INTRODUCTION:

The diversity of nature flow develops the various rheological properties of liquids. The different industrial processes like plastic material production, paints production, preventative coating, lubricants performance and many others involve the rheological liquids of various characteristics. All the rheological aspects of liquids cannot be described by simple theory of Navier-Stokes. For the exact description of rheological liquids, different fluid models like power-law models, Powell-Eyring liquid, Jeffrey, Maxwell and Oldroyd-B fluids, Casson fluid etc. have been formulated according to physical characteristics of non-Newtonian materials. The present research dealt with Jeffery liquid model which is proposed by suspension of cylindrical shaped particles in liquid.

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 convective derivatives, which make it more amenable for numerical simulations. Recently the Jeffery model has received considerable attention. Interesting studies employing this model include peristaltic Magneto hydrodynamic non-Newtonian flow by Kothandapani and Srinivas [1], MHD Free Convection Flow over an Exponentially Moving Vertical Plate by Srinivasa Raju et al., [2], variable-viscosity peristaltic flow Nadeem and Akbar [3], convective-radiative flow in porous media Hayat et.al. [4] and stretching sheet flows by Hayat and Alsaedi [5] and Nadeem et.al. [6]. Pop et al. [7] simulated numerically the steady laminar forced convection boundary layer of power-law non-Newtonian fluids on a continuously moving cylinder with the surface maintained at a uniform temperature or uniform heat flux.

Recently, Amanulla et al. [8] provided the numerical solution for Jeffery’s fluid flow over an inclined vertical plate. The extended work of [8] was presented by Amanulla et al. [9]. With that they highlighted magnetic field and slip effects acts as a controller in the flow field. Nadeem et al. [10] deliberated steady stagnation point flow of Jeffery fluid towards a stretching surface. Amanulla et al. [11-12] obtained the numerical solutions for the boundary layer flow past an isothermal sphere with convective boundary conditions.

In the present work, a mathematical model is developed for steady, forced convection boundary layer flow in a non-Newtonian viscoelastic fluid external to an inclined plate with presence of Navier slip and Newtonian heating effects. A numerical solution is obtained for the transformed nonlinear two-point boundary value problem subject to physically appropriate boundary conditions at the plate surface and in the free stream. The impact of the emerging thermo-physical parameters i.e. non-Newtonian fluid parameters, Navier (velocity) slip and Newtonian heating on velocity, temperature, wall shear stress function and rate of heat transfer, are presented graphically.

FLOW MODEL:

Consider steady two dimensional laminar forced convection flow over an inclined vertical surface in a viscous fluid of temperature T_∞(x), (see Fig. 1) is considered. The buoyancyforces 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 and to couple in this way the temperature and fields to the flow field. Under these assumptions, the governing boundary layer equations can be expressed as:

Fig. 1: non-Newtonian heat transfer over a pl

COMPUTATIONAL FINITE DIFFERENCE SOLUTIONS:

The Computational finite difference method (Keller-Box) is employed to solve the transformed, coupled boundary layer problematic defined by eqns. (6) - (7) under (8). Keller-Box method is the most versatile technique available for engineering analysis. This technique has been described succinctly in Cebeci and Bradshaw [13] and Keller [14]. It has been used recently in polymeric flow dynamics by Subba Rao et al. [15-17], Amanulla et al [18,19] for viscoelastic models and Beg et al. [20] for viscoplastic fluid flows with Convective conditions.

RESULTS AND INTERPRETATION:

The nonlinear boundary value problem solved in the previous section is dictated by an extensive number of thermal and hydrodynamic parameters. In order to gain a clear insight into the physical problematic, numerical calculations for distribution of the Skin friction and rate of heat transfer for different values of these parameters is conducted with graphical illustrations (Figs. 2-5). For the purpose of our computation, we adopted the following default parameters:

are initial values (unless otherwise stated).

Fig.2-3 presents the evolution velocity , temperature, skin friction and in local Nusselt number function, Nu, with transverse coordinate with variation in hydrodynamic slip parameter, Nf. Velocity profile and boundary layer thickness increase for the higher values of hydrodynamic slip as shown in figure 2a. it's determined that the increase in Nf causes a reduction in temperature profiles and additionally as thermal boundary layer thickness. Rate of Heat transfer profiles constantly growth monotonically from a most extreme at the plate surface to the free stream. All profiles meet at a vast estimation of transverse coordinate, again presentation that a sufficiently large infinity boundary condition has been utilized in the numerical computations. Greater momentum slip substantially increases heat transfer rate in the boundary layer and therefore also elevates thermal boundary layer thickness. The regime is therefore coolest when slip is absent (Nf=0 i.e. no-slip classical case) and hottest with strong hydrodynamic wall slip. This generates a strong increase in skin friction (acceleration) and an enhancement in Rate of heat transfer i.e. Nusselt number function.

Figure4a, b depicts the have an impact on of the Newtonian heating, , on the dimensionless skin friction coefficient and rate of heat transfer at the plate surface. The skin friction on the plate floor is visible to beautify significantly with rising , that is mainly because of the lower in Grashof (forced convection) variety which leads to an acceleration inside the boundary layer flow, as elaborated by Subba Rao et al. [16]. Rate of Heat transfer (local Nusselt number) is likewise improved with increasing , as shown in Fig. 4b.

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Fig. 2 Effect of Nf on (a) velocity profiles (b) temperature profiles

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Fig. 3 Effect of Nf on (a) skin friction (b) Nusselt number

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Fig. 4 Effect of on (a) skin friction (b) Nusselt number

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Fig. 5 Effect of on (a) skin friction (b) Nusselt number

Figures 5a-b presents the influence of on the wall heat transfer rate i.e. Nusselt number, . An increase in non-Newtonian fluid parameter () induces a substantial increase in the magnitude of . With increasing values, more heat is transferred from the boundary layer regime to the plate surface (the fluid is heated and the plate surface is cooled). This manifests in an increase in Nusselt numbers with greater Viscoelastic effect (largervalues).

Figures 6a-b presents the effect of the plate inclination on the dimensionless skin friction and heat transfer rate. Taking angle of the plate ( ) i.e. rotating plate with negative angle , in fig.6a, the skin friction is reduced. Conversely in fig.6b, with inclination the rate of heat transfer decreases slightly. similarly, more heat transfer rates are improved marginally with positive inclination of the plate.

Figures 7a–bdepict 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 7a). 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 (q), as observed in figure 6b, is generated throughout the boundary layer with increasing x values. The temperature field decays monotonically. Temperature is maximized at the cylinder surface (h=0) and minimized in the free stream (h= 8). Although the behavior at the upper stagnation point (x~p) is not computed, the pattern in figure 7b suggests that temperature will continue to progressively grow here compared with previous locations on the cylinder surface (lower values of x).

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Fig. 6 Effect of on (a) skin friction (b) Nusselt number

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Fig. 7 Effect of on (a) velocity (b) temperature

CONCLUSIONS:

In this work, Computational results have been presented for the boundary-driven navier slip flow of non-Newtonian fluid from an inclined vertical plate with considering under the effect of Newtonian heating at plate surface. The governing equations are solved numerically the finite difference method. Numerical results are reported for various values selected parameters interest. When takes the values larger than 0.5. The transformed boundary layer equations for heat and momentum conservation have been solved using a finite difference method. The present investigation has shown that:

1) Increasing the velocity (navier) slip parameter (Nf) reduces the skin friction near the plate surface and increases the Nusselt number i.e. enhances momentum boundary layer thickness and decreases thermal boundary layer thickness.

2) Increasing Newtonian heating parameter, () consistently accelerates the skin friction and also heat transfer rate (and thermal boundary layer thickness).

3) Increasing transverse coordinate () generally decelerates the flow near the plate surface and reduces momentum boundary layer thickness whereas it enhances temperature. Heat transfer rate is also maximized at the lower stagnation point (= 0).

ACKNOWLEDGEMENT:

The authors are grateful to the authorities of Madanapalle Institute of Technology and Science, Madanapalle for providing research facilities in the campus.

CONFLICT OF INTEREST:

The authors declare no conflict of interest.

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