Flow through Renal Tubules: An Application through Porous Walled Ducts

Abstract

The study of fluid flow through porous walled channels and ducts has many applications in biomechanics and in industry. In industry, processes such as desalination, reverse osmosis and flow through various tubular nanostructures (see, for example, [1]). However, such fluid flow studies have gained much popularity due to their applications in biology and biomechanics. For example, a few dozens of blood flows through capillaries and arterioles each day. The blood pressure in arterioles is much lower in comparison with that in the main arteries. Therefore, the pressure gradient gives rise to Poiseuille like flow in such structures. However, the Poiseuille like flow cannot be maintained in the renal tubules of a kidney. Kidneys are vital organs in an organism and perform the function of filtration of fluid through the body. Kidneys balance the amount of water in the body apart from getting rid of metabolic waste. Each day, about 200 liters of blood passes through human kidneys in order to filter out about 2 liters of excessive water containing waste products in the form of urine [2]. Blood enters a kidney through renal arteries for purification where the metabolic waste in the blood enters glomerular filtrate (urine). Kidney contains more than a million small filtration units called nephrons. The nephron can structurally be divided into two parts-the Bowman’s capsule and the renal tubule. Absorption of useful substances like glucose, sodium, bicarbonate, potassium, phosphate, calcium and amino acids from the filtrate takes place in the nephron. This reabsorption takes place through small pores among the surface cells on the tube walls. The glomerular filtrate, after the reabsorption process is completed, enters the bladder through ureters for excretion. There have been several mathematical studies on the analysis of fluid flow through renal tubules, both in plane channel geometry and in cylindrical tube geometry. Researchers have assumed several variations in the type of variation that takes place through the tubule walls. Some discussions of renal tubule models were presented by Wesson [3] & Burgen [4]. These studies were theoretical in nature and the authors assumed a constant rate of reabsorption. There have also been studies on a purely mathematical basis that address the analysis of flow through porous walled channels and ducts. These works, however, by Berman [5-9] do not include the application of flow through renal tubules. The idea of these studies is to establish the nature of the flow as a two-dimensional flow. This is caused by a transverse velocity component that arises due to the suction/absorption that takes place at the surface of channel walls. Thus, the velocity pro les of such flows differs greatly from simple Poiseuille flow.