In this chapter we start by reviewing literature on flow of Casson fluid, its application in blood flow and diffusion of chemically reactive species in fluid flow. We then briefly discuss the literature on the stretching surface, magnetohydrodynamic effects, and porous medium effects already discussed in Chapters 3 and 4. We then end by discussing the numerical
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method used in this chapter.
”Casson fluid can be defined as a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear” (Dash et al., 1992). Casson fluid is classified as a non-Newtonian fluid due to its unique rheological characteristics. These characteristics show shear stress- strain relationships that are significantly different from Newtonian fluids and other non-Newtonian fluids. The study of non-Newtonian fluids has not been thoroughly covered due to the complex representation of their constitutive equations (Makinde, 2009).
The nonlinear Casson’s constitutive equation was derived by Casson (1959), It describes the properties of many polymers over a wide range of shear rates (Vinogradov and Malkin, 1979).
At low shear rates when blood flows through small vessels, the blood flow is described by the Casson fluid model (McDonald, 1974); Shaw et al.,2009). These constitutive equations will be fully described by mathematical equations in section 5.3.
an unsteady stretching surface in porous medium in the presence of a magnetic field
Casson fluid is a non-Newtonian fluid as discussed in Chapter 4. The study of Casson fluid has attracted attention to many researchers due to its application in the field of met- allurgy, food processing, drilling operations and bio-engineering operations. Its application extends to the manufacturing of pharmaceutical products, coal in water, china clay, paints, synthetic lubricants, biological fluids such as synovial fluids, sewage sludge, jelly, tomato sauce, honey, soup and blood due to its contents such as plasma, fibrinogen and protein, making the study of Casson fluid important in fluid dynamics (Pramanik, 2013). Some studies in Casson fluid flow include the work of Mukhopadhyay and Vajravelu (2013) who studied chemical reaction in Casson fluid but did not study heat transfer, Mukhopadhyay et al. (2013) investigated heat transfer on Casson fluid flow over a stretching sheet, Pramanik (2013) also studied heat transfer on Casson fluid flow, but none of these studies investigated momentum, heat and mass transfer of Casson fluid flow.
One of the most important applications of Casson fluid flow is the study of blood flow.
The flow of blood in humans needs to be thoroughly understood as it can be used to save human lives. Recent studies include the work of Sibanda and Shaw (2014) in which a magnetic field is used to direct nanoparticles to cancerous cells. Studies that involved blood as Casson fluid include among others the work of Rohlf and Tenti (2001) who investigated the role of Womerseley number in pulsatile blood flow a theoretical study of the Casson model.
Sankar and Lee (2008) and Sankar and Lee (2010) investigated two-fluid nonlinear model for flow in catheterized blood vessels and two-fluid Casson model for pulsatile blood flow through stenosed arteries respectively. Shaw et al. (2009) studied Pulsatile Casson fluid flow through stenosed bifurcated artery. In relation to blood flow there are other research works that were done in different geometries such as flows in micro-slit channels, slightly curved channels and peristaltic transport (Mernone et al.,2002; Das and Batra, 1993; Ng,2013). The theoretical representations in this chapter also apply to blood flow.
When some chemicals interact with certain fluids, a chemical reaction takes place and affects the flow characteristics. These reactions can result in a constructive reaction in which more solute species are added to the fluid or a destructive reaction in which the species are removed from the fluid. The study of fluid flows with chemical reactions was investigated by Mukhopadhyay and Vajravelu (2013) who studied diffusion of chemically reactive species in
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Casson fluid. Other studies that investigated fluid flow with chemical reactions include the work of Kameswaran et al. (2013) who investigated homogeneous-heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. Shaw et al. (2013) studied homogeneous- heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. Chamkha and Mansour (2010) investigated similarity solutions for unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and chemical reaction effects.
The study of fluid flow on a stretching surface has been studied in Chapter 3 in which we considered an exponentially stretching sheet. In this chapter we consider an unsteady stretching sheet; the velocity of the stretching sheet depends on time and position from the extrusion slit. This model is considered to be a more practical representation of how an actual flow takes place making it necessary to conduct this study. Studies on unsteady stretching surfaces have been done by among others Mukhopadhyay et al. (2013) who investigated Casson fluid flow over an unsteady stretching surface in which the mass transfer equations was not considered. El-Aziz (2013) studied mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation, in which a similar stretching velocity was considered. In this chapter we extended the work of Grubka and Bobba (1985) who investigated heat transfer characteristics of a continuous stretching surface with variable temperature in which we introduced magnetohydrodynamics (MHD), porous medium and chemical reaction effects. Further to the studies mentioned in section 3.1 we consider the work of Sharidan et al. (2006) who studied similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet; Nadeem et al. (2012) exponentially stretching sheet, Nadeem et al. (2014) Maxwell fluid flow past a stretching sheet, Ahmed and Nazar (2011) also studied Casson fluid over a stretching sheet and in their work they assumed that the velocity of the stretching surface is linearly proportional to the distance from fixed origin.
Magnetohydrodynamics (MHD) affect the flow of a fluid as discussed in Chapter 4. A porous medium also affects the momentum, heat and mass transfer as discussed in section 4.1. Consequently, to make sure that this is indeed true, we control the permeability of the porous medium. Earlier work in the effect of varying permeability of a porous medium
an unsteady stretching surface in porous medium in the presence of a magnetic field
in Casson fluid include the work of Dash et al. (1996) who studied Casson fluid flow in a pipe filled with homogeneous porous medium. Nadeem et al. (2013) who considered MHD three dimensional Casson fluid flow past a porous linearly stretching sheet. Ramachandra et al. (2013a) investigated flow and heat transfer of Casson fluid from a horizontal circular cylinder with partial slip in a non-Darcy porous medium, In their work they considered slip conditions at the wall. Tripathi (2013) investigated the transient peristaltic heat flow through a finite porous channel. More recently a study by Pramanik (2014) studied Casson fluid flow and heat transfer past an exponentially porous stretching surface in the presence of thermal radiation.
As can be seen from the literature cited above, it appears that no analysis has yet been published on diffusion of chemically reactive species in Casson fluid flow for the momentum, heat and mass transfer, under the given boundary conditions. The velocity and temperature on the stretching sheet depends on time. The free stream velocity is considered to be zero.
The model considered in this chapter is based on the work put forward by Mukhopadhyay and Vajravelu (2013) and Pramanik (2013) in the study of heat and mass transfer in Casson fluid flow. We extend the models to include heat and mass transfer, magnetohydrodynamic effects and porous medium.
In conclusion, this section has shown that it is necessary to study the effects of magnetic field, porous medium and chemical reaction on Casson fluid flow. In this chapter we inves- tigate the effect of varying unsteadiness parameter, Casson, Schmidt and Prandtl numbers and the reaction rate parameter on the velocity, temperature and concentration profiles with the use of graphical illustrations. The numerical method used to solve the equations is the successive linearization method (SLM) and the results are validated by comparing them to those obtained by the Matlab bvp4c and to other previously published results in the liter- ature. These aspects will be considered in the formulation of the problem of diffusion of chemically reactive species in Casson fluid flow.
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