )

0.6

0.4

-0.15

0.2

-0.2

0

-0.25

-0.2

0

1

2

3

4

5

-0.3

6

λ=1.0;α =-0.5;Pr=0.7;Nr=1.0;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

0

1

2

3

4

5

6

η

η

Fig. 3. Primary velocity profiles for different values of M

Fig. 4. Secondary velocity profiles for different values of M 0.05

1.2 α = -1.0 α = -0.5 α = 0.0

1

0

α = 0.5

0.8

-0.05 f ' (η )

α = -1.0 α = -0.5 α = 0.0

g(η )

M=1.0;λ=1.0;Pr=0.7;Nr=1.0;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

0.6

-0.1

α = 0.5

-0.15

M=1.0;λ=1.0;Pr=0.7;Nr=1.0;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

0.4

0.2

0

-0.2

-0.2 0

1

2

3

4

5

6

0

1

2

3

Fig. 5. Primary velocity profiles for different values of ߙ

5

6

Fig. 6. Secondary velocity profiles for different values of ߙ

1

0.7 Bi=0.1 Bi=1.0 Bi=5.0 Bi=10.0 Bi=100.0

0.9 0.8 0.7

Pr=0.7 Pr=1.0 Pr=2.0 Pr=3.0 Pr=7.0

0.6

0.5

M=1.0;α =-0.5;λ=1.0;Pr=0.7;Nr=1.0;Q=0.5;Sc=0.5;γ =0.1;

M=1.0;α =-0.5;λ=1.0;Nr=1.0;Q=0.5;Sc=0.5;γ =0.1;Bi=1

0.4 θ (η )

0.6 θ (η )

4

η

η

0.5

0.3

0.4 0.3

0.2

0.2

0.1

0.1 0

0

1

2

3

4

5

6

7

8

9

10

η

Fig. 7. Temperature profiles for different values of Bi

0

0

1

2

3

4

5

6

7

8

9

η

Fig. 8. Temperature profiles for different values of Pr

10

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Sreelakshmi K et al. / Procedia Engineering 127 (2015) 678 – 685 1

0.8 Nr=0.0 Nr=0.5 Nr=1.0 Nr=1.5 Nr=2.0

0.7 0.6

0.8 0.7

0.5

M=1.0;α =-0.5;λ=1.0;Pr=0.7;Nr=1.0;Q=0.5;Bi=1.0;γ =0.1

0.6 φ (η )

θ (η )

Sc=0.5 Sc=1.0 Sc=1.5 Sc=2.0

0.9

M=1.0;α =-0.5;λ=1.0;Pr=0.7;Q=0.5;Sc=0.5;;Bi=1.0;γ =0.1

0.4

0.5 0.4

0.3

0.3

0.2 0.2

0.1 0

0.1

0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

4

5

6

7

8

9

10

η

η

Fig. 9. Concentration profiles for different values of Nr

Fig. 10. Concentration profiles for different values of Sc 0.5

-1.2 λ = 1.0

-1.3 -1.4

λ = 1.5 λ = 2.0

0.45

λ = 2.5

0.4 0.35 Nu x/Re 1/2 x

CfxRe 1/2 x

-1.5 -1.6

0.3

-1.7

0.25

-1.8

0.2

-1.9 -2 0.1

M=1.0;λ=1.0;Nr=1.0;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

α = -1.0 α = -0.5

0.2

0.3

0.4

0.5

0.6

0.7

α = 0.0

0.15

α =-0.5;Pr=0.7;Nr=1.0;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

0.8

0.9

α = 0.5

0.1 0.1

1

0.2

0.3

0.4

0.5

Fig. 11: Variation of Skin friction coefficient with M for different values of ߣ

0.7

0.8

0.9

1

Fig. 12: Variation of Nusselt number with Pr for different values of ߙ

0.45

2

M=1.0;α =-0.5;Pr=0.7;Q=0.5;Sc=0.5;Bi=1.0;γ =0.1

Sc=0.5 Sc=1.0 Sc=1.5 Sc=2.0

1.8

0.4 1.6 1.4 Sh x/Re 1/2 x

0.35 Nux/Re 1/2 x

0.6 Pr

M

0.3

1.2 1 0.8

Nr=0.0 Nr=0.5 Nr=0.5 Nr=1.0

0.25

0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Q

Fig. 13: Variation of Nusselt number with Q for different values of Nr

0.9

0.6 M=1.0;α =-0.5;λ=1.0;Pr=0.7;Nr=1.0;Q=0.5;Bi=1.0

1

0.4 0.1

0.2

0.3

0.4

0.5

0.6

0.7

γ

Fig. 14: Variation of Sherwood number with ߛ for different values of Sc

0.8

0.9

1

Sreelakshmi K et al. / Procedia Engineering 127 (2015) 678 – 685

4. Conclusions ¾ ¾ ¾ ¾ ¾ ¾

From the above analysis it is observed that The Lorentz force retards primary velocity while it accelerates the secondary velocity and magnetic parameter prevents the flow reversal in the x – direction. The temperature is enhanced with increasing magnetic field, Biot number and thermal radiation and increasing values of heat absorption parameter produce thinner thermal boundary layers. The Schmidt number and chemical reaction parameter suppress the species concentration significantly. The rotation parameter reduces skin friction coefficients. The rate of heat transfer is increased by magnetic field, thermal radiation parameter and rotation parameter. The rotation parameter and time enhance the rate of mass transfer.

Acknowledgement The authors are thankful to the reviewers for their suggestions to improve the quality of the paper. References [1] [2] [3] [4] [5] [6] [7]

Wang, C. Y. (1988) “Stretching a surface in a rotating fluid”, Journal of Applied Mathematics and Physics (ZAMP), 39, 177 Gorla, R. S. R., Dakappagari, V., Pop, I. (1995) “Stretching a Surface in a Rotating Power-Law Fluid”, Z. angew. Math. Mech. (ZAMM ), 75(11), 876 – 878 Takhar, H. S., Nath, G. (1998) “Unsteady flow over a stretching surface with a magnetic field in a rotating fluid”, Zeitschrift fur angewamdte Mathematic and Physik (ZAMP), 49, 989 – 1001 Takhar, H. S., Chamkha, A. J., Nath, G. (2003) “Flow and heat transfer on a stretching surface in a rotating fluid with a magnetic field”, International Journal of Thermal Sciences, 42, 23 – 31 Nazar, R., Amin, N., Pop, I., (2004) “unsteady boundary layer flow due to a stretching surface in a rotating fluid”, Mechanics Research Communications, 31, 121 – 128 Seddeek, M. A. (2002) “Effects of radiation and variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic field in the case of unsteady flow”, Int. J. Heat Mass Transfer, 45, 931 – 5 Grubka, L. J., Bobba, K. M. (1985) “Heat transfer characteristics of a continuous stretching surface with variable temperature”, ASME J. Heat Transfer, 107, 248

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