Generalized Brinkman Type Dusty Fluid Model for Blood Flow

Introduction

In fluid dynamics, the study of biological fluids (BF) under the influence of a magnetic field is referred to as bio-magnetic fluid dynamics (BFD). This area in fluid dynamics attracted numerous researcher (Carlton et al., 2001; Haik et al., 1999; Liu et al., 2001; Voltairas et al., 2002) due to abundance of application proposed by medical science and bio-engineering which include development of magnetic tracers, targeted transport of drugs using magnetic particles as drug carriers, development of magnetic devices for cell separation, provocation of occlusion of the feeding vessels of cancer tumors, cancer tumor treatment causing magnetic hyperthermia and reduction of bleeding during surgeries (Tzirtzilakis, 2005). The BF exists in all living organisms and their flow is significantly affected by the incidence of the magnetic field. Among different biological fluids, the one electrically conducting fluid is blood. The human blood is composed of plasma and red blood cells (RBC’s) (Boyd, 1961). The RBC’s consists of a high concentration of hemoglobin, the oxide of iron while the magnetic characteristics of blood are due to the state of oxygenation (Sharma et al., 2015).

To study bio-magnetic fluids particularly blood, the mathematical BFD model developed by Haik et al. (1999) is taken into consideration. They suggested that Ferro-hydrodynamics (FHD) is like BFD in which the flow of electrically conducting fluid is influenced by the magnetization of the fluid in the presence of magnetic field and no induce current due to small Reynold number. Hence, in the mathematical model of BFD, the magnetization of the fluid is assumed in the formulation that corresponds to magnetohydrodynamics (MHD) (Haik et al., 2002). Tzirtzilakis (Tzirtzilakis, 2005) considered blood as a Newtonian fluid and develop a mathematical model for blood flow under the effect of magnetic field in a rectangular duct taking into consideration the idea of Haik et al. (1999) . Sharma et al. (Sharma et al., 2015) reported a comprehensive study on MHD blood flow with magnetite particles in a cylindrical tube. Ghasemi et al. (2015) driven analytical and numerical results for blood flow in the porous artery. The blood is assumed as a non-Newtonian third-grade fluid together with heat transfer and thermophoresis effect. Ellahi et al. (2015) considered Prandtl fluid as blood in stenosed arteries with the porous wall. They developed a two-dimensional model for blood flow with slip condition in the cylindrical coordinate system and generated analytical solutions via perturbation technique. Bhatti et al. (2016b) analyzed blood with suspension particles considering endoscopic and slip effect in the non-uniform annulus. In another study, Bhatti et al. (2016a) extended his work by taking Sisko fluid as blood with magnetite titanium nanoparticles in the uniform tube. A theoretical study on MHD blood flow is carried out by Sinha and Shit (2015) in a capillary with the combined effect of radiative heat transfer.