IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 5 Ser. IV (Sep. – Oct. 2020), PP 20-28 www.iosrjournals.org DOI: 10.9790/5728-1605042028 www.iosrjournals.org 20 | Page A mathematical model on two phased capillary blood flow during lung cancer Dheerendra Kumar 1 ,JitendraPratap Singh 2 , Virendra Upadhyay 3 1 Deptt. of Mathematics and Comp. Sci., Rani DurgavatiVishwavidyalaya, Jabalpur, India. 2 Deptt. of Applied Science, Noida Institute of Engineering &Technology, Greater Noida, India. 3 Deptt. Physical Sciences, Mahatma Gandhi ChitrakootGramodayaVishwavidyalaya, India. Abstract: The present piece of work has been devoted to human circulatory system. The structure and function of pulmonary capillary has been discussed so as to develop a mathematical model in tensorial form. The resulting governing equations have been transformed in to cylindrical polar form taking in view the shape of capillaries. The relationship between blood pressure drop and hematocrit has been derived to interpret a clinical data into graphical approach. Keywords: Pulmonary blood flow, Hematocrit, Blood pressure drop, Lung cancer. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 05-10-2020 Date of Acceptance: 19-10-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction The circulatory system of individual (human being) is controlled by homeostatic mechanism which constantly screens (continuously screens) and modifies with conditions in the human body and their conditions. It is as now demonstrated that it clarifies the physical laws which deal with all courses of blood in the circulatory system. Circulatory system divided in two sections pulmonary and systemic flow in human body. Some previous researchers have done few works in this field one phase and two phase but they have considered one or both phase solid particles not fluids. There was we have considered both phase liquids which is one of red blood cells and other is plasma. We have used power law model and suitable its boundary conditions there micro circulation in human pulmonary capillaries blood vessels during lung cancer patient. Fahraeus R. and Lindqvist, 1931, have already considered the pulmonary capillaries are narrow and thin enough. The velocity of blood flow becomes already very slow in the capillaries; the blood flow is possible in capillaries due to Fahreaus- Lindqvits effect. Blood flows in two separate layers when moving through capillaries according to this effect. Fast no red blood cells are present in the plasma layer. The second layer is that of the blood cells that flow along the capillary axis. The successful blood viscosity in this cycle depends on the size of the capillary. The effective viscosity decreases as the radius of capillary and thus the blood flow becomes possible. Core layer may be supposed to be non-Newtonian power law because here the ratio of blood cells is too high in the comparison to plasma [Upadhyay V., 2000]. According to Fung 1984, “the aorta and thoracic arteries have nonlinear stress strain curves. The pulmonary arteries and veins, in contrast, have linear pressure diameter relationship. The capillary blood vessels of the mesentery appear to be rigid-without measurable change in diameter when blood pressure changes over a range of 100 . But the capillary blood vessels in the lung are very distensible in which the variation of the thickness of the capillary sheet with the transmural pressure∆. The pulmonary capillaries are closely knit into a dense network which occupies about 90% of the total space in the interalveolar septa”. II. Hypothesis of two phase blood volume Blood has always held a special position in human though. The quantity of blood in the body is substantial, making up about 7% of the total body weight. Blood function in the transport of blood, oxygen, waste material and hormones in the regulation of temperature and in the control of disease [Upadhyay V., 2000]. According to Bessonovet al., 2016; “The human blood is a concentrated suspension of several formed cellular elements. The human blood cells volume more than 99% of all blood cells and total volume concentration of leukocytes and thrombocytes is only about 1%”. Which i s ignorable; so we have selected two phases where one phase-plasma and another is red blood cells phase. Plasma is a fluid that comprises packets of RBC semi- permeable membranes. Blood activity is about Newtonian at high shear rates, while blood is stressful and non- Newtonian at low shear levels. The presence of blood cells is making blood flow valuable. That effect is directly proportional to the volume of the received blood cells. Let the volume part of blood cell-covered unit volume be