# variable viscosity

As Pr increases, the thermal diffusivity decreases and thus, the heat is diffused away from the heated surface slowly, which results in higher heat transfer at the surface. Full pictures of the effect of γ towards the surface shear

stress f ″ 0ð Þ and the local Nusselt number −θ′(0) when

β¼0, 0.5 and 2, Pr¼0.7, λ¼1 with M¼1 and 3 are depicted in Figures 1 and 2, respectively. Both Table 2 and Figures 1, 2 conclude that an M increment will lead to both decrement of surface shear stress f ″ 0ð Þ and the local Nusselt number −θ′(0). Supplementary evidence is found in Figures 3 and 4 at fixed value of λ and γ. Further, it is noted from Figures 1 and 2 that the surface shear stress f ″ 0ð Þ and the local Nusselt number −θ′(0) are also found to decrease with the increment of γ for fixed value of β, M, Pr and λ. These behaviours are consistent with the results plotted in Figures 3 and 4. The effect of the mixed convection parameter λ is seen to increase both f ″ 0ð Þ and −θ′(0), with the increment of λ as depicted in Figures 3 and 4. This is because the existence of the buoyancy force

induces a favourable pressure gradient that enhances the flow (increases the velocity f ′ ηð Þ) and heat transfer in the boundary layer). This is in line with the velocity profile f ′ ηð Þ plotted in Figure 9, which is evidenced in the behaviour of the fluid motion.

The resulting profiles of the dimensionless velocity f ′ ηð Þ and the temperature distribution θ(η) for various values of the Deborah number β and γ when Pr¼0.7, λ¼1 and M¼1, are displayed in Figures 5 and 6, respectively. It is observed that the velocity and boundary layer thickness are increasing functions of the Deborah number β. It should be pointed out that β¼0 represents Newtonian fluid and β40 represents the Jeffrey fluid parameter. However, opposing phenomenon is observed for the temperature profile. The

Figure 4 Variations of -θ′(0) with λ at selected values of M and γ when Pr¼0.7 and β¼1.

Figure 1 Variations of f″(0) with γ at selected values of M and β when Pr¼0.7 and λ¼1.

Figure 2 Variations of -θ′(0) with γ at selected values of M and β when Pr¼0.7 and λ¼1.

Figure 3 Variations of f″(0) with λ at selected values of M and γ when Pr¼0.7 and β¼1.

effect of γ is found to decrease the velocity distribution and increase temperature distribution, respectively.

The effects of the MHD parameter M on the velocity f ′ ηð Þ and the temperature profiles θ(η) are shown in Figures 7 and 8, respectively. Velocity is found to decrease with the increase of M. The introduction of the transverse magnetic field will result in a restrictive force (Lorenz force), which tends to resist the motion of the fluid flow and hence, lead to the decrement of velocity. However, the opposite trend is observed in the increment of M, which results in the increment of temperature distribution across the boundary layer. The effect of the porous medium γ on flow velocity and temperature can also be garnered from the same figures. It is obvious that an increase in the porosity γ causes greater obstruction to the fluid flow, which culmi- nates in the decrement of velocity, whereas the opposite

Kartini Ahmad, Anuar Ishak274

trend occurs for the temperature profile θ(η), i.e the increment of γ results in an increment in temperature and thermal boundary layer thickness, as shown in Figures 7 and 8, respectively. Figures 9 and 10 present the velocity and temperature

profiles when Pr¼0.7 and 6.8 for few values of the mixed convection parameter λ, respectively. It is well known that λ¼0 corresponds to pure forced convection and the presence of thermal buoyancy (λ≠0) will lead to stronger buoyancy force, which induces more flow along the surface. The consequences can be seen in the increase of the velocity f ′ ηð Þ as λ increases. However, this phenomenon is more pronounced for flow with low Pr numbers compared to flow with high Pr numbers. An overshoot peak in the velocity profile is observed near the surface for flow with low Pr number and for large values of the mixed

Figure 7 Velocity profiles f′(η) for some values of M and γ when Pr¼0.7 and λ¼β¼1.

Figure 8 Temperature profiles θ(η) for some values of M and γ when Pr¼0.7 and λ¼β¼1.

Figure 5 Velocity profiles f′(η) for some values of β and γ when Pr¼0.7 and λ¼M¼1.

Figure 6 Temperature profiles θ(η) for some values of β and γ when Pr¼0.7 and λ¼M¼1.

convection parameter (λ¼10) where the free convection is dominant. At the beginning of the motion (0rηr0.5), the velocity increases until it reaches a certain value and gradually decreases until it goes to 0 at the outside of the boundary layer, whereas the velocity for other profiles produce lower velocities toward the edge of the boundary layer starting from the beginning.

Figure 10 depicts the graph of the temperature distribu- tions for the same data used in Figure 9. The tabulated temperature is more noticeable for different values of λ when Pr¼0.7 compared to Pr¼6.8. The aim of the increasing the values of λ is to decrease the thickness of the thermal boundary layer and reduce temperature. How- ever, this phenomenon does not happen for Pr¼6.8, i.e the variations of λ appear not to influence temperature distribu- tion as they are seen to have similar profiles.

Figure 9 Velocity profiles f′(η) for some values of M and γ when Pr¼0.7 and λ¼β¼1.

Figure 10 Temperature profiles θ(η) for some values of M and γ when Pr¼0.7 and λ¼β¼1.

Irrespective of the value of the parameters in this study, all the plotted velocity and temperature profiles satisfied the boundary conditions (8) asymptotically.