3D radiative Oldroyd-B fluid flow towards Convectively Heated Surface with Irregular Heat Source and Activation Energy

3D radiative Oldroyd-B fluid flow towards Convectively Heated Surface with Irregular Heat Source and Activation Energy

Y. Veeranna

Department of Studies and Research in Mathematics, Government Science College (Autonomous), Bengaluru-560 001, Karnataka, India.

Abstract: Three dimensional MHD (magnetohydrodynamic)flow of radiative Oldroyd-B fluid over a convectively heated stretched sheet is investigated. The influence of irregular heat source, suction and activation energy are also studied. The Appropriate transformations are used reduce a system of nonlinear partial differential equations into a system of ordinary differential equations. The numerical solution are reported by RKF method. The diverse liquid flow characteristics are examined for various involved parameters through graphs and tables. It is exposed that, the Deborah numbers have quite opposite behaviour on flow field. The liquid temperature is increase when increasing in irregular heat sources. The effect of activation energy is favourable to concertation profile.

Keywords: Oldroyd-B fluid; irregular heat sources; activation energy; thermal radiation,

Introduction

Fluid streams are existing universally in nature and vital to plants, animals and people living on earth. In current years, the study of non-Newtonian fluids has gained a great attention of engineers and scientists due to its significant applications in industries and technology. These applications can be found mainly in chemical and nuclear industries, food and pharmaceutical industries, material processing, geophysics and biomedical engineering. As a consequence of different physical structures of non-Newtonian fluids, there is not a single constitutive model which can predict all its salient features. Generally, there are three non-Newtonian fluid models, namely empirical models, differential models and integral models. In present work, we have studied the second model and consider its subclass known as Oldroyd-B fluid. The major properties of the Oldroyd-B model was proposed to determine the relaxation time and retardation time, which cannot be explored by the Maxwell fluid model. On the other hand, it has a measurable retardation time and can relate the viscoelastic manner of dilute polymeric solutions under general flow conditions. The idea of the boundary layer flow of the Oldroyd-

B liquid induced by the linear stretching of a surface was first initiated by Sajid et al. [1]. They presented that the numerical solution of boundary layer flow of an Oldroyd-B fluid in the region of stagnation point past a stretching sheet using finite difference method. Many investigators have extended the work of Sajid et al. [1]. Shehzad et al. [2] deliberated the three-dimensional flow of the Oldroyd-B fluid with the effects of temperature-dependent thermal conductivity, and considered that the fluid flow generating due to the bidirectional stretching of the surface. The effects of the temperature stratification in a steady-state Oldroyd-B liquid flow at the stagnation point with mixed convection was investigated by Hayat et al. [3]. They reported the analytic solutions for velocity and temperature using homotopic analysis method. Motsa et al. [4] addressed the temperature dependent three- dimensional flow of Oldroyd-B fluid with heat source/sink effects by spectral relaxation technique. Hayat et al. [5] have discussed convergence of series solutions for the steady flow of an Oldroyd-B fluid induced by an exponentially stretched surface using homotopy analysis technique. Mahanthesh et al. [6] addressed the effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow and the heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. Some recent investigations on Oldroyd-B fluid flows can be found in Refs. [7]-[10].

The influence of thermal radiation significantly affects the rate of heat transfer and the temperature distributions in the boundary layer flow of participating fluid. So, the radiation effect have many industrial and engineering applications, such as space technology, nuclear power plants, in aircraft and also in gas turbines and controlling heat transfer processes in engineering fields. There are many research have made to predict the effect of thermal radiation with different aspects (see Makinde [11] – Hayat et al. [17]). However, all these works deals with influence of linear thermal radiation, but in recent time; the idea of nonlinear radiation has attracted many researchers. For the reason that, linear radiation is valid only for small temperature differences within the fluid, it is not sufficient in scientific and engineering applications in which the working fluids having the small temperature differences. In fact, the researchers have interested to consider nonlinear thermal radiation influence by assuming large temperature difference. In view of this, Pantokratoras and Fang

[18] were first to investigated the effect of nonlinear radiation on the Sakiadis flow. They have analyzed great temperature differences within the fluid. They found that, nonlinear radiation effect significantly affects the formation thermal boundary layer when comparison with linear radiation. Mushtaq et al. [19] reported the numerical solutions for nonlinear

radiative flow and heat transfer of nanofluid due to solar energy. Cortell [20] investigated the fluid flow and heat transfer over a stretching sheet with considered the Rosseland approximations to non-linearize the thermal radiation term in energy equation.Shehzad et al.

[21] have scrutinised the nonlinear thermal radiation effect on non-Newtonian Jeffery fluid flow with suspended nanoparticles. Hayat et al [22] presented the analytical solutions for magnetohydrodynamic flow and radiative heat transfer of nanofluid under velocity slip boundary by HAM method. Sampath et al. [23] presented the numerical solutions for nonlinear radiative rotating flow of Ferrofluid with Joule heating and convective condition using RKF technique. Furthermore, the nonlinear thermal radiation effect on different problems of the fluid flows have been treated by various investigators(Mahanthesh et al. [24],Farooq et al. [25]. Laxmi and Shankar [26]. Jayachandra et al. [27]).

In this work, reported the similarity solutions for nonlinear radiative heat transfer in three-dimensional flow of an Oldroyd-B fluid induced by the bidirectional stretching surface. We also consider the irregular heat source, suction and activation energy effects in the heat transfer expressions. The convective boundary condition is employed at the boundary surface. With the help of similarity transformations, the governing PDE system is reduced to nonlinear ODE system then the mathematical model is undertaken through the 4-5th order RK-Fehlberg techniques via the shooting algorithm. The physical consequence of scheming parameters on the velocity, temperature and concentration fields are inspected and deliberated graphically.

Mathematical formulation

We consider three dimensional radiative flow of an Oldroyd-B fluid over a stretching sheet. The sheet coincides with the plane at = zero and it establish in to the space > zero as shown in Fig. 1. By keeping origin fixed and the sheet is elaborate in to bidirectional and with velocity and . The sheet is heated by a hot fluid with temperature which is related with the heat transfer coefficients and and as temperature and concentration far away from the surface.

Figures 1: Geometry of the problem

The simplified three dimensional boundary layer equations of continuity, momentum, heat and variation of concentration with subjected to the boundary layer assumption are;

(1)

(2)

(3)

(4)

(5)

Where and are the velocities along the and -directions, – temperature, and are the relaxation and retardation time, -kinematic viscosity , andare the volumetric thermal expansion coefficients, is fluid thermal diffusivity of the

fluid , -thermal conductivity , -density of the fluid , -specific heat of the fluid , – heat generation / absorption coefficient and -radiative heat flux.

The thermal radiation heat flux expression through the Rosseland approximation is

(6)

Where -mean absorption coefficient, , and -Stefan Boltzmann constant From equation (4), we obtained

(7)

For the present problem corresponding boundary conditions are

at

as (8)

The above partial differential equations transformed into the corresponding nonlinear ordinary differential equations by setting correct similarity variables as follows

(9)

Here the prime denoted by derivative with respective to . in the above equations We get the following set of non-linear ordinary differential equations.

(10)

(11)

Where temperature Heat source,

Exponential space dependent Heat source. The boundary conditions for the present problem are

Where the dimensionless parameters are

(12)

(13)

(14)

(15)

Here we observe from the above equations and are the relaxation time and retardation

time. -kinamatic viscosity,-density of the fluid,B0 strength of magnetic field , – Magnetic parameter, -specific heat -fluid temperature, – velocity power index parameter, – reaction rate, – non-dimensional energy , – activation energy, – Temperature ratio parameter , -Schmidt number , –

temperature heat source, exponential space dependent heat source ,and are the Deborah numbers, -stretching ratio parameter, ,

Reynolds number, is the thermal radiation parameter, temperature ratio parameter, is the heat source / sink parameter, prandtl number and

is the Biot number.

In the engineering applications the physical quantity of the boundary value problems, the skin frication, local Nusselt number and Sherwood number are denoted by

Nusselt number (15)

Where is the surface heat flux with the similarity variables, we obtain

(16)

, (17)

Where is the concentration flux. with the similarity variables, we obtain

(18)

Results and Discussion

A set of nonlinear ordinary differential equations (2.10)-(2.13) with boundary conditions are regenerate to a set of linear ordinary differential equations by acceptable substitutions. Then resultant equations are treated as numerically using RKF technique. During this theme, it's most important to set the suitable finite values of . In accordance with standard practice in the boundary layer theory the asymptotic boundary conditions at

arereplaced by . In our numerical computations, the step size is select as

and therefore the convergence criteria were set to. In this section devoted to analyze the effect of the assorted physical parameters involved in the fluid ow problem, such as magnetic parameter, Deborah numbers , stretching ratio parameter, Biot number , temperature and exponentially space dependent heat source parameters , radiation parameter, temperature ratio parameter, temperature difference parameter , activation energy parameter (, Schmidt number and dimensionless reaction rate parameter .

The variation of velocity profiles and for is displayed in Fig. 2.The strong magnetic field produces a Lorentz force (resistive force). The resistive force opposes to liquid flow at the surface, as a result the velocity profiles with related boundary layer thickness decreases. Impact of on velocity profiles is illustrated in Fig. 3. It is observed

that, the and are reduces for higher . This mechanism due to the fact that the Deborah number depends on relaxation time. The increase in the relaxation time leads to diminish the liquid flow and their boundary layer thickness. Fig. 4indicated that the influence of the on and . It is confirm that augment the liquid velocity for higher .Physically the Deborah numbergenerates the retardation time. The high retardation time diminishes the viscous property of working liquid which causes increase in and .

The effect of (stretching ratio parameter) on the and is displayed in Fig.

5. It is remark that, decay the velocity fieldwhile the velocity field is increases.

The reason behind that an increase in is leads to the movement of the lateral surface in the -direction that corresponds to increase the momentum boundary layer thickness. Impact of

on and is plotted in Fig. 6. It is clearly shows that, the higher values of is

enhance the but quit opposite trend has been observed in .Behaviour of and

on temperature field and concentration field are depicted in figure 7 and 8 respectively. It is noted that, the liquid concentration is reduces while the liquid temperature is augmented via higher values of and . By enhancing the values of and provides extra heat from surface towards working fluid (Oldroyd-B fluid), in fact the fluid temperature and their related thermal boundary layer thickness is increase.

Figures 9-10 indicate the impact of and on and . It clearly shows that, the higher values of and as increase the temperature distribution but opposite behaviour is occurs in concentration profile. Heat creates in liquid flow through radiation mechanism and the surface become warm for higher, as a result the thermal boundary layer become thicker. The variation of temperature difference parameter on is plotted in Fig. 11. It is noted that the higher values of is diminishes .The reason behind that, the higher ( increases the wall temperature and decreases the ambient

temperature. However, the liquid concentration decrease. Fig. 12 represent the concentration profiles for various values of . It is reported that the impact of activation energy leads to increases in . Due to higher values of leads to lesser reaction rate constant and consequently slow down the chemical reaction, as a result increases in .The impact of and on are illustrated in Fig. 13 and 14 respectively. We can see that diminishing in concentration boundary layer when and are increased. Physically, an higher generates the lower solute diffusivity then reduction accurse in .

Numerical solutions of and for different parametric values are reported in table 1. It is clearly shows that, the friction force is more effective for larger but quit opposite behaviour is seen in . The rate of heat transfer is high for and but conflicting for .The mass transfer rate is high for and higher values of have less mass transfer rate. Table 2 present the numerical data of and

with varying and .It is reported that no difference is found in drag force for and . For exact opposite behaviour is occurs in and with varying and .The is increase with higher and . The Sharwood number is enhances for and but decrease for.

Concluding remarks

3D nonlinear radiative Oldroyd-B fluid flow over an elongated sheet with influence of irregular heat source and activation energy is scrutinized in this paper. The effects of magnetic field, suction and convective condition are also elaborated. The following observations are found;

• Higher activation energy leads to increase in concertation profile.

• The momentum boundary layer become thinner for high magnetic field strength.

• The impact of and are quite opposite behaviour on liquid field characteristics.

• Heat and mass transfer rate is high for and .

• Choose lesser values of thermal radiation to liquid cooling process.

• Increae the liquid temperature for irregular heat sources.

Table 1: Numerical results of and for different values of and .

0.4	2	0.5	1.2	0.1	0.1	0.4	2	-1.20971	-0.66238	0.651255	1.101407
0.4								-1.20971	-0.66238	0.651255	1.101407
0.6								-1.20971	-0.66238	0.716853	1.161491
0.8								-1.20971	-0.66238	0.746018	1.197950
0.4	0							-1.20971	-0.66238	0.651255	1.973478
	2							-1.20971	-0.66238	0.651255	1.101407
	4							-1.20971	-0.66238	0.651255	0.672215
	2	0						-1.04730	-0.55430	0.706060	1.084009
		0.5						-1.20971	-0.66238	0.651255	1.101407
		0.8						-1.29846	-0.72026	0.619114	1.112353
		0.5	0.7					-1.20971	-0.66238	0.447421	1.194781
			1.2					-1.20971	-0.66238	0.651255	1.101407
			1.5					-1.20971	-0.66238	0.736385	1.060731
			1.2	0				-1.20971	-0.66238	0.803187	1.035922
				0.1				-1.20971	-0.66238	0.651255	1.101407
				0.2				-1.20971	-0.66238	0.512402	1.159935
					0			-1.20971	-0.66238	0.759943	1.054070

					0.1			-1.20971	-0.66238	0.651255	1.101407
					0.2			-1.20971	-0.66238	0.465141	1.181487
						0.4		-1.20971	-0.66238	0.651255	1.101407
						0.6		-1.20971	-0.66238	0.742923	1.153528
						0.8		-1.20971	-0.66238	0.787803	1.194280
						0.4	1.2	-1.20971	-0.66238	0.328135	1.028997
							1.6	-1.20971	-0.66238	0.469168	1.058742
							2	-1.20971	-0.66238	0.651255	1.101407

Table 2: Numerical results of and for different values of

and .

0.5	0.2	0.2	0.6	1	0.8	5	-1.20971	-0.66238	0.651255	1.101407
0.5							-1.20971	-0.66238	0.651255	1.101407
1							-1.20971	-0.66238	0.651255	1.609301
2							-1.20971	-0.66238	0.651255	2.328783
	0.05						-1.15868	-0.63681	0.683357	1.090295
	0.1						-1.17584	-0.64542	0.672732	1.093920
	0.2						-1.20971	-0.66238	0.651255	1.101407
		0.05					-1.33876	-0.73451	0.596338	1.120828
		0.1					-1.29127	-0.70791	0.616924	1.113429
		0.2					-1.20971	-0.66238	0.651255	1.101407
			0.4				-1.19320	-0.41064	0.579046	1.118408
			0.5				-1.20160	-0.53311	0.617276	1.108962
			0.6				-1.20971	-0.66238	0.651255	1.101407
				1			-1.20971	-0.66238	0.651255	1.101407
				2			-1.20971	-0.66238	0.651255	1.421285
				3			-1.20971	-0.66238	0.651255	1.701805
					0.4		-1.20971	-0.66238	0.521741	1.069771
					0.6		-1.20971	-0.66238	0.604609	1.081346

					0.8		-1.20971	-0.66238	0.651255	1.101407
						2	-1.20971	-0.66238	0.651255	0.772735
						3	-1.20971	-0.66238	0.651255	0.895297
						5	-1.20971	-0.66238	0.651255	1.101407

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Fig 2: Impact of on and .

Fig 3: Impact of on and .

Fig 4: Impact of on and .

Fig 5: Impact of on and .

Fig 6: Impact of on and .

Fig 7: Impact of on and .

Fig 8: Impact of on and .

Fig 9: Impact of on and .

Fig 10: Impact of on and .

Fig 11: Impact of on .

Fig 12: Impact of on.

Fig 13: Impact of on .

Fig 14: Impact of on .