Numerical Exploration of Non-Newtonian Polymeric Boundary Layer Flow over an Isothermal Sphere

Syed Fazuruddin^1,2*, S Sreekanth¹, GSS Raju²

¹Department of Mathematics, Sreenivasa Institute of Technology and Management Studies, Chittoor-517001, India.

²Department of Mathematics, JNTUA College of Engineering, Pulivendula-516390, Andhra Pradesh, India.

*Corresponding Author E-mail: fazuruddinsyed@gmail.com

ABSTRACT:

This paper examines the nonlinear steady state boundary layer flow and heat transfer of an incompressible Jeffery non-Newtonian fluid from an isothermal sphere. The governing partial differential equations of the flow field are converted to a system of non-linear coupled non-similarity ordinary differential equations. Finite difference technique followed by Keller Box method, the system is solved numerically. The numerical code is validated with previous studies. The effects of the various physical parameters countered in the flow field on the velocity, temperature as well as the skin friction coefficient and the rate of heat transfer near the wall are computed and illustrated graphically. It is found that increasing suction decelerates the flow and also cools the boundary layer i.e. reduces temperatures. With increasing tangential coordinate the flow is also decelerated whereas the temperatures are enhanced. The simulation is relevant to polymer coating thermal processing. Polymeric enrobing flows are important in industrial manufacturing technology and process systems. Such flows are non-Newtonian. Motivated by such applications, we did the present problem.

KEYWORDS: Polymers, heat transfer, skin friction, Deborah number, Suction.

1. INTRODUCTION:

Recent era is transferring of heat through lubricants due to isothermal sphere plays major role in the extrusion process of fiber technology, making of polymer sheets and plastic films, manufacturing paper, glass blowing, spinning of metals and plastic films, etc. Livescu et al. [1] include petroleum drilling muds, biological gels (Hron et al. [2]). Polymer processing (Loix et al. [3]) and food processing (Kechichian et al. [4]). Most commonly, the viscosity of non-Newtonian fluids is dependent on shear rate. One subclass of non-Newtonian fluids known as the Jeffery fluid is particularly useful owing to its simplicity. This fluid model is capable of describing the characteristics of relaxation and retardation times which arise in complex polymeric flows.

Furthermore, the Jeffrey type model utilizes time derivatives rather than converted derivatives, which make it more amenable for numerical simulations. Recently the Jeffery model has received considerable attention. Interesting studies employing this model include peristaltic Magnetohydrodynamic non-Newtonian flow [5], Magnetohydrodynamic peristaltic flow in Eccentric Cylinders [6], Unsteady Hydromagnetic Flow [7] and stretching sheet flows [8,9]. Recently, Amanulla et al. [10] provided the numerical solution for Jeffery’s fluid flow over an inclined vertical plate. The extended work of [10] was presented by Amanulla et al. [11]. With that they highlighted magnetic field and slip effects acts as a controller in the flow field. Nadeem et al. [12] deliberated steady stagnation point flow of Jeffery fluid towards a stretching surface. Amanulla et al. [13-14] obtained the numerical solutions for the boundary layer flow past an isothermal sphere with convective boundary conditions.

The main objective of the present investigation is to study the natural convection flow of Jeffery’s fluid past an isothermal sphere with uniform suction effect. The method of solution involves non-similarity transformation which reduces the partial differential equations into a set of non-linear ordinary differential equations. These non-linear ordinary differential equations have been solved by applying finite difference method (Keller Box Method) with help of Newton’s method. The velocity and temperature profiles for different values of flow parameters are presented in the figures. It is observed from all figures that the boundary conditions are satisfied asymptotically in all the cases which support the accuracy of numerical results.

2. MATHEMATICAL ANALYSIS

Fig. 1: Schematicdiagramoftheproblem

A steady two dimensional laminar free convection flow over an isothermal sphere in a viscous fluid of temperature T_∞ (see Fig. 1) is considered. The buoyancy forces arise due to the variations in temperature of fluid. The Boussinesq approximation is invoked for the fluid properties to relate the density changes to temperature to couple in this way the temperature field to the flow field. Under these assumptions, the governing boundary layer equations can be expressed as:

(1)

(2)

(3)

with the relevant boundary conditions

As (4)

The following similarity transformations are introduced to Eqs. (1) – (3),

(5)

where is the stream function defined in the usual form as, Eq. (1) is identically satisfied and the velocity components u and v are given by ,and is the radial distance from the symmetrical axis to the surface of the sphere. Therefore, substituting the new variables into Eqs. (1) – (3) are reduced to the following set of ordinary differential Eqs.,

(6)

(7)

where prime denotes differentiation with respect to .

Along the transformed boundary conditions (4) are

As (8)

The skin-friction coefficient (Sphere surface shear stress) and the local Nusselt number (Sphere surface heat transfer rate) can be defined, respectively, using the transformations described above with the following expressions:

(9)

(10)

3. RESULTS AND DISCUSSION:

The system of coupled extremely non-linear ordinary differential equations (6) – (7) subject to the boundary conditions (8) is resolved numerically using finite difference technique (Keller Box). This method has been successfully used by the present authors [15-19] to resolve numerous problems associated with boundary layer flow and heat and mass transfer. During this method, the end of the boundary layer has been chosen as η = 10 that is sufficient to realize the far field boundary conditions attention for all values of the parameters considered. Comprehensive numerical parametric computations are carried out for various physical parameters values then the results are reported in terms of graphs. Numerical solutions obtained for the problem are expressed in terms of graphs for various ranges and for various choices of the flow parameters. Impact of the ratio of relaxation to retardation times parameter, Deborah number, Prandtl number and Suction parameter on flow, temperature, friction at wall and rate of heat transfer profiles are mentioned. In order to test the accuracy, the present results are compared with those of Huang and Chen [20]for the appropriate reduced cases, and found that there is an excellent agreement, as presented in Table 1.

Table 1: Values of the local heat transfer coefficient for various values of with

(degrees)
(degrees)	Huang and Chen [20]	Present Solutions	Huang and Chen [20]	Present Solutions
0	1.2276	1.2279	0.5165	0.5155
30	1.2031	1.2033	0.5065	0.5061
60	1.1296	1.1300	0.4768	0.4763
90	1.0071	1.0075	0.4276	0.4272

In Figures 2(a)–2(b), the evolution of velocity and temperature functions with a variation in Deborah number, De, is depicted. Dimensionless velocity component (fig. 2a) is considerably reduced with increasing De near the sphere surface and for some distance into the boundary layer. For polymers, larger De values imply that the polymer becomes highly oriented in one direction and stretched. Generally, this arises when the polymer takes longer to relax in comparison with the rate at which the flow is deforming it. When such fluids are stretched there is a delay in their return to the unstressed state. For very large Deborah numbers, the fluid movement is too fast for elastic forces to relax and the material then acts like a purely elastic solid. Large Deborah numbers are therefore not relevant to the present simulations. In fig. 2b, an increase in Deborah number is seen to considerably enhance temperatures throughout the boundary layer regime.

(a) (b)

Fig. 2. (a) Velocity and (b) Temperature profiles for various values of De

Figures 3(a) - 3(b)illustrates the effect of the ratio of relaxation to retardation times i.e. λ on the velocity and temperature distributions through the boundary layer regime. Velocity is significantly decreased with increasing λ, in particular close to the sphere surface. The polymer flow is therefore considerably decelerated with an increase in relaxation time (or decrease in retardation time). Conversely temperature is depressed slightly with increasing values of λ.

(a) (b)

Fig. 3. (a) Velocity and (b) Temperature profiles for various values of λ

Figures 4(a)-4(b)present typical profiles for velocity and temperature for various values of Prandtl number, Pr. It is observed that an increase in the Prandtl number Pr massively reduces the velocity i.e. decelerates the polymeric boundary layer flow. With increasing Prandtl number the dynamic viscosity of the fluid is strongly elevated and this is representative of non-Newtonian polymers. The flow is therefore retarded and momentum boundary layer thickness is decreased. At high Prandtl number, thermal conduction heat transfer dominates over thermal convection heat transfer and thermal boundary layer thickness is decreased. Conversely for lower Prandtl numbers the opposite behavior is observed. Effectively a rise in Prandtl number decreases fluid temperatures and cools the regime.

(a) (b)

Fig. 4. (a) Velocity and (b) Temperature profiles for various values of Pr

(a) (b)

Fig. 5. (a) Velocity and (b) Temperature profiles for various values of S

(a) (b)

Fig. 6. (a) Velocity and (b) Temperature profiles for various values of x

Figures 5(a) - 5(b)present typical profiles for velocity and temperature for various values of the transpiration parameter, S. As in all other graphs, only the case of wall suction is studied (S> 0). It is observed that an increase in the suction parameter significantly decelerates the flow for all values of radial coordinate. The boundary layer thickness is reduced and suction causes the boundary layer to adhere closer to the wall. Similarly increasing wall suction is found to lower temperatures in the boundary layer regime and strongly decreases thermal boundary layer thickness.

Figures 6(a)– 6(b)depict the velocity and temperature distributions with radial coordinate, for various transverse (stream wise) coordinate values, x. Generally, velocity is noticeably lowered with increasing migration from the leading edge i.e. larger x values (figure 6a). The maximum velocity is computed at the lower stagnation point (x~0) for low values of radial coordinate (h). The transverse coordinate clearly exerts a significant influence on momentum development. A very strong increase in temperature, as observed in figure 6b, is generated throughout the boundary layer with increasing x values. The temperature field decays monotonically.

(a) (b)

Fig. 7. (a)Skin friction profiles and (b) Nusselt number profiles for various values of De

Figures 7(a)– 7(b) presents the influence of increasing Deborah number parameter, De on skin friction and heat transfer rate, along with a variation in the transverse coordinate, x. With increasing Deborah number, the skin friction is generally decreased, and heat transfer rate is also decreased. This trend is sustained for all values of transverse coordinate.

(a) (b)

Fig. 8. (a)Skin friction profiles and (b) Nusselt number profiles for various values of λ

Figures 8(a)– 8(b) depict the skin friction and rate of heat transfer with variation of ratio of relaxation to retardation times i.e. λ. With an increase in λ, both skin friction and heat transfer rates are increased. This implies that as the relaxation time is reduced (and the retardation time increased) the polymer flows faster and also transfers heat more efficiently from the sphere surface.

Figures 9(a)– 9(b) depict the skin friction and rate of heat transfer with variation of suction parameter. An increase in suction (S > 0) reduces skin friction at higher values of the transverse coordinate. Further from the vicinity of the lower stagnation point therefore the polymer flow is decelerated with suction.

(a) (b)

Fig. 9. (a)Skin friction profiles and (b) Nusselt number profiles for various values of S

4. CONCLUSIONS

A mathematical model has been framed to examine the influence of variable thermal conductivity on MHD flow past a stretching cylinder imbedded in a thermally stratified medium with internal heat sour or sink. The following conclusions drawn are as follows:

1. Increasing the viscoelasticity parameter i.e. Deborah number (De), reduces the velocity, skin friction and heat transfer rate whereas it enhances temperature.

2. Increasing the parameter ratio of relaxation and retardation times (λ), increases velocity, skin friction coefficient and heat transfer rate whereas it reduces temperature.

3. Increasing suction at the sphere surface (S>0) decelerates the flow, skin friction and also strongly depresses temperature and enhance the heat transfer from the fluid.

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Received on 13.09.2017 Modified on 28.10.2017

Research J. Science and Tech. 2017; 9(4): 525-531.

DOI: 10.5958/2349-2988.2017.00090.0