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Viscous Fluids 9 Viscous Fluids 9 In the previous chapter on fluids, we introduced the basic ideas of pressure, fluid flow, the application of conservation of mass and of energy in the form of the continuity equa- tion and of Bernoulli’s equation, respectively, as well as hydrostatics. Throughout those discussions we restricted ourselves to ideal fluids, those that do not exhibit any frictional properties. Often these can be neglected and the results of the previous chapter applied without any modifications whatsoever. Clearly mass is conserved even in the presence of viscous frictional forces and so the continuity equation is a very general result. Real fluids, however, do not conserve mechanical energy, but over time will lose some of this well-ordered energy to heat through frictional losses. In this chapter we consider such behavior, known as viscosity, first in the case of simple fluids such as water. We study the effects of viscosity on the motion of simple fluids and on the motion of suspended bodies, such as macromolecules, in these fluids, with special attention to flow in a cylinder, the most important geometry of flow in biology. The complex nature of blood as a fluid is studied next leading into a description and physics perspective of the human circulatory system. We conclude the chapter with a discussion of surface tension and capillarity, two important surface phenomena in fluids. In Chapter 13 we return to the general notion of the loss of well-ordered energy to heat in the context of thermodynamics. 1. VISCOSITY OF SIMPLE FLUIDS Real fluids are viscous, having internal attractive forces between the molecules so that any relative motion of molecules results in frictional, or drag, forces. The work done by these drag forces, in turn, results in a loss of mechanical energy due to slight heat- ing. We can think of viscosity as a measure of the resistance of a liquid to flowing, so that liquids such as paint or maple syrup have much higher viscosities than water. A quantitative definition of viscosity can be introduced using the example of laminar flow of a liquid between two parallel plates (Figure 9.1), the lower one fixed and the upper one pulled by an external force to move with a constant velocity v parallel to the surface of the plate. Clearly in the absence of drag forces the constant external force would lead to uniform acceleration of the top plate, but due to the drag forces the top plate quickly reaches a steady-state constant velocity. Because the liquid is viscous, it tends to stick to the surfaces of the plates, forming a boundary layer. Therefore the liq- uid layer at the fixed plate is at rest, whereas the liquid layer at the top plate moves with velocity v. For laminar flow, the velocity of the liquid varies linearly in the trans- verse direction (y-direction in Figure 9.1) from 0 to v over the separation distance between the plates of area A. Planar layers of fluid slide over one another. Viscosity can be defined through the relation between the shear stress, or force per unit area F/A, needed to keep the upper plate moving with a constant velocity J. Newman, Physics of the Life Sciences, DOI: 10.1007/978-0-387-77259-2_9, © Springer Science+Business Media, LLC 2008 V ISCOSITY OF S IMPLE F LUIDS 231 y and the rate of variation of the velocity between the plates, ⌬v/⌬y (known as the rate of strain), F ¢v ϭ h , (9.1) v A ¢y where ␩ is the viscosity of the liquid. Contrast this with the stress–strain relation discussed in Chapter 3 for solids where the strain ⌬x/⌬y appeared on the right-hand side and not the rate of strain, appropriate here for fluids. Strain and rate of strain are connected in the usual way because the time rate of change of strain is given by (⌬x/⌬y)/⌬t ϭ (⌬x/⌬t)/⌬y ϭ⌬v/⌬y. The SI unit for viscosity is the Pa-s, but another commonly used unit is the poise ϭ FIGURE 9.1 A fluid sandwiched (P; 1 P 10 Pa-s). Table 9.1 lists viscosities of water and blood. Equation between two plates with the bot- (9.1) can be taken as the definition of viscosity, originally due to Sir Isaac tom plate fixed and the top plate Newton. Fluids that obey this relation are said to be Newtonian fluids. The proportion- moving at a constant velocity v. ality of the shear stress and rate of strain usually holds only at lower strain rates. Water and salt solutions are Newtonian, whereas blood, whose behavior does not follow Equation (9.1), is said to be a non-Newtonian fluid and is discussed in the next section. Table 9.1 Viscosities of Water and Blood Fluid Temperature Viscosity (10Ϫ3 Pa-s) Water 0 1.8 20 1.0 37 0.7 Whole blooda 37 4.0 Blood plasma 37 1.5 a Varies greatly with hematocrit, or red blood cell content. When a solid is put under shear stress, with an external force applied in a partic- ular direction, it deforms and, for small stresses F/A, the strain, or response of the solid, is proportional to the stress. Once the stress is removed, the solid returns to its original shape (unless it has some plasticity, in which case it may flow). In a Newtonian liquid, however, a constant applied shear stress results in a constant rate of strain (Equation (9.1)) rather than constant strain. The larger the rate of strain, meaning the more abruptly the velocity changes with transverse distance, the greater the viscous force, and in turn, the greater the applied shear stress needed to keep the top plate moving at the same constant velocity. At higher shear stress there are devi- ations from this relation, and at still higher stress, turbulence will occur. Example 9.1 A sheet of plywood is covered with a 1 mm thick layer of tile adhesive and a square piece of ceramic tile measuring 30 cm on a side is placed on it. If a force of 10 N is applied parallel to the surface, find the velocity with which the tile slides. Assume laminar Newtonian flow and use a viscosity of 50 Pa-s for the adhesive. Solution: We first calculate the stress as F/A ϭ 10/(.3)2 ϭ 110 N/m2. Dividing this stress by the adhesive viscosity, the rate of strain is found to be 2.2 sϪ1, so that the velocity of the tile is given as ¢v v ϭ y ϭ (2.2 sϪ1)(1 mm) ϭ 2.2 mm/s. ¢y FIGURE 9.2 Laminar capillary flow showing a concentric layer of fluid that flows at the same velocity The capillary tube is a very common geometry for fluid flow in biology. It is along the length of the tube. relevant for blood flow, for example, as well as for viscometry, the methodology of 232 V ISCOUS F LUIDS viscosity measurement. When a liquid flows through a tube 1 without obstacles, the flow at low velocities is laminar with lay- 0.75 ers of liquid in concentric cylinders (Figure 9.2). The outermost 0.5 layer is the boundary layer that remains at rest and the fastest v/vo flowing liquid lies at the center of the tube. The actual velocity 0.25 profile across the tube is parabolic as indicated in Figure 9.3. 0 The velocity varies across the capillary tube; thus in order to 0 0.2 0.4 0.6 0.8 1 find the volume flow rate, Q (ϭ vA when the velocity was r/ro assumed uniform in the absence of viscosity), an average must FIGURE 9.3 The velocity profile be calculated across the cross-sectional area. This was first done in 1835 by across a capillary tube of radius ro. Poiseuille, a French physician interested in blood flow (the viscosity unit poise is taken from his name), who found pPr4 Q ϭ , (9.2) 8hL where P/L is the applied pressure per unit length of the tube and r is the tube radius. Equation (9.2) is known as Poiseuille’s law. If we rewrite this equation in the form 8hL ¢P ϭ Q, (9.3) a pr4 b where we write ⌬P as the pressure difference across the tube of length L, then we can interpret the equation as follows. For a given ⌬P across the tube, the resulting flow Q depends on the resistive term in parentheses. The larger this term, the slower the flow rate is for a given applied pressure. With a constant resistive term (fixed tube length, radius, and fluid viscosity), the greater the pressure difference acting on the liquid, the greater is the expected flow rate. A longer tube or larger viscosity provides a greater resistance to flow as might be intuitively expected. The very strong dependence of the resistive term on tube radius rϪ4 is surprising and extremely significant in controlling the flow rate of a liquid in a capillary tube. The resistance to fluid flow increases dramatically as the tube radius gets smaller. This can lead to important effects in the flow of blood in arteries because a partially clogged artery will require a much higher pressure differential to supply the same fluid flow rate. Example 9.2 In giving a transfusion, blood drips from a sealed storage bag with a 1 m pressure head through capillary tubing of 2 mm inside diameter, passing through a hypodermic needle that is 4 cm long and has an inside diameter of 0.5 mm.
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