Unsteady Boundary Layer Flow of Viscous Fluid Along a Vertical Surface with Viscous Dissipation and Thermal Radiation

Unsteady Boundary Layer Flow of Viscous Fluid Along a Vertical Surface with Viscous Dissipation and Thermal Radiation

M. C. Kemparaju

Department of mathematics, Gulbarga University, Kalaburagi -585105 Karnataka, INDIA.

Department of Mathematics, Jyothy Institute of Technology, Bangalore- 560082, Karnataka, INDIA.

M. Subhas Abel

Department of mathematics, Gulbarga University, Kalaburagi -585105 Karnataka, INDIA.

Mahantesh M. Nandeppanavar Department of UG and PG Studies and Research in Mathematics, Government College Kalaburagi- 585105, Karnataka, INDIA

Abstract – The present work is devoted to investigate the heat transfer analysis of a two dimensional unsteady boundary layer flow of viscous fluid at a vertical surface in the presence of viscous dissipation and thermal radiation. The modeled equations are converted into non-linear ordinary differential equations by using similarity transformations. The coupled differential equations are solved numerically by using Runge- Kutta method with shooting technique. The effects of various parameters on velocity and temperature profiles are discussed.

Keywords – Unsteady boundary layer, viscous dissipation, thermal radiation, convective boundary conditions.

Nomenclature

a,b,c Empirical Constants

A Unsteady parameter

Skin friction

Specific heat at constant pressure

Dimensionless stream function

Surface mass transfer parameter

() Thermal conductivity

Thermal conductivity at the sheet

Thermal conductivity far away from the sheet

Kummers function

Nusselt number

Thermal radiation parameter

Prandtl number

Eckert number

Local heat flux at the sheet

Radiative heat flux

Fluid temperature

Kinematic viscosity

Thermal expansion coefficient

µ Dynamic Viscosity

Stream function

Density

* Stephan-Boltzman constant k* Mean absorption coefficient

xy Shear stress

Dimensionless temperature variable

Free convection or buoyancy parameter

1. INTRODUCTION

   The study of two dimensional boundary layer flow and heat transfer of incompressible viscous fluid over a continuous stretching and heated surfaces has acquired momentum due to its various applications such as extrusion of polymer in industry ,wire and fiber coating, design of heat exchangers and chemical process. The analysis of momentum and thermal transports within the fluid on a continuously stretching surface is important for gaining of some fundamental understanding of such processes. The first among to analyze the problem on boundary layer flow over acontinuous solid surface moving with constant velocity by Sakiadis[1]. Mahmoud and Waheed [2] have studied MHD flow and heat transfer of a micro-polar fluid over a stretching surface with heat generation (absorption) and slip. Aziz [3] has studied flow and heat transfer over an unsteady stretching surface with hall effect. Qin et al. [4] have analyzed the

   Given temperature at the sheet

   cauchy problem for a 1D compressible viscous micro-polar

   fluid model: analysis of the stabilization and the

   Constant temperature of the fluid far away from the sheet

   Velocity in x direction

   Velocity of the stretching surface

   Suction/ blowing velocity

   Greek Symbols

   Thermal diffusivity

   T Sheet temperature

   Thermal conductivity parameter

   regularity.Khan and Pop [5] have discussed boundary layer flow of a nano-fluid past a stretching sheet.Chamkha et al. [6] discussed similarity solution for unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction / injection and chemical reaction effects.

   Mukhopadhyay [7] was studied the heat transfer analysis of unsteady flow over a porous stretching surface embedded in a porous medium in presence of thermal radiation

   .Bhattacharyya et al. [8] have analyzed slip effects an

   unsteady boundary layer stagnation point flow and heat

   T

   T T

   T q u 2

   transfer towards a stretching sheet. Mukhopadhyay [9] has

   Cp

   u v K (T )

   r

   (3)

   analyzed the effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching

   t x y y y y y

   Subjected to the boundary conditions

   surface.Vajravelu et al. [10] have studied unsteady

   u Uw , v vw , T Tw at y 0

   (4)

   convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties. Hamad et al. [11] have

   u 0,

   T T ,

   as y

   studied magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate.

   Sarkar [12] has discussed mixed convective heat transfer of nanofluids past a circular cylinder in cross flow in unsteady regime. Kousar and Liao [13] have discussed unsteady non-similarity boundary-layer flows caused by an impulsively stretching flat sheet. Rohni et al. [14] have analyzed unsteady mixed convection boundary-layer flow with suction and temperature slip effects near the stagnation point on a vertical permeable surface embedded in a porous medium. Chen [15] has studied Mixed convection unsteady

   Where u and v are the velocity components in the x and y directions, respectively. ( / ) is the kinematic

   viscosity. g is the acceleration due to gravity. is the coefficient of thermal expansion, T is the fluid temperature, T is the ambient temperature, is the density, Cp is the specific heat at constant pressure. K (T) is the variable thermal conductivity. vw (t) v0 1 ct is the suction/injection velocity and qr is the radiative heat flux. The thermal conductivity is assumed to vary linearly with temperature

   stagnation-point flow towards a stretching sheet with

   slip.Hunsain et al. [16] have made study on heat and mass

   K (T ) K 1 (T

   (T T )

   T )

   (5)

   transfer analysis in unsteady boundary layer flow through porous media with variable viscosity and thermal. Raju and Varma [17] made a study on unsteady MHD free convection

   w

   The radiative heat flux con be expressed as

   oscillatory couette flow through a porous medium with periodic wall temperature.Hossain et al. [18] made a note on solution of natural convection boundary layer flow above a

   4 *

   qr

   3k *

   T 4

   y

   (6)

   1

   1

   semi-infinite porous horizontal plate under similarity

   We introduce new dimensionless variables:

   1

   1

   transformations with suction and blowing. Hossain et al. [19]

   a

   2

   a

   2

   have studied similarity solution of unsteady combined free

   2

   y,

   va

   xf (),

   v(1 ct ) (1 ct)

   and force convective laminar boundary layer flow about a vertical porous surface with suction and blowing. Rahman et

   ()

   (T T ) ,

   (7)

   al. [20] have analyzed thermophoresis particle deposition on unsteady two-dimensional forced convective heat and mass transfer flow along a wedge with variable viscosity and variable Prandtl number. Ishak et al. [21] have made study on boundary layer flow and heat transfer over an unsteady stretching vertical surface.

2. MATHEMATICAL FORMULATION

   (Tw T )

   Using Eqs (7) ,(6) and () in to (1), (2) and(3) reduces to

   2

   2

   f ''' ff '' f '2 A f ' 1 f '' 0 (8)

   1 Nr '' '2 Pr f ' Pr f ' Pr Ecf ''2 Pr A 1 ' 2 0

   2

   Consider the unsteady laminar two dimensional boundary layer flow of incompressible viscous fluid flow on a

   stretching permeable surface. The sheet is stretched with a

   (9)

   ax

   velocityUw (1 ct)

   , where a and c are constants, in

   Where Pr is the Prandtl number,

   1

   1

   the positive x direction. Here a 0, b 0 and t . The

   c

   k ,

   Cp

   fw

   v0 ,

   a

   fw is the surface mass

   16 *T 3

   sheet surface temperature

   T (x, t) T bx varies

   w (1 ct)2

   transfer parameter,

   Nr

   3K k *

   is the thermal

   with the distance x .

   radiation parameter,

   g b

   a2

   free convection parameter,

   The boundary layer governing equations are: C U 2

   u v 0

   (1)

   A unsteady parameter, and Ec w is the

   x y

   a (Tw T )Cp

   u u u v u u

   u u u v u u

   2

   g T T

   (2)

   local Eckert number. The transformed boundary conditions are

   t x y y2

   f ' () 1, f () f w , () 1, at 0

   f ' () 0

   () 0

   as

   (10)

   Figs.8-10 shows effect of thermal conductivity

   The quantities Skin-Friction Coefficient (rate of shear stress) and Nusselt number are (rate of heat transfer).The local skin friction coefficient and Nusselt number is defined as

   x

   x

   C w , Nu xqw

   parameter on temperature profiles of the flow. It is evident from the figures that an increase in thermal conductivity parameter enhances the temperature profiles. Figs. 11-13 depict the influence of thermal radiation parameter on temperature profiles. It is clear from the figures that an

   w

   w

   f U 2 2

   K (Tw

   T )

   (11)

   increase in the thermal radiation parameter enhances the temperature profiles. The influence of viscous dissipation

   Where the skin friction and heat transfer from the sheet are

   given by

   parameter (Ec) on temperature profiles is shown in figs 14-

   16. It observed from the figure that increasing in the Eckert

   w y

   w y

   u

   y 0

   q K T

   w y

   w y

   y 0

   number enhances the temperature profiles of the flow. This is due to the fact that an increase in the dissipation causes to improve the thermal conductivity of the flow. This helps to enhance the thermal boundary layer thickness.

   Figs. 17 and 18 depict the influence of the skin friction

   TABLE 1: COMPARISON OF VALUES OF '(0) FOR VARIOUS

   VALUES OF A, and Pr

   A		Pr	Present	Vajravelu et al[10]	Ishak et al[21]
   0	0	0.01	0.019723	0.019723	0.0197
   0	0	0.72	0.8088341	0.808836	0.8086
   0	0	1	1.000008	1000000	100000
   0	0	3	1.923679	1.923687	1.9237
   0	0	10	3.720671	3.720788	3.7207
   0	0	100	12.294081	12.30039	12.2941
   1	0	1	1.681993	1.681921	1.6820
   1	1	1	1.703913	1.703910	1.7039
   0	1	1	1.087275	1.087206	1.0873
   0	2	1	1.142336	1.142298	1.1423
   0	3	1	1.185289	1.185197	1.1853

3. RESULTS AND DISCUSSION

   The nonlinear ordinary differential Eqs. (8) – (9), subject to the boundary conditions Eqs.(10) are solved numerically using shooting technique. The obtained results show the effects of the various non-dimensional governing parameters on the velocity and temperature fields.

   parameter on velocity and temperature profiles. It is evident from the figures that an increase in skin friction parameter enhances the velocity and temperature profiles.

   Fig.1 Velocity profile for different values of with

   Pr 1, 0.1, Nr 0.1, Ec 0.1

   Comparison of values of

   '(0) for various values of A,

   and Pr in respective figures and tables.

   Figs.1-5 displays the effects of thermal buoyancy parameter on velocity and temperature profiles. It is evident from the figures that increase in buoyancy parameter enhances the velocity profiles but decreases the temperature profiles.

   Figs. 6 and 7 shows the effect of Prandtl number on temperature profiles. From these figures, the temperature profiles are decreasing with increasing value of Prandtl number. Because, an increase in Prandtl number results a decrease of the thermal boundary layer thickness and in general lower average temperature within the boundary layer. The reason is that smaller values of Pr are equivalent to increase in thermal conductivity of the fluid and therefore, heat is able to diffuse away from the heated surface more rapidly for higher values Pr. Hence in the case of smaller Prandtl number as the thermal boundary layer is thicker and the rate of heat transfer is reduced.

   Fig.2 Velocity profile for different values of with

   Pr 1, 0.1, Nr 0.1, Ec 0.1

   Fig.3 Temperature profile for different values of with

   Pr 1, 0.1, Nr 0.1, Ec 0.1

   Fig.4 Temperature profile for different values of with

   Pr 1, 0.1, Nr 0.1, Ec 0.1

   Fig.5 Temperature profile for different values of with

   Pr 1, 0.1, Nr 0.1, Ec 0.1

   Fig.6 Temperature profile for different values of Pr with

   1, 0.1, Nr 0.1, Ec 0.1

   Fig.7 Temperature profile for different values of Pr with

   1, 0.1, Nr 0.1, Ec 0.1

   Fig.8 Temperature profile for different values of with

   1, Pr 1, Nr 0.1, Ec 0.1

   Fig.9 Temperature profile for different values of with

   1, Pr 1, Nr 0.1, Ec 0.1

   Fig.10 Temperature profile for different values of with

   1, Pr 1, Nr 0.1, Ec 0.1

   Fig.11 Temperature profile for different values of Nr with

   1, Pr 1, 0.1, Ec 0.1

   Fig.12 Temperature profile for different values of Nr with

   1, Pr 1, 0.1, Ec 0.1

   Fig.13 Temperature profile for different values of Nr with

   1, Pr 1, 0.1, Ec 0.1

   Fig.14 Temperature profile for different values of Ec with

   1, Pr 1, 0.1, Nr 0.1

   Fig.15 Temperature profile for different values of Ec with

   -0.80

   -0.85

   -0.90

   -0.95

   -1.00

   -1.05

   -1.10

   -1.15

   '(0)

   '(0)

   -1.20

   -1.25

   -1.30

   -1.35

   -1.40

   -1.45

   -1.50

   -1.55

   -1.60

   -1.65

   -1.70

   A=0,0.5,1

   A=0,0.5,1

   0 1 2 3 4 5

   1, Pr 1 0.1, Nr 0.1

   Fig.16 Temperature profile for different values of Ec with

   1, Pr 1, 0.1, Nr 0.1

   A=0,0.5,1.0

   A=0,0.5,1.0

   1.0

   0.5

   f''(0)

   f''(0)

   0.0

   -0.5

   Fig.18 Variation of heat flux '(0) vs for different values of A and

   fw

4. CONCLUSIONS

This study presented the heat transfer analysis of a two dimensional unsteady convective boundary layer flow of a viscous fluid at a vertical stretching in the presence of viscous dissipation and thermal radiation. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary and coupled differential equations and are solved numerically by using Runge-Kutta method with shooting technique. The effects of various parameters on velocity and temperature fields are discussed and presented through graphs. Also, the Skin friction, local Nusselt and Prandtl numbers are analyzed and presented through tables. The conclusions of the present study are made as follows:

• An increase in the thermal radiation parameter decreases the heat transfer rate.

• An increase in the buoyancy parameter enhances the velocity profile and decreases temperature profile.

• An increase in Prandtl number decreases the heat transfer rate.

• An increase in skin friction parameter enhances the velocity.

-1.0

-1.5

0 1 2 3 4 5

Fig.17 Variation of Skin friction f ''(0) vs for different values of A and fw

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