Essay: Boundary layer and heat transfer

Many researchers have done studies on boundary layer and heat transfer over different types of surfaces. One of the famous researchers is Afzal [1]. In Afzal [1] study, he has investigated about the heat transfer from a stretching surface. In his study, it is said that the heat transfer from a stretching surface is one of interest in polymer extrusion processes after the object passing through a die, it enters the fluid for cooling below a certain temperature.
In the study of heat transfer in a liquid film on an unsteady stretching surface by Andersson et al. [3], he stated that the temperature was observed to increase monotonically from the elastic sheet towards the free surface except in the high diffusivity limit Pr’0 in which the surface temperature approached the variable sheet temperature. At sufficient high for Pr, the surface temperature will became equal to T_0 and thus make both of position and time independent. The influence of the unsteadiness parameter S on the heat flux from the liquid film to the stretching sheet was more pronounced at low Prandtl numbers than for Pr>1, the surface temperature was affected by S at intermediate Prandtl numbers.
Heat transfer in a liquid film over an unsteady stretching sheet studied by Nandeppanavar et al. [11] stated that the effect of transverse magnetic field on the flow is to suppress the velocity field which in turn causes the enhancement of the temperature field. The effect of viscous dissipation is to enhance the temperature in the thermal boundary layer. Nandeppanavar et al. [11] also stated that the effect of the thermal radiation parameter produces a significant increase in the thickness of the thermal boundary layer of the liquid film and the temperature increase in the presence of a non-uniform heat source/sink.
Arnold et al. [4] in his study entitled ‘Heat transfer in a viscoelastic boundary layer flow over a stretching sheet’ has investigated about viscoelastic fluid flow and heat transfer characteristics over a stretching sheet with frictional heating and internal heat generation or absorption. Arnold et al. [4] stated that the momentum equation is decoupled from the energy equation for the present incompressible boundary layer flow problem with constant physical parameter. In Arnold et al. [4] study, it is also said that the magnitude of the non-dimensional surface velocity gradient is found to increase with the increase in the viscoelastic parameter while the magnitude of the non-dimensional surface temperature gradient increases with the Prandtl number (Pr). Increase in Pr can cause reduction in the thickness of the thermal boundary layer.
Bhattacharyya [5] has investigated the boundary flow and heat transfer over an exponentially shrinking sheet. The result has shown that the temperature at a point decreased when the Prandtl number increased. The velocity will increased with mass suction and decreased for the second solution. Due to the increasing Prandtl number, the thermal boundary layer thickness has become thinner.
In the study of heat transfer of a continuous stretching surface with suction or blowing by Chen and Char [6], the sheet with prescribed wall temperature and the heat flux need to be considered in the cases. The result of Chen and Cha [6] study showed that the temperature at a given point decreases with the increase in Pr. For fixed values of ??,Pr,and r, the smaller the m, the larger is the thermal boundary-layer thickness. This implies that the thermal boundary-layer thickness in suction is thinner than that is in blowing since m in suction is larger than m in blowing.
Elbashbeshy and Bazid [7] in their study about heat transfer over an unsteady stretching surface with internal heat generation mentioned that the skin friction f^” (1,0) and the dimensionless coefficient ‘??^’ (1,0) decreases with increases ?? for all constant A and Pr. Their study also showed that the velocity f^’ (1,??) and temperature ??(1,??) decreases with increasing A and the temperature ??(1,??) increases with ?? and decreases with Pr.
The study of boundary layer flow and heat transfer over an unsteady stretching vertical surface done by Ishak et al. [8] showed that for assisting flow, the solution exist for all values of buoyancy parameter whereas there exist the magnitude of the buoyancy parameter is small only if opposing flow occured. Ishak et al. [8] also mentioned that an assisting buoyant flow produces an increase in the skin friction coefficient, while an opposing buoyant flow produces the opposite.
Another study done by Ishak et al. [9] entitled ‘Heat transfer over an unsteady stretching permeable surface with prescribe of wall temperature’ stated that a stretching permeable surface causes the similarity solutions of the unsteady boundary layer flow and heat transfer. Ishak et al. [9] also found that the heat transfer rate at the surface ‘??'(0) increase with A,?? and Pr.
Nadeeem et al. [10] have done a study on heat transfer analysis of water-based nanofluid over an exponentially stretching sheet. Nadeem et al. [10] stated that the velocity profile (x and y) decrease with the increase in nano particles volume fraction while the temperature profile increase. As the stretching parameter increase, the skin frictions and the heat transfer rate also increase.
In all of the mentioned studies, none of them have considered the study of heat transfer over an unsteady stretching surface. Therefore, the aim of this research paper is to investigate the effect of velocity profile and temperature profile with variations of parameter in the study of heat transfer over an unsteady stretching surface.

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