Laminar flow occurs when fluid flows in parallel layers without disruption. The study of this type of fluid flow is important as it leads to understanding of flows with disruption within its layers referred to as turbulent flow. A stretching surface is a region held at one end and moves as a result of a pull at one end. Laminar boundary layer fluid flow occurs in many industrial situations, such as glass fibre and paper production; polymer extrusion from dyes; drawing, tinning and annealing of copper wires, and bath cooling of metallic plates.
Consequently, the study of laminar boundary layer flow over a stretching sheet was studied as far back as the 1970s, and this interest still continues. Many authors have investigated the flow of a linear stretching sheet for example Crane (1970), Gupta and Gupta (1977), Grubka and Bobba (1985), Dutta and Gupta (1987), Siddappa and Abel (1985), Chen and Char (1988), Laha et al. (1989), Chakrabarti and Gupta (1979), Andersson et al. (1992), Siddheshwar and Mahabaleswar (2005), Abel and Mahesha (2008), Abel et al. (2009a), and Abel et al. (2009b) In this chapter we focus on nonlinear systems, in particular exponential stretching sheet.
to an exponentially stretching sheet
Many practical situations involve a nonlinear stretching sheet. With this in mind it is often necessary to consider the velocity of the sheet to vary exponentially with the distance from the extrusion slit. Early studies on exponentially stretching sheets by Magyari and Keller (1999) showed that fluid flow and heat transfer characteristics derived this way could be compared with those from well- known literature. The Elbashbeshy (2001) considered a perforated sheet, and examined the effect of wall mass suction on the fluid flow and heat transfer over an exponentially stretching surface. The influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet was studied by Sajid and Hayat (2008). Khan (2006) presented an elegant solution of the viscoelastic boundary layer flow over an exponentially stretching sheet, giving it in terms of Whittaker’s function. The characteristic of an industrial extrusion product will depend on the rates of stretching and cooling the sheet. Consequently, to ensure the desired characteristic, external means of controlling the flow are necessary, as could be done with a magnetic field. Early work in external control by means of magnetic fields was directed at controlling plasmas (Tonks, 1939). Sanjayanand and Khan (2006) studied the heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet.
In fluid flow, we can consider two types of fluids, Newtonian fluids and Non-Newtonian fluids. In this chapter we investigate the flow of a Newtonian fluid. Newtonian fluids are whose viscous stresses are linearly proportional to the local strain rate. Newtonian fluids have the simplest mathematical models than non-Newtonian fluids. There are no real fluids that conform to these mathematical models but, however water and air are assumed to be Newtonian fluids (Bar-Meir, 2009). We describe non-Newtonian fluids in Chapter 4. Studies on Newtonian fluids include the work of Alim et al. (2006) who studied pressure work effect in Newtonian fluid, Awad et al. (2011) studied convection from a cone in Newtonian fluid, Boivin et al. (2000) investigated Navier-Stokes equations on incompressible flows, Chen (2004) investigated heat and mass transfer in a Newtonian fluid. Gebhart (1962) studied viscous dissipation on natural convection in a Newtonian fluid.
Magnetohydrodynamics (MHD) is the study of magnetic properties of electrically con- ducting fluids; these include electrolytes, liquid metals and plasmas. When conductive fluids flow across a magnetic field, a current is induced polarizing the fluid and in turn change the
magnetic field itself Radiation is the transmission of energy in the form of waves or particles through space or through a material medium. As already mentioned, the temperature and rate of cooling of extrusion product will affect its properties. Analysis has been directed at thermal radiation magnetohydrodynamics. Until recently, radiation effects on exponentially stretching sheets had received little attention. However, if polymer extrusion takes place in a thermally controlled environment, thermal radiation and magnetic field effects will affect fluid flow characteristics. Studies on thermal radiation and magnetohydrodynamics include the work of Reddy et al. (2012) who investigated radiation effects on MHD flow past an ex- ponentially accelerated isothermal vertical plate with uniform mass diffusion in the presence of heat source. They observed that the velocity of fluid flow decreases with an increase in the magnetic parameter. This is due to a resistive drag force that tends to resist the fluid flow, thus reducing the fluid flow velocity. Ishak (2011) studied the MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. Pavlov (1974) considered the magnetohydrodynamic flow of an incompressible viscous fluid over a linearly stretching surface. Sarpakaya (1961) extended Pavlov’s work to non-Newtonian fluids. Most of the earlier work neglected radiation effects making it necessary to conduct this study. If the polymer extrusion process is placed in a thermally controlled environment, radiation be- comes important. Many researchers have considered the effect of thermal radiation on flows over stretching sheets (see, for instance, Raptis, 1988; Raptis and Perdikis, 1998). Bidin and Nazar (2009) studied the boundary layer flow over an exponentially stretching sheet with thermal radiation. Reddy and Reddy (2011) studied the effect of thermal radiation on magnetohydrodynamic flow due to an exponentially stretching sheet. Elbashbeshy and Dimian (2002) analyzed boundary layer flow in the presence of radiation and heat transfer over the wedge with viscous dissipation. Raptis et al. (2004) studied the effect of thermal radiation on the magnetohydrodynamic flow of a viscous fluid past semi-infinite stationary plate and Hayat et al. (2007) extended the analysis to a second grade fluid.
Analysis has also been directed to viscous dissipation, which may affect the rate of cooling. Viscous dissipation is the energy produced by work done between fluid layers.
This heat energy produced affect the fluid temperature and therefore important to consider.
In most studies in fluid flow, viscous mechanical dissipation is neglected. A number of authors have considered viscous heating effects on Newtonian flows. Zueco (2007) studied
to an exponentially stretching sheet
the dissipation effects on unsteady free convection over vertical porous plate. The influence of viscous dissipation on a grey absorbing-emitting fluid flowing past moving vertical plate has been studied by Suneetha et al. (2009). Kameswaran et al. (2012) studied the effect of the viscous dissipation on magnetohydrodynamic nanofluid flow due to a stretching or shrinking sheet. With regard to viscous dissipation, we note the work done by Partha et al. (2005) who studied the effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. They observed a rapid growth in the non-dimensional skin friction coefficient with the mixed convection parameter. In addition to a magnetic field and thermal radiation, one has to consider the viscous dissipation effects due to frictional heating between fluid layers. Gebhart (1962) and Gebhart and Mollendorf (1969) investigated the effect of viscous dissipation in natural convection processes, they observed that the effect of viscous dissipation is predominant in both vigorous natural convection and mixed convection processes. Vajravelu and Hadjinicalaou (1993) studied the heat transfer characteristics over a stretching surface with viscous dissipation in the presence of internal heat generation or absorption. Abel et al. (2009a) investigated viscous dissipation in a Newtonian fluid; in this study it was revealed that viscous dissipation increase thermal energy in the fluid.
In brief, this section has shown that firstly, it is necessary to study radiation effects on magnetohydrodynamic fluid flow over an exponentially stretching sheet, secondly, consid- ering viscous dissipation on Newtonian fluid requires is required and finally, the use of an appropriate numerical method to solve the governing differential equations. In this chapter we investigate the effects of magnetic, radiation, and viscous dissipation parameters on the fluid flow and heat transfer characteristics of an exponentially stretching sheet. These as- pects will be considered in the formulation of the problem of fluid flow on an exponentially