ABSTRACT
The cause and development of many arterial diseases are related to the flow characteristics of blood and the mechanical behavior of the blood vessel walls. The abnormal and unnatural growth in the arterial wall thickness at various locations of the cardiovascular system is medically termed “stenosis.” Its presence in one or more locations restricts the flow of blood through the lumen of the coronary arteries into the heart leading to cardiac ischemia. A systematic study on the rheological and hemodynamic properties of blood and blood flow could play a significant role in the basic understanding, diagnosis, and treatment of many cardiovascular, cerebrovascular, and arterial diseases. It is well known that stenosis (narrowing in the local lumen in the artery) is responsible for many cardiovascular diseases. The high blood pressure and the arterial constriction increase flow velocity and shear stress and decrease pressure substantially leading to thrombus formation. If this disease takes a severe form, it may lead to serious circulatory disorders, morbidity, or even fatality. The fact that the hemodynamic factors play a commendable role in the genesis and growth of the disease has attracted many researchers to explore modern approach and sophisticated mathematical models for investigation on flow through stenotic arteries. In most of the investigations relevant to the domain under discussion, the Newtonian behavior of blood (single-phase homogeneous viscous fluid) was accepted. Sankar et al. suggested that this model of blood is acceptable for high shear rate in case of a flow through narrow arteries of diameter ≤1,000 μm on the basis of experimental observations Bernett and White more [1] suggested that blood behaves like a non-Newtonian fluid under certain conditions. H-B fluid model and Casson fluid models are used in the theoretical investigation of blood flow through narrow arteries. Investigations have mentioned that blood obeys Casson fluid equation at low shear rates when flowing through a tube of diameter of 0.095 mm or less and represent fairly closely occurring flow of blood in arteries. In narrow arteries, at a time, the arterial transport becomes much larger as compared to axial transport, and it contributes to the development of atherosclerotic plaques, greatly reducing the capillary diameter. The problem of flow and diffusion becomes much more difficult through a capillary with stenosis at some region. The theoretical study of Scott Blair and Spanner [2] pointed out that blood obeys the Casson’s equation only in the limited range, except at very high and very low shear rate and that
there is no difference between the Casson’s plots and the Herschel-Bulkley plots of experimental data over the range where the Casson’s plot is valid. Also, he suggested that the assumptions included in the Casson’s equation are unsuitable for cow’s blood and that the Herschel-Bulkley equation represents fairly closely what is occurring in the blood. As the HerschelBulkley equation contains one more parameter than as compared to Casson’s equation, it will be expected that more detailed information about blood properties could be obtained by the use of the Herschel-Bulkley equation. It has been demonstrated by Blair [3] and Copley [4] that the Casson fluid model is adequate for the representation of the simple shear behavior of blood in narrow arteries. Casson [5] examined the validity of Casson fluid model in his studies pertaining to the flow characteristics of blood and reported that at low shear rates the yield stress for blood is nonzero. It has been established by Merrill et al. [6] that the Casson fluid model predicts satisfactorily the flow behaviors of blood in tubes with the diameter of 130-1,000 μm. Charm and Kurland [7] pointed out in their experimental findings that the Casson fluid model could be the best representative of blood when it flows through narrow arteries at low shear rates and that it could be applied to human blood at a wide range of hematocrit and shear rates. Aroesty and Gross [8] developed a Casson fluid theory for pulsatile blood flow through narrow uniform arteries. Chaturani and Samy [9] analyzed the pulsatile flow of Casson fluid through stenosed arteries using the perturbation method. Misra and Chakraborty [10] developed a mathematical model to study unsteady flow of blood through arteries treating blood as a Newtonian viscous incompressible fluid paying due attention to the orthotropic material behavior of the wall tissues. The analysis explored the wall stress in the stenotic region and the shear stress at stenotic throat. The tapered blood vessel segment having a stenosis in its lumen is modeled as a thin elastic tube with a circular cross-section containing a non-Newtonian incompressible fluid. Nanda and Basu Mallik [11] presented a theoretical study for the distribution of axial velocity for blood flow in a branch capillary emerging out of a parent artery at various locations of the branch. The results are computed for various values of r and the angle made by the branch capillary with the parent artery. A mathematical analysis of MHD flow of blood in very narrow capillaries in the presence of stenosis has been studied by Jain et al. [12]. It is assumed that the arterial segment is a cylindrical tube with time-dependent multistenosis. In the proposed investigation, an attempt will be made to deal with a
problem, considering hemodynamic and cardiovascular disorders due to non-Newtonian flow of blood in multistenosed arteries. The application of magneto hydrodynamics principles in medicine and biology is of growing interest in the literature of bio-mathematics [13-15]. Bali and Awasthi [16] suggested that by Lenz’s law, the Lorentz’s force will oppose the motion of conducting fluid. As blood is an electrically conducting fluid, the MHD principles may be used to decelerate the flow of blood in a human arterial system, and thereby, it is useful in the treatment of certain cardiovascular disorders and in the diseases that accelerate blood circulation, such as hemorrhages, hypertension, etc. [17]. This provides us an opportunity to consider the problem of blood flow through a stenosed segment of an artery where the rheology of blood is described by Casson fluid model under the influence of externally applied magnetic field. A quantitative analysis will be done based on numerical computations by taking the different values of material constants and other parameters. The variation of shear stress and skin-friction with different radial distance in the region of the stenosis is presented graphically with respect to externally applied magnetic field on stenosed arterial segment. The qualitative and quantitative changes in the skin-friction, shear stress, and volumetric flow rate at different stages of the growth of the stenosis have also been presented in the presence of an applied magnetic field.
