View of EFFECT ON UNSTEADY MHD BOUNDARY LAYER FLOW PAST OVER A SHRINKING SHEET WITH HEAT TRANSFER AND MASS SUCTION

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, ISSN No. 2456-1037

Vol.02, Issue 03, March 2017, Available Online: www.ajeee.co.in/index.php/AJEEE

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EFFECT ON UNSTEADY MHD BOUNDARY LAYER FLOW PAST OVER A SHRINKING SHEET WITH HEAT TRANSFER AND MASS SUCTION

Bhagwan Singh

Govt. Mahila College Budaun U P Sooraj Pal Singh

Department of Mathematics Govt. P G College Bazpur Uttrakhand

Abstract - The purpose to the present problem is to study the unsteady effect on MHD boundary layer flow past over a shrinking sheet with mass transfer and mass suction The generating equation for boundary layer flow and energy are reduced into linear differential equation by means similar transformation. Solution of resulting equation are solved and obtain various parameter are present and discussed. The velocity and temperature profile, concentration and skin friction profile are shown graphically with the help of tables.

Keywords: MHD boundary layer flow, Magnetic parameter, Prandtl number, Specific heat of fluid, Suction parameter, Magnetic field parameter, etc.

1 INTRODUCTION

The problem of MHD boundary layer flow has attracted in Engineering science as Aerodynamics, Astrophysics, geophysics and Chemical engineering MHD fluid flow and heat sources in channels with constant wall temperature was discussed by Ostrach⁽¹⁾. In recent year MHD flow problem have become more important in industrial field. MHD boundary layer behavior over a shrinking surface a significant type of flow having considerable. Practical application used in chemical engineering. Boundary layer flow in porous medium owing to combined heat and mass transfer. Lai and Kulacki⁽²⁾. The study of two dimensional flow through a porous medium bounded by vertical infinite surface With constant heat flux in presence of free convective current was studied. Sharma and Kumar⁽³⁾ Have studied unsteady effect on MHD free Convection and mass transfer through porous medium With constant suction. The boundary layer flow of an incompressible viscous fluid over a shrinking sheet has received considerable attention modern day researchers because of increasing applications to many engineering system Wong⁽⁴⁾ first pointed out the flow over a shrinking sheet. Then Miklavcic⁽⁵⁾ obtained an analytic, solution for unsteady viscus hydrodynamics flow over a permeable shrinking sheet then Hayat⁽⁶⁾ derived both exect and series solution describing the MHD boundary layer flow of second grade fluid over a shrinking sheet

―Nadeen and Awis⁽⁷⁾ studied thin film flow of an unsteady shrinking sheet through porous medium with variable value

velocity. Viscus flow over an unsteady shrinking sheet with mass transfer was studied by ―Fang and Zhang⁽⁸⁾ investigated the heat transfer characteristic of the shrinking sheet problem with a linear velocity.

―Midya⁽⁹⁾ studied MHD Viscus flow and heat transfer over a linearly shrinking porous sheet. Effect of chemical reaction heat and mass transfer of non-linear boundary layer past a porous shrinking sheet in presence of suction was discussed numerically by ―Muhaimin⁽¹⁰⁾. MHD boundary layer past a porous shrinking sheet with suction. Rajesh [12]

investigates chemical reaction and radiation effects on the transient MHD free convection flow of dissipative fluid past an infinite vertical porous plate with ramped wall temperature. Paper, we consider the problem of a laminar electrically conducting fluid as a boundary layer flow past a stretching plate and heat Transfer. Kumar Anuj, Manoj [13] investigates MHD boundary layer flow past a stretching plate with heat transfer. In the present we consider the Effect of unsteady MHD boundary layer flow past a stretching plate with heat transfer.

MHD boundary layer flow past over a shrinking sheet with heat transfer and mass suction was studied by

―Jhankal and Kumar⁽¹¹⁾. In this section we have considered the problem effect on unsteady MHD boundary layer flow past over a shrinking sheet with heat transfer and mass suction.

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2 1.1 Used Symbols

𝜇 = Viscosity of the fluid 𝜌 = Density of the fluid Qe= Electrical conductivity

= Dimensionless Variable K = Thermal conductivity

𝜗 = Kinematic Viscosity of the fluid 𝜃 = Temperature of the fluid

B0 = Constant applied magnetic field c = Shrinking Constant

M = Magnetic failed parameter

Q =Volumetric rate of the heat generation S = Suction parameter

Pr = Prandtl number

Tw = Temperature of the wall 𝑇_∞= Free stream temperature

𝑢, 𝑣 = Velocity component of the fluid along X and Y directions.

2 MATHEMATICAL FORMULATION Let we suppose that two dimensional unsteady MHD boundary layer flow over a shrinking sheet. The X axis is taken along the sheet in the upward direction and Y axis is taken in normal to it. A transverse constant magnetic field applied in direction of Y axis. The governing equation of continuity, equation of motion and energy equation for MHD flow. The Reynolds number taken small and taken v^* = - v0

𝜕𝑉^∗

𝜕Y∗ = 0

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𝜕u∗

𝜕t∗ + ^𝜕u∗

𝜕y∗ =𝜗 ^∂_∂y∗²^u∗₂ —^ℚB⁰²

𝜌 u^* (2)

𝜕T∗

𝜕t∗ + v^* ^𝜕T∗

𝜕y∗ = ^𝐾

𝜌_𝐶𝑝

∂²T∗

∂y∗² + ^ℚ

𝜌_𝐶𝑝 (T — 𝑇∞ ) (3) Where 𝜌 be the density, 𝜗 be the kinematic viscosity and B0 is the magnetic induction K be the thermal conductivity then boundary condition of this problem are

u^* = 𝑢_𝑤 , T^*= T𝑤 at y^*= 0 , T^* = 0 (4) u^* → 0, T^* = 𝑇∞ at y^* = ∞, T^* → 0 On introducing following non dimensional quantities.

y^* = ^𝜗

V₀𝑦 , u^* = u v0 , T^* = t ^V_𝜗⁰² , T^* — 𝑇_∞ = 𝜃(T𝑤 — 𝑇_∞ ) (5) Solving equation (2) & (3) with help of (5) We have

∂²u

∂y² + ^∂u_∂y — ^∂u_∂t —Mu = 0 (6)

∂²θ

∂y² + Pr∂θ

∂y —Pr∂θ

∂t = 0 (7) And corresponding boundary conditions are.

u = ℚ, 𝜃 =1 at y = 0, t = 0 (8) u → 0, 𝜃 = 0 at y → ∞, t = 0

Where M = ^{ℚ𝜗 𝐵}_𝜌𝑉 ⁰²

02 , Pr = ^𝜌𝐶_𝐾^𝑝 Now we assume the solution of these equations U(y, t) = u0 (y) 𝑒^−𝑛𝑡 and 𝜃 (y,t)= 𝜃0( y) 𝑒^−𝑛𝑡

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By solving equation (6),(7) with the help of (9)

u0′′ + u0’ —u0 ( M—n ) = 0

(10) 𝜃₀^′′ + Pr 𝜃₀^′ +Pr 𝑛𝜃0 = 0 (11) And corresponding boundary conditions are

U0 = ℚ, 𝜃0 = 1 at y = 0, (12) U0 → 0, 𝜃0 = 0 at y → ∞,

Now solving equation (10) &(11) then U0 = 𝑒Â¹^𝑦 and u = 𝑒Â¹^𝑦𝑒^−𝑛𝑡 (13) 𝜃0 = 𝑒Â²^𝑦 and 𝜃 = 𝑒Â²^𝑦𝑒^−𝑛𝑡 (14) Where

A1 =−1± 1+4( 𝑀−𝑛)

2 = [1± 1+4( 𝑀−𝑛)

2 ] (15) A2 = ^{−Pr± Pr}₂²^{−4( 𝑛Pr)} = [^{Pr± Pr}²₂^{−4( 𝑛Pr)} ]

(16) Skin friction coefficient at the plate is given by

𝜏 = [ ^𝜕𝑢_𝜕𝑦]at y= 0 = A1𝑒^−𝑛𝑡 (17) Table -1 according to equation (13)

Y A1 M n t u

0 1.96628 2 0.1 2 0.818730 1 1.96628 2 0.1 2 5.849054 2 1.96628 2 0.1 2 41.78594 3 1.96628 2 0.1 2 298.83153 4 1.96628 2 0.1 2 2132.60603

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3 Fig. 1 Now we take different values of t

then according to (13) Table -2

y A1 M n t u

0 1.96628 2 0.1 2 0.818730 1 1.96628 2 0.1 4 4.788417 2 1.96628 2 0.1 6 28.00827 3 1.96628 2 0.1 8 163.8251 4 1.96628 2 0.1 10 958.26082

Fig.-2

Again we take different value of t then table-3 according to (13)

y A1 M n t u

0 1.96628 2 0.1 5 1.64872 1 1.96628 2 0.1 10 2.62814 2 1.96628 2 0.1 15 11.38731 3 1.96628 2 0.1 20 49.34517 4 1.96628 2 0.1 25 213.81261

Fig.-3

Table 4 According to equation (14)

y A2 Pr n t 𝜃

0 1.89442 2 0.1 2 0.818730 1 1.89442 2 0.1 2 5.44348 2 1.89442 2 0.1 2 36.19062 3 1.89442 2 0.1 2 240.61544 4 1.89442 2 0.1 2 1599.74577

Fig. 4

Again differen value of t and Pr table-5

y A2 Pr n t 𝜃

0 1.18920 1 0.1 5 1.64872 1 1.18920 1 0.1 10 1.20828 2 1.18920 1 0.1 15 2.40704 3 1.18920 1 0.1 20 4.79512 4 1.18920 1 0.1 25 9.55247

Fig. 5

Graph between y and 𝜏 for Skin friction coefficient according equation (17)

Table -6

Y A1 M n t 𝝉

0 1.96628 2 0.1 2 1.60985 1 1.96628 2 0.1 2 11.49995 2 1.96628 2 0.1 2 82.15793 3 1.96628 2 0.1 2 586.95222 4 1.96628 2 0.1 2 4193.30058

Fig. 6 3 RESULT AND DISCUSSION

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4 The effect of temperature and the effect of velocity both are depend on the value of y and t. If the value of y increase then decreasing the u by fig. 1. Parameter M and n change the velocity of unsteady boundary layer also change, the dimensionless temperature profile as well as the thermal unsteady boundary layer thickness quickly reduces as increasing Pr. Similarly thermal boundary layer thickness decreases for some higher values of heat source parameter heat absorption occurs at the sheet. The rate of heat transfer increases with Prandtl number.

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