Fundamental Elements of Hemorheology I. Rheology is the study of the flow and deformation of matter. A. Biorheology is a subfield concerning the flow and deformation of biological materials(e.g. skin and artificial skin). 1. Hemorheology is the study of the flow and deformation of elements within the circulatory system and, particularly, of blood and of the constituents of blood in various states. A closely aligned field is fluid mechanics. Blood is actually an aqueous suspension of cells. There are 3 main groups of cells including erythrocytes(or red cells), leukocytes(white blood cells) and platelets. The function of red cells is to deliver oxygen to the tissues and help remove carbon dioxide. Leukocytes are active in the body’s defense from invasion. Platelets assist in the formation of blot clots. At large scales, blood may be treated as a simple homogeneous fluid while smaller scales may require us to take into account the properties of particular components.

I. Flow in Large Vessels - Blood treated as a simple fluid Primary Factors in Fluid Flow Problems 1. Geometry - shape, dimensions, network structure, boundary (e.g., vasoconstriction) 2. Kinematics, driving force - flow rate, velocity and velocity gradients* (e.g., tachycardia) 3. Material Properties of the Fluid - viscosity (e.g., thrombosis and hemostasis) In the circulatory system, these factors are dynamic and not independent. *In general, the velocity of a moving fluid will change with position. For example, the velocity in the x-direction vx may vary as we move in the y-direction. In symbolic form, vx = vx(y). If we follow the distance moved by fluid elements, variation in velocity will be reflected in the “velocity profile.” Slow Fast The curve indicates the velocity profile. Slow

------------------------- Viscosity--What is it? ------------------------- An ideal fluid in terms of its deformation properties is said to be Newtonian. A Newtonian liquid has constant viscosity. Qualitatively, viscosity is often described as the “thickness” of a liquid or its resistance to flow. More precisely, viscosity is a measure of the ability of a fluid to transmit frictional or sliding forces. ---------- V Fluid flowing in parallel layers or laminae. As one layer slides past the next frictional forces transfer momentum. Wall vx(y) Unit Depth d y x The frictional force per unit area is called the shear stress . For a Newtonian liquid, the stress is proportionate to the rate of deformation or rate of shear. The rate at which the material is being deformed is described by the velocity gradient or shear rate dvx. dy Hence,  =  dvx Newton’s “Law” of Viscosity

--------------------- Viscosity--Units ---------------------  is the common symbol for viscosity with units M L-1 t -1 and dvx/dy is the shear rate with units t -1. Thus,  has the units of M L t -2/ L2 ( or force per unit area). The common unit of viscosity is the poise = 1 g cm -1 s -1 . τ =  dvx dy Note the higher the viscosity, the greater the force that is required to move the material. Similarly, the faster we try to deform some material, the greater the force required. Compare, for example, the behavior of water with that of honey. Water readily pours out of a jar or is easily stirred; honey, on the other hand, is difficult to stir or will pour out of a jar only very slowly. In the sketch above, the force exerted on the wall will depend on the particular fluid. The shear rate is dvx = vx(d) - vx(0) = V   = V dy d - 0 d d