Radiation and chemical reaction effects on MHD ...

Alexandria Engineering Journal (2016) 55, 583–595

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Radiation and chemical reaction effects on MHD Casson fluid flow past an oscillating vertical plate embedded in porous medium Hari R. Kataria a, Harshad R. Patel b,* a b

Department of Mathematics, Faculty of Science, The M.S. University of Baroda, Vadodara, India Applied Science & Humanities Department, Sardar Vallabhbhai Patel Institute of Technology, Vasad, India

Received 29 October 2015; revised 4 January 2016; accepted 16 January 2016 Available online 9 February 2016

KEYWORDS Magnetohydrodynamics; Casson fluid; Ramped temperature; Oscillating plate

Abstract Analytic expression for unsteady free convective hydromagnetic boundary layer Casson fluid flow past an oscillating vertical plate embedded through porous medium in the presence of uniform transverse magnetic field, thermal radiation and chemical reaction is obtained. Both isothermal and ramped wall temperatures are taken into account. The governing equations are solved using Laplace transform technique and the solutions are presented in closed form. The numerical values of Casson fluid velocity, temperature and concentration at the plate are presented graphically for several values of the pertinent parameters. Effect of governing parameters on Skin friction, Nusselt number and Sherwood number is also discussed. Casson parameter c is inversely proportional to the yield stress and it is observed that for the large value of Casson parameter, the fluid is close to the Newtonian fluid where the velocity is less than the Non-Newtonian fluid. It is seen that velocity increases and Temperature decreases with increase in thermal radiation R. Radiation parameter R signifies the relative contribution of conduction heat transfer to thermal radiation transfer. Concentration decreases tendency with chemical reaction parameter R0 . Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Research in the magnetohydrodynamics has been advanced significantly during the last few years in natural sciences and engineering disciplines after the pioneer work of Hartmann [1] in liquid metal duct flows under the influence of a strong * Corresponding author. Tel.: +91 9727413159. E-mail addresses: [email protected] (H.R. Kataria), [email protected] (H.R. Patel). Peer review under responsibility of Faculty of Engineering, Alexandria University.

external magnetic field. The study of magnetohydrodynamic (MHD) flow of non-Newtonian fluid in a porous medium has attracted many researchers due to its application in the optimization of solidification processes of metals, alloys, the geothermal sources investigation and nuclear fuel debris treatment. Alfven [2] has studied Existence of electromagnetichydrodynamic waves. Singh et al. [3] have studied natural convection in different physical conditions in the presence of uniform/radial varying magnetic field. Khan [4] has studied MHD Mixed Convection Flow of Ferro fluid along a Vertical Channel. Magneto-Nanofluid flow past an impulsively started porous flat plate in a rotating frame was investigated by

http://dx.doi.org/10.1016/j.aej.2016.01.019 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

584

H.R. Kataria, H.R. Patel

Nomenclature u T C g k1 cp K1 D sc pr Gm Gr t

velocity of the fluid in the x direction temperature concentration acceleration due to gravity permeability specific heat of the fluid at constant pressure chemical reaction coefficient mass diffusivity Schmidt number Prandtl number Grashof number of mass transfer Grashof number time

Makinde [5]. Nadeem [6,7] discussed on MHD oblique stagnation-point flow of a viscoelastic fluid over a convective surface. Khan [8–14] has studied unsteady MHD conjugate flow for different fluids and different wall temperatures in a porous medium. Recently Kataria [15] has studied Mathematical model for velocity and temperature of gravity-driven convective optically thick nanofluid flow past an oscillating vertical plate in the presence of magnetic field and radiation. Kataria [16] studied effect of magnetic field on unsteady natural convective flow of a micropolar fluid between two vertical walls. Many applications for the MHD flows of non-Newtonian fluids in a porous medium are encountered in biological systems, heat-storage beds, irrigation problems, process of petroleum, paper, textile and polymer composite industries. Numerous studies have been presented on various aspects of MHD flows of non-Newtonian fluid flows passing through a porous medium. In nature, some non-Newtonian fluids behave like elastic solid that is, no flow occurs with small shear stress. Casson fluid is one of such fluids. This fluid has distinct features and is quite famous recently. Casson fluid model was introduced by Casson in 1959 for the prediction of the flow behavior of pigment-oil suspensions [17] so, for the flow, the shear stress magnitude of Casson fluid needs to exceed the yield shear stress. Dash [18], consider Casson fluid flow in a pipe filled with a homogeneous porous medium. Hayat [19], consider Soret and Dufour effects on MHD flow of Casson fluid. Recently Nadeem [20,21] has studied MHD flow of a Casson fluid over an exponentially/linearly shrinking sheet. Animasaun [22] has studied MHD dissipative Casson fluid flow with suction and nth order of chemical reaction. Nadeem [23,24] has discussed on Casson fluid past a linearly stretching sheet with convective boundary condition. Recently Akbar [25] has studied Metachronal beating of cilia under the influence of Casson fluid and magnetic field and again Akbar [26–28] has investigated the magnetic field effects on EyringPowell/Casson fluid flow toward a stretching sheet, asymmetric channel and Plumb Duct. The purpose of this paper was to analyze the important role of radiation and chemical reaction in MHD flow of a Casson fluid flow past over an oscillating vertical plate with ramped wall temperature in a porous medium. We find the analytic solution using Laplace transform technique.

R R0

radiation chemical reaction

Greek alphabets c Casson parameter b volumetric coefficient of thermal expansion r electric conductivity of the fluid q density ; porosity of the fluid k thermal conductivity lb plastic dynamic viscosity m kinematic viscosity

2. Mathematical formulation of the problem We consider Casson fluid past an impulsively started vertical plate with concentration in porous medium. In Fig. 1 the flow being confined to y > 0, where y is the coordinate measured in the normal direction to the plate and x axis is taken along the wall in the upward direction. Initially, at time t ¼ 0, both the fluid and the plate are uniform temperature T1 and the concentration near the plate is assumed to be C1 at all the points respectively. At time t > 0, the plate is given an oscillatory motion in the vertical direction against gravitational field with velocity U0 coswt or U0 sinwt with constant mass flux, T1 þ ðTw þ T1 Þt=t0 when t 6 t0 and Tw when t > t0 respectively and concentration near the plate is raised linearly to Cw which is thereafter maintained constant Tw and Cw . A uniformly distributed transverse magnetic field of strength B0 is applied in the y axis direction. Induced magnetic field produced by the fluid motion is negligible in comparison with the applied one as the magnetic Reynolds number is small enough to neglect the effects of applied magnetic field. We assume that, rigid plate, incompressible flow, one dimensional flow, non-Newtonian fluid, free convection and viscous dissipation term in the energy equation are neglected. The constitutive equation for the Casson fluid can be written as

Figure 1

Physical sketch of the problem.

Radiation and chemical reaction effects on MHD Casson fluid flow w = 0 Ramped Temperture

Pr = 0.71, 7, 25

585 K = 0.5, 1.5, 5

1

w = pi/2

w = 0 Ramped Temperture 0.9

w = 0 Isothermal Temperture

1

w = pi/2

w = pi/2 w = 0 Isothermal Temperture

0.8

w = pi/2 0.8

0.7

Velocity

Velocity

0.6 0.6

0.5 0.4

0.4 0.3 0.2

0.2

0.1 0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

y

Figure 2 Velocity profile u for different values of y and Pr at M = 0.5, Sc = 0.6, Gm = 5, Gr = 10, k = 1, t = 0.6, R = 0.5 and R0 ¼ 2.

@C @2C ¼ D 2 K 1 ðC C 1 Þ @t @y

Gamma = 0.2, 0.5, 1 w = 0 Ramped Temperture 1

Figure 4 Velocity profile u for different values of y and k at M = 0.5, Sc = 0.6, Gm = 5, Gr = 10, Pr = 25, c ¼ 0:6, t = 0.6, R = 0.5 and R0 ¼ 2.

ð3Þ

w = pi/2

With initial and boundary conditions


u ¼ 0;

w = pi/2

T ¼ T1 ;

C ¼ C1 ;

as y > 0 and t < 0

u ¼ U sin ðwtÞ or UHðtÞ cos ðwtÞ; T1 þ ðTw T1 Þt=t0 if 0 < t < t0 ; T¼ Tw if t P t0

0.6

0.4

C ¼ C1 þ ðCw C1 Þ;

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

Figure 3 Velocity profile u for different values of y and c at M = 0.5, Sc = 0.6, Gm = 5, Gr = 10, Pr = 7, k = 1, t = 0.6, R = 0.5 and R0 ¼ 2.

8 > ffi eij < 2 lB þ pPyffiffiffi 2p sij ¼ > : 2 lB þ pPy ffiffiffiffiffi eij 2pc

þ qgbC ðC C1 Þ @T @ 2 T @q ¼k 2 r @y @t @y

ð4Þ

Introducing the following dimensionless quantities: pffiffiffiffi U U2 t T T1 t0 y¼ u; h ¼ y; t ¼ ; u¼ ; vt0 vt0 Tw T1 U C C1 wv lcp s C¼ ; s¼ 2 ; w ¼ 2 ; Pr ¼ qu Cw C1 k U

1

p > pc

M = 0.5, 1, 2 w = 0 Ramped Temperture w = pi/2 w = 0 Isothermal Temperture w = pi/2

0.9

p < pc

0.8

where p ¼ eij eij and eij is the ði; jÞth component of the deformation rate, p is the product of the component of deformation rate with itself, pc is a critical value of this product based on the non-Newtonian model, lB is plastic dynamic viscosity of the non-Newtonian fluid and Py is yield stress of fluid. Under these condition we get the following partial differential equation with initial and boundary conditions are given below. @u 1 @2u lu ¼ lB 1 þ q rB20 u u þ qgbðT T1 Þ @t c @y2 k1

qcp

as t P 0 and y ¼ 0

u ! 0; T ! T1 ; C ! C1 ; as y ! 1

0.2

ð1Þ ð2Þ

0.7 0.6

Velocity

Velocity

0.8

0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

Figure 5 Velocity profile u for different values of y and M at k = 1, Sc = 0.6, Gm = 5, Gr = 10, Pr = 25, c ¼ 0:6, t = 0.6, R = 0.5 and R0 ¼ 2.

586

H.R. Kataria, H.R. Patel 1

R = 0.5, 5, 10

w = 0 Ramped Temperture


1

Sc = 0.1, 2, 5

0.9

w = pi/2



w = pi/2

0.8

w = pi/2

0.8

0.6

0.6

Velocity

Velocity

0.7

0.4

0.5 0.4 0.3

0.2

0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.1

2

y

0 0

Figure 6 Velocity profile u for different values of y and R at k = 1, Sc = 0.6, Gm = 5, Gr = 10, Pr = 7, c ¼ 0:6, t = 0.6, M = 0.5 and R0 ¼ 2.

rB20 1 vu2 vgbðTw T1 Þ t; ¼ ; Gr ¼ ; 2 0 k k1 U 2 qU U3 pffiffiffiffiffiffiffi l 2pc m vgbc mðCw C1 Þ c¼ B ; Sc ¼ ; Gm ¼ D Py U3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

Figure 8 Velocity profile u for different values of y and Sc at k = 1, R0 ¼ 2, Gm = 5, Gr = 10, Pr = 25, c ¼ 0:6, t = 0.6, M = 0.5 and R ¼ 0:5.

M2 ¼

R¼

@C 1 @ 2 C ¼ R0 c @t sc @y2

4I1 v2 0 vK1 ;R ¼ 2 ; U0 kU20

With initial and boundary conditions u ¼ h ¼ C ¼ 0; 0

In the Eqs. (1)–(4) dropping out the ‘‘ ” notation (for simplicity) we get Where @u 1 @2u 1 2 ¼ 1þ u þ Gr h þ Gm C M þ ð5Þ @t c @y2 k @h 1 @ h R ¼ h @t Pr @y2 Pr 2

1

ð6Þ

y > 0; t < 0

u ¼ sin ðwtÞ or HðtÞ cos ðwtÞ; h¼

t; 0 < t 6 1 1

t>1

¼ tHðtÞ ðt 1ÞHðt 1Þ; C ¼ t at y ¼ 0; t P 0

u ! 0; h ! 0; C ! 0 at y ! 1

ð8Þ

Where H() is Heaviside unit step function.

R’ = 0.1, 10, 25 w = 0 Ramped Temperture

0.9

Gm =1, 5, 10

w = pi/2

1

w = 0 Ramped Temperture w = pi/2 w = 0 Isothermal Temperture w = pi/2


0.8

0.9

w = pi/2 0.7

0.8

0.6

0.7

0.5

0.6

Velocity

Velocity

ð7Þ

0.4

0.5 0.4

0.3 0.3 0.2 0.2 0.1 0

0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Figure 7 Velocity profile u for different values of y and R0 at k = 1, Sc = 0.6, Gm = 5, Gr = 10, Pr = 25, c ¼ 0:6, t = 0.6, M = 0.5 and R ¼ 0:5.

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

Figure 9 Velocity profile u for different values of y and Gm at k = 1, R0 ¼ 2, Sc = 0.6, Gr = 10, Pr = 25, c ¼ 0:6, t = 0.6, M = 0.5 and R ¼ 0:5.

Radiation and chemical reaction effects on MHD Casson fluid flow

587 1

1

t = 0.2, 0.4, 0.6

Gr = 1, 7, 15


0.9

0.9 w = 0 Ramped Temperture w = pi/2 w = 0 Isothermal Temperture w = pi/2

0.8 0.7


0.8

w = pi/2

0.7 0.6

0.5

Velocity

Velocity

0.6

0.4

0.5

0.3

0.4

0.2

0.3

0.1

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.1

2

y 0

Figure 10 Velocity profile u for different values of y and Gr at k = 1, R0 ¼ 2, Sc = 0.6, Gm = 5, Pr = 25, c ¼ 0:6, t = 0.6, M = 0.5 and R ¼ 0:5.

3. Solution Taking Laplace transform of Eq. (6) and (7) with initial and boundary condition Eq. (8) hðy; sÞ ¼ F9 ðy; sÞð1 es Þ

ð9Þ

C ¼ F11 ðy; sÞ

0

ð1 es ÞG2 ðy; sÞ a10 F10 ðy; sÞ a12 F11 ðy; sÞ a12 F12 ðy; sÞ

ð11Þ

0.8

1

1.2

1.4

1.6

1.8

2

1 1 ucos ðy; sÞ ¼ F1 ðy; sÞ þ F2 ðy; sÞ þ ð1 es ÞG1 ðy; sÞ 2 2 þ a10 F4 ðy; sÞ þ a11 F5 ðy; sÞ þ a12 F6 ðy; sÞ ð1 es ÞG9 ðy; sÞ a10 F12 ðy; sÞ a11 F11 ðy; sÞ a12 F12 ðy; sÞ pffiffiffiffi sþb ey a F1 ðy; sÞ ¼ s þ iw pffiffiffiffi sþb ey a F2 ðy; sÞ ¼ s iw

ð14Þ

curves 1 2 3

Cosine Oscillating Isothermal Temperture

0.4

w 0 pi/2 pi

4 5 6

0 pi/2 pi

7 8 9

0 pi/2 pi

Sine Oscillating 10 Isothermal Temperture 11 12

0 pi/2 pi

Sine Oscillating Ramped Temperture

0.6

ð12Þ ð13Þ

Cosine Oscillating Ramped Temperture

0.8

Velocity

0.6

Figure 11 Velocity profile u for different values of y and t at k = 1, R0 ¼ 2, Sc = 0.6, Gm = 5, Pr = 25, c ¼ 0:6, Gr = 10, M = 0.5 and R ¼ 0:5.

3, 6, 7, 10, 2, 5, 8, 11, 9, 12, 1, 4

1

0.4

y

ð10Þ

i i usin ðy; sÞ ¼ F1 ðy; sÞ F2 ðy; sÞ þ ð1 es ÞG1 ðy; sÞ 2 2 þ a10 F4 ðy; sÞ þ a11 F5 ðy; sÞ þ a12 F6 ðy; sÞ

0.2

0.2

0

−0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

Figure 12 Velocity profile u for different values of y and w at k = 1, R0 ¼ 2, Sc = 0.6 Gm = 5, Pr = 25, c ¼ 0:6, Gr = 10, M = 0.5, t = 0.6 and R ¼ 0:5.

588

H.R. Kataria, H.R. Patel 1

1 Ramped Temperture 0.9

Isothermal Temperature

0.9

Ramped Temperture

Pr = 0.71, 7, 15, 25


0.8

0.8

0.7

0.7

t = 0.2, 0.4, 0.6, 0.8

0.6

Temperature

Temperature

0.6 0.5 0.4 0.3

0.5 0.4 0.3

0.2

0.2

0.1

0.1

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

0

0.1

0.2

0.3

0.4

y

0.5

0.6

0.7

0.8

0.9

1

y

Figure 13 Temperature profile h for different values of y and Pr at t = 0.6 and R ¼ 0:5.

Figure 15 Temperature profile h for different values of y and t at Pr ¼ 25 and R ¼ 0:5.

0.6 1

R’ = 0.1, 5, 10, 15, 25

Ramped Temperture 0.9

0.5


0.8

R = 1, 5, 10, 20

0.4

Concentration

0.7

Temperature

0.6 0.5 0.4 0.3

0.3

0.2

0.1

0.2 0

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Figure 14 Temperature profile h for different values of y and R at t = 0.6 and Pr ¼ 25.

G1 ðy; sÞ ¼ a7 F3 ðy; sÞ þ a8 F4 ðy; sÞ þ a9 F5 ðy; sÞ

ð15Þ

pffiffiffiffi sþb

F3 ðy; sÞ ¼

ey a s þ a3

ð16Þ

pffiffiffiffi sþb

F4 ðy; sÞ ¼

ey

a

s pffiffiffiffi sþb

ey a F5 ðy; sÞ ¼ s2 pffiffiffiffi sþb ey a F6 ðy; sÞ ¼ s þ a6 G2 ðy; sÞ ¼ a7 F7 ðy; sÞ þ a8 F8 ðy; sÞ þ a9 F9 ðy; sÞ

ð17Þ

Figure 16 Concentration profile C for different values of y and R0 at Sc ¼ 0:6 and t ¼ 0:6.

pffiffiffiffiffiffiffiffiffi ey Rþpr s F7 ðy; sÞ ¼ s þ a3 pffiffiffiffiffiffiffiffiffi ey Rþpr s F8 ðy; sÞ ¼ s pffiffiffiffiffiffiffiffiffi ey Rþpr s F9 ðy; sÞ ¼ s2

ð20Þ

ð22Þ

ð23Þ

1 pffiffiffiffiffiffiffiffiffiffiffiffi 0 F10 ðy; sÞ ¼ ey sc ðR þsÞ s

ð24Þ

F11 ðy; sÞ ¼

1 ypffiffiffiffiffiffiffiffiffiffiffiffi sc ðR0 þsÞ e s2

ð25Þ

F12 ðy; sÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffi 1 0 ey sc ðR þsÞ ðs þ a6 Þ

ð26Þ

ð18Þ

ð19Þ

ð21Þ

Taking inverse Laplace transform of Eqs. (9)–(26) we get


589

0.6

Ramped Temperture Isothermal Temperature

25 0.5 Sc = 0.2, 2, 5, 7, 9

20

Pr = 0.71, 7, 15, 25

0.4

Skin friction

Concentration

15

0.3

10

5

0.2

0 0.1 −5 0.1

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

2

y

Figure 17 Concentration profile C for different values of y and Sc at R0 ¼ 2 and t ¼ 0:6.

Figure 19 Skin friction for different values of t and Pr at k = 1, Sc = 0.6, Gm = 5, Gr = 2, R = 0.5, R0 ¼ 1, c ¼ 0:1 and M = 0.5.

hðy; tÞ ¼ f9 ðy; tÞ f9 ðy; t 1ÞHðt 1Þ

ð27Þ

4. Solutions for plate with constant temperature

Cðy; tÞ ¼ f11 ðy; tÞ

ð28Þ

i i usin ðy;tÞ ¼ f1 ðy;tÞ f2 ðy;tÞ þ g1 ðy;tÞ 2 2 g1 ðy;t 1ÞHðt 1Þ þ a10 f4 ðy;tÞ þ a11 f5 ðy;tÞ þ a12 f6 ðy;tÞ g2 ðy;tÞ þ g2 ðy;t 1ÞHðt 1Þ a10 f10 ðy;tÞ a11 f11 ðy;tÞ a12 f12 ðy;tÞ

ð29Þ

1 1 ucos ðy; tÞ ¼ f1 ðy; tÞ þ f2 ðy; tÞ þ g1 ðy; tÞ 2 2 g1 ðy; t 1ÞHðt 1Þ þ a10 f4 ðy; tÞ

In order to understand the effects of ramped temperature of the plate on the fluid flow, we must compare our results with constant temperature. In this case, the initial and boundary conditions are the same excluding Eq. (8) that becomes h ¼ 1 at y ¼ 0; t P 0. We find the isothermal temperature hðy; tÞ and velocity profile u (y, t) using Laplace transform technique. hðy; tÞ ¼ f8 ðy; tÞ

ð31Þ

Cðy; tÞ ¼ f11 ðy; tÞ

ð32Þ

i i usin ðy;tÞ ¼ f1 ðy;tÞ f2 ðy;tÞ þ ða9 þ a10 Þf4 ðy;tÞ a9 f3 ðy;tÞ 2 2 þ a11 f5 ðy;tÞ þ a12 f6 ðy;tÞ a9 f8 ðy;tÞ þ a9 f7 ðy;tÞ

þ a11 f5 ðy; tÞ þ a12 f6 ðy; tÞ g2 ðy; tÞ þ g2 ðy; t 1ÞHðt 1Þ a10 f10 ðy; tÞ a11 f11 ðy; tÞ a12 f12 ðy; tÞ

ð30Þ

a10 f10 ðy;tÞ a11 f11 ðy;tÞ a12 f12 ðy;tÞ

1

1

0.9

0

0.8

−1

ð33Þ

Pr = 0.71, 7, 15, 25

−2

0.7 t = 0.2, 0.4, 0.6, 0.8, 1.0

Nusselt Number

Concentration

0.6 0.5 0.4 0.3

−3 −4 Ramped Temperture

−5

Isothermal Temperature −6 −7

0.2

−8

0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

Figure 18 Concentration profile C for different values of y and t at R0 ¼ 2 and Sc ¼ 0:6.

−9 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

Figure 20 Nusselt Number for different values of t and Pr at k = 1, Sc = 0.6, Gm = 5, Gr = 2, R0 ¼ 1, c ¼ 0:1, R = 0.5 and M = 0.5.

590

H.R. Kataria, H.R. Patel Skin friction variation for air (Pr = 0.71 and sinwt = 0).

Table 1 t

c

Sc

Gr

Gm

R

R0

M

k

Skin friction s for Ramped temperature

Skin friction s for isothermal temperature

0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.6

0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.6 0.6 0.6 0.7 0.8 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

2 2 2 2 2 3 5 2 2 2 2 2 2 2 2 2 2 2 2

5 5 5 5 5 5 5 7 9 5 5 5 5 5 5 5 5 5 5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

1 1 1 1 1 1 1 1 1 1 1 0.9 0.8 1 1 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.9 0.5 0.5 0.5 0.5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.1 1.2 0.5 0.5

2.7934 2.8531 3.0731 2.7197 2.6561 3.4741 4.8355 3.3663 3.9391 2.0626 1.8636 2.4719 1.7080 2.9362 3.0545 2.7289 2.6711 2.5466 2.1748

0.3121 0.2179 0.0556 0.2383 0.1747 0.2479 1.3679 0.8849 1.4577 0.3399 0.3533 0.0095 0.7733 0.2960 0.1787 0.3044 0.2928 0.0261 0.4287

Skin friction variation for water (Pr = 7 and sinwt = 0).

Table 2 t

c

Sc

Gr

Gm

R

R0

M

k



0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.6

0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.6 0.6 0.6 0.7 0.8 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

2 2 2 2 2 3 5 2 2 2 2 2 2 2 2 2 2 2 2

5 5 5 5 5 5 5 7 9 5 5 5 5 5 5 5 5 5 5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

1 1 1 1 1 1 1 1 1 1 1 0.9 0.8 1 1 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.9 0.5 0.5 0.5 0.5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.1 1.2 1 1

10.1279 12.4865 16.2812 10.0541 9.9906 14.4758 23.1716 10.7007 11.2735 7.0476 6.1810 9.8063 9.0425 10.6974 11.4713 9.9141 9.7365 10.8952 11.4679

0.9596 0.7843 0.5758 0.8858 0.8223 0.7234 0.2509 1.5324 2.1053 0.9612 0.9620 0.6381 0.1258 0.9367 0.8105 0.9546 0.9452 0.6697 0.3041

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eiwt ypffiffiffiffiffiffiffiffiffiffiffi y 1ðbþiwÞ a e erfc pffiffiffiffi ðb þ iwÞt 2 2 at pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y 1aðbþiwÞ erfc pffiffiffiffi þ ðb þ iwÞt þe 2 at

1 1 ucos ðy; tÞ ¼ f1 ðy; tÞ þ f2 ðy; tÞ þ ða9 þ a10 Þf4 ðy; tÞ 2 2 a9 f3 ðy; tÞ þ a11 f5 ðy; tÞ þ a12 f6 ðy; tÞ

f2 ðy; tÞ ¼

a9 f8 ðy; tÞ þ a9 f7 ðy; tÞ a10 f10 ðy; tÞ a11 f11 ðy; tÞ a12 f12 ðy; tÞ

ð34Þ

Where

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eiwt ypffiffiffiffiffiffiffiffiffiffiffi y 1ðbiwÞ a f1 ðy; tÞ ¼ e erfc pffiffiffiffi ðb iwÞt 2 2 at pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y 1aðbiwÞ þe erfc pffiffiffiffi þ ðb iwÞt 2 at

g1 ðy; tÞ ¼ a7 f3 ðy; tÞ þ a8 f4 ðy; tÞ þ a9 f5 ðy; tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ea3 t ypffiffiffiffiffiffiffiffiffiffiffi y 1ðba Þ 3 a e erfc pffiffiffiffi ðb a3 Þt 2 2 at pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y 1aðba3 Þ þe erfc pffiffiffiffi þ ðb a3 Þt 2 at

ð36Þ ð37Þ

f3 ðy; tÞ ¼ ð35Þ

ð38Þ


591

Skin friction variation for (Pr = 25 and sinwt = 0).

Table 3 t

c

Sc

Gr

Gm

R

R0

M

k



0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.6

0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.6 0.6 0.6 0.7 0.8 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

2 2 2 2 2 3 5 2 2 2 2 2 2 2 2 2 2 2 2

5 5 5 5 5 5 5 7 9 5 5 5 5 5 5 5 5 5 5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

1 1 1 1 1 1 1 1 1 1 1 0.9 0.8 1 1 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.7 0.9 0.5 0.5 0.5 0.5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.1 1.2 1 1

19.0090 24.2296 32.4505 18.9352 18.8716 27.7974 45.3743 19.5818 20.1546 12.9509 11.2459 18.6874 17.9236 20.1065 21.6913 18.6104 18.2847 20.8744 22.4534

1.1659 0.9779 0.7616 1.0922 1.0286 1.0329 0.7667 1.7388 2.3116 1.1662 1.1663 0.8444 0.0805 1.1420 1.0147 1.1613 1.1522 0.8981 0.5518

Table 4

Nusselt number variation for air (Pr = 0.71).

t

R

Nusselt number for Ramped Temperature

Nusselt number for isothermal Temperature

0.4 0.4 0.4 0.5 0.6

0.5 0.7 0.8 0.5 0.5

0.2026 0.2721 0.3047 0.2758 0.3534

0.1803 0.0166 0.1092 0.0495 0.0518

f4 ðy; tÞ ¼

pffib pffiffiffiffi pffiffiffiffi 1 ypffiba y y e erfc pffiffiffiffi bt þ ey a erfc pffiffiffiffi þ bt 2 2 at 2 at ð39Þ

pffi pffiffiffiffi 1 y y b t pffiffiffiffiffi ey a erfc pffiffiffiffi bt f5 ðy; tÞ ¼ 2 2 at 2 ab pffi pffiffiffiffi y y y ba þ t þ pffiffiffiffiffi e erfc pffiffiffiffi þ bt 2 at 2 ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ea6 t ypffiffiffiffiffiffiffiffiffiffiffi y 1ðba Þ 6 a e erfc pffiffiffiffi ðb a6 Þt 2 2 at pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 1 þey aðba6 Þ erfc pffiffiffiffi þ ðb a6 Þt 2 at

ð40Þ

f6 ðy; tÞ ¼

Table 6

Nusselt number variation for (Pr = 25).

t

R



0.4 0.4 0.4 0.5 0.6

0.5 0.7 0.8 0.5 0.5

0.0379 0.0530 0.0606 0.0530 0.0696

4.3536 4.3111 4.2899 3.8702 3.5113

g2 ðy; tÞ ¼ a7 f7 ðy; tÞ þ a8 f8 ðy; tÞ þ a9 f9 ðy; tÞ " pffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ea3 t ypffiffiffiffiffiffiffiffiffiffiffi y Pr R RPra3 pffiffi a3 t e erfc f7 ðy; tÞ ¼ Pr 2 2 t pffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# pffiffiffiffiffiffiffiffiffiffiffi y Pr R y RPra3 pffiffi þ erfc þe a3 t Pr 2 t f8 ðy;tÞ ¼

ð43Þ

" pffiffiffiffiffiffi rffiffiffiffiffiffiffiffi! pffiffiffiffiffiffi rffiffiffiffiffiffiffiffi!# pffiffiffi 1 ypffiffiRffi y Pr R y Pr R pffiffi pffiffi þ erfc e t þ ey R erfc t 2 Pr Pr 2 t 2 t ð44Þ

ð41Þ

Table 7 Table 5

ð42Þ

Sherwood Number variation.

Nusselt number variation for water (Pr = 7).

t

R0

Sc

Sherwood Number

0.4 0.4 0.4 0.4 0.4 0.5 0.6

1 1.1 1.2 1 1 1 1

0.6 0.6 0.6 0.7 0.8 0.6 0.6

0.2532 0.2745 0.2952 0.2734 0.2923 0.3414 0.4334

t

R



0.4 0.4 0.4 0.5 0.6

0.5 0.7 0.8 0.5 0.5

0.0711 0.0991 0.1130 0.0991 0.1299

2.1601 2.0813 2.0422 1.8879 1.6834

592

H.R. Kataria, H.R. Patel "

pffiffiffiffiffiffi rffiffiffiffiffiffiffiffi! 1 yPr ypffiffiRffi y Pr R pffiffi f9 ðy; tÞ ¼ erfc t pffiffiffiffi e t 2 Pr 2 t 2 R pffiffiffiffiffiffi rffiffiffiffiffiffiffiffi!# pffiffiffi yPr y Pr R pffiffi þ t þ t þ pffiffiffiffi ey R erfc Pr 2 t 2 R f10 ðy;tÞ ¼

For ramped wall temperature

ð45Þ

i i ssin ðy; tÞ ¼ h1 ðtÞ h2 ðtÞ þ h3 ðtÞ h3 ðy; t 1ÞHðt 1Þ 2 2 þ a10 h5 ðtÞ þ a11 h6 ðtÞ þ a12 h7 ðtÞ h8 ðtÞ þ h8 ðt 1ÞHðt 1Þ a10 h12 ðtÞ a11 h13 ðtÞ

pffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffi 1 ypffiffiffiffiffiffi y Sc y Sc 0 0 pffiffi R0 t þ ey R Sc erfc pffiffi þ R0 t e R Sc erfc 2 2 t 2 t

ð46Þ

pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffi ffi 1 y Sc ypffiffiffiffiffiffi y Sc ScR0 pffiffi R0 t f11 ðy; tÞ ¼ erfc t pffiffiffiffiffi0 e 2 2 t 2 R pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffi y Sc pffiffiffiffiffiffiffi0 y Sc pffiffi þ R0 t ð47Þ þ t þ pffiffiffiffiffi0 ey ScR erfc 2 t 2 R pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ea6 t ypffiffiffiffiffiffiffiffiffiffiffiffiffiffi y Sc ScðR0 a6 Þ pffiffi ðR0 a6 Þt erfc f12 ðy; tÞ ¼ e 2 2 t pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y Sc y ScðR0 a6 Þ pffiffi þ ðR0 a6 Þt erfc ð48Þ þe 2 t

a12 h14 ðtÞ 1 1 scos ðy; tÞ ¼ h1 ðtÞ þ h2 ðtÞ þ h3 ðtÞ 2 2 h3 ðy; t 1ÞHðt 1Þ þ a10 h5 ðtÞ þ a11 h6 ðtÞ þ a12 h7 ðtÞ h8 ðtÞ þ h8 ðt 1ÞHðt 1Þ a10 h12 ðtÞ a11 h13 ðtÞ a12 h14 ðtÞ

i i ssin ðy; tÞ ¼ h1 ðtÞ h2 ðtÞ þ ða9 þ a10 Þh5 ðtÞ a9 h4 ðtÞ 2 2 þ a11 h6 ðtÞ þ a12 h7 ðtÞ a9 h10 ðtÞ þ a9 h9 ðtÞ a10 h12 ðtÞ a11 h13 ðtÞ a12 h14 ðtÞ

ð49Þ

Using the Eq. (27), we obtained the Nusselt number for Ramped wall temperature is Nu ¼ ½h11 ðtÞ h11 ðt 1ÞHðt 1Þ

ð50Þ

Using the Eq. (31), we obtained the Nusselt number for Ramped wall temperature is Nu ¼ ½h10 ðtÞ

ð51Þ

a10 h12 ðtÞ a11 h13 ðtÞ a12 h14 ðtÞ Where iwt

h1 ðtÞ ¼ e

rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b iw ebt erf ðb iwÞt þ pffiffiffiffiffiffiffi a pat

sh ¼ ½h13 ðtÞ

h3 ðtÞ ¼ a7 h4 ðtÞ þ a8 h5 ðtÞ þ a9 h6 ðtÞ

ð62Þ

h4 ðtÞ ¼ e

rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a3 ebt erf ðb a3 Þt þ pffiffiffiffiffiffiffi a pat

ð64Þ

rffiffiffi pffiffiffiffi pffiffiffiffi tebt 1 b erf bt þ pffiffiffiffiffiffiffi h6 ðtÞ ¼ pffiffiffiffiffiffiffiffi erf bt t a pat 4ab

ð65Þ

7. Skin friction

rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a6 ebt erf ðb a6 Þt þ pffiffiffiffiffiffiffi a pat

h8 ðtÞ ¼ a7 h9 ðtÞ þ a8 h10 ðtÞ þ a9 h11 ðtÞ Expressions of skin-friction for both cases are calculated from a3 t

h9 ðtÞ ¼ e

Where s¼

@u

@y y¼0

ð55Þ

ð63Þ

rffiffiffi pffiffiffiffi b ebt h5 ðtÞ ¼ erf bt þ pffiffiffiffiffiffiffi a pat

ð53Þ

ð54Þ

ð60Þ

ð61Þ

h7 ðtÞ ¼ ea6 t

Eqs. (29), (30), (33), and (34) using the relations 1 s ðy; tÞ ¼ lB 1 þ s c

ð59Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b þ iw ebt h2 ðtÞ ¼ e erf ðb þ iwÞt þ pffiffiffiffiffiffiffi a pat iwt

a3 t

Using the Eqs. (28) and (32), we obtained the Sherwood Number for Ramped wall temperature and isothermal temperature is

ð58Þ

1 1 scos ðy; tÞ ¼ h1 ðtÞ þ h2 ðtÞ þ ða9 þ a10 Þh5 ðtÞ a9 h4 ðtÞ 2 2 þ a11 h6 ðtÞ þ a12 h7 ðtÞ a9 h10 ðtÞ þ a9 h9 ðtÞ

6. Sherwood number Sherwood Number is defined and denoted by the formula @C sh ¼ ð52Þ @y y¼0

ð57Þ

For isothermal temperature

5. Nusselt number The Nusselt number Nu can be written as v @T Nu ¼ U0 ðT T1 Þ @y y¼0

ð56Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R Pra3 erf

pffiffiffiffi h10 ðtÞ ¼ R erf

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! rffiffiffiffiffiffi R Pr PrR t a3 t þ e Pr pt

rffiffiffiffiffiffiffiffi! rffiffiffiffiffiffi R Pr PrR t t þ e Pr pt

ð66Þ ð67Þ ð68Þ

ð69Þ

Radiation and chemical reaction effects on MHD Casson fluid flow rffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffi! pffiffiffiffi Pr R R h11 ðtÞ ¼ pffiffiffiffi erf t Rt erf t Pr Pr 2 R rffiffiffiffiffiffiffi tPr PrR t e þ p pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi h12 ðtÞ ¼ R0 Sc erf R0 t þ

rffiffiffiffiffi Sc R0 t e pt

rffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Sc R0 t t ScR0 erf R0 t 0 erf 4R rffiffiffiffiffiffiffi tSc R0 t þ e p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h14 ðtÞ ¼ ea6 t ScðR0 a6 Þ erf ðR0 a6 Þt

ð70Þ ð71Þ

h13 ðtÞ ¼

rffiffiffiffiffi Sc R0 t e þ pt

ð72Þ

ð73Þ

Where 1 1 a ¼ 1 þ ; b ¼ M2 þ ; a1 ¼ aR b; a2 ¼ aPr 1; c k a1 a3 ¼ ; a4 ¼ a Sc R0 b a2 a4 Gr ; a7 ¼ ; a5 a2 a23 Gr a4 Gr a6 ; a9 ¼ a8 ¼ a2 ð1 þ a3 Þ 1 þ a3 a2 a3 a5 ¼ aSc 1; a6 ¼

a10 ¼

Gm a9 Gm Gm a8 ; a11 ¼ ; a12 ¼ a5 ð1 þ a6 Þ a5 a26 a5 a6 1 þ a6

8. Results and discussion In order to get a clear insight into the physics of the problem, a parametric study is performed and the obtained numerical results are elucidated with the help of graphical illustrations. We have presented the non-dimensional fluid velocity, fluid temperature and concentration for several values of Prandtl number Pr, Grashof number Gr, Grashof number for mass transfer Gm, Casson parameter c, magnetic parameter M, radiation parameter R, chemical reaction parameter R0 , permeability of porous medium K and time t in Figs. 2–18. It is observed that magnitude of velocity and heat in case of ramped temperature are less than the isothermal temperature. Fig. 2 exhibits the velocity profiles for different values of Prandtl number Pr, when the other parameters are fixed. It is observed that velocity of the fluid decreases with increasing Prandtl number. The influence of Casson fluid parameter on velocity profiles is shown in Fig. 3. It is found that velocity increases with increasing value of c. It is important to note that an increase in Casson parameter makes the velocity boundary layer thickness shorter. It is further observed from this graph that when the Casson parameter c is large enough, the nonNewtonian behaviors disappear and the fluid purely behaves like a Newtonian fluid. Thus, the velocity boundary layer thickness for Casson fluid is larger than the Newtonian fluid. It occurs because of plasticity of Casson fluid. When Casson parameter decreases the plasticity of the fluid increases, which

593

causes the increment in velocity boundary layer thickness. In Fig. 4, the profiles of velocity have been plotted for various values of permeability parameter K by keeping other parameters fixed. It is observed that for large values of K, velocity and boundary layer thickness increase which explains the physical situation that as K increases, the resistance of the porous medium is lowered which increases the momentum development of the flow regime and ultimately enhances the velocity field. Fig. 5 displays the effect of magnetic parameter M on the velocity profiles. It is observed that the amplitude of the velocity as well as the boundary layer thickness decreases when M is increased. Physically, it may also be expected due to the fact that the application of a transverse magnetic field results in a resistive type force (called Lorentz force) similar to the drag force, and upon increasing the values of M, the drag force increases which leads to the deceleration of the flow. Figs. 6 and 7 show that velocity decreases with radiation parameter R and chemical reaction parameter R0 respectively. The graphical results for Sc are shown in Fig. 8. It is observed that the fluid velocity decreases with increase in Sc. It is depicted from Fig. 9 that, velocity increases with increasing values of Gm. Fig. 10 illustrates the profiles of velocity for different values of Gr. It is observed that velocity increases with increasing values of Gr. The flow is accelerated due to the enhancement in the buoyancy forces corresponding to the increasing values of Grashof number, i.e., free convection effects. In Fig. 11, the influence of dimensionless time t on the velocity profiles is shown. It is found that the velocity is an increasing function of time t for phase angle x ¼ 0 whereas it decreases with increase in t for x ¼ p2. The graphical results for phase angle x are shown in Fig. 12. It is observed that the fluid velocity decreases with increase in x for u ¼ cos ðwtÞ and otherwise for u ¼ sin ðwtÞ. It is depicted from Fig. 13 that, the temperature decreases as the Prandtl number Pr increases. It is justified due to the fact that thermal conductivity of the fluid decreases with increasing Prandtl number Pr and hence decreases the thermal boundary layer thickness. It is depicted from Fig. 14 that temperature decreases with increasing values of radiation parameter R. Fig. 15 is plotted to show the effects of the dimensionless time t on the temperature profiles. Obviously the temperature increases with increasing time t. This graphical behavior of temperature is in good agreement with the corresponding boundary conditions of temperature profiles as shown in Eq. (8). It is depicted from Fig. 16 that, concentration decreases with increasing values of chemical reaction parameter R0 . It is depicted from Fig. 17 that, concentration decreases with increasing values of Sc. Fig. 18 exhibits the concentration profiles for different values of t. It is observed that concentration increases with t. Fig. 19 exhibits the skin friction for different values of Pr. It is observed that skin friction increases with increase in Pr. Skin friction is more for ramped wall temperature compared to isothermal temperature. Fig. 20 shows the effect of Pr on Nusselt number. It is seen that Nusselt number decreases tendency with Pr. The variation of the skin friction, Nusselt Number, Sherwood Number for air (Pr = 0.71), water (Pr = 7) and (Pr = 25) is shown in Tables 1–7 for various values of the governing parameters. Skin friction increases in a Ramped wall temperature while it decreases in isothermal temperature with increase in c, Gr and M. Skin friction decreases in case of Ramped wall temper-

594 ature while it increases in isothermal temperature with increase in R. For both thermal cases, Skin friction decreases with increase in Sc and k, increases with increase in Gm and R0 , and Nusselt number increases with increase in R and t. Sherwood number increases with increase in R0 , Sc and t. Effect of all parameters c, Gr, M, R, Sc, k, Gm and R0 on skin friction and Nusselt number is similar in air and water. For air, Skin friction decreases with increase in t. For water, Skin friction increases in a Ramped wall temperature while it decreases in isothermal temperature with increase in t. From Tables 1–6, we observed that Magnitude of Skin friction and Nusselt Number increases with increase in Pr. It is seen that magnitude of Skin friction and Nusselt Number is more for Casson fluid compared with water and air. 9. Conclusion The purpose of this study was to obtain exact solutions for the unsteady natural convective Casson fluid flow past over an oscillating vertical plate in the presence of a transverse uniform magnetic field. The expressions for the velocity, the temperature and concentration have been obtained in closed form with the help of the Laplace transform technique. The effects of the pertinent parameters on velocity, concentration and temperature profiles are presented graphically. The most important concluding remarks can be summarized as follows: The fluid velocity decreases with increase in Prandtl number Pr, magnetic parameter M, radiation parameter R, chemical reaction parameter R0 , permeability of porous medium K, Schmidt number Sc and time t. Velocity increases with increasing values of Grashof number Gr, Grashof number for mass transfer Gm, Casson parameter c, and time t. Temperature decreases as the Prandtl number Pr and radiation parameter R increase. Temperature increases with increasing time t. Concentration increases with increase in t. Concentration decreases with increase in Schmidt number Sc and chemical reaction parameter R0 . Skin friction increases while Nusselt number decreases with increase in Pr. Sherwood number increases with increase in R0 , Sc and t.

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