University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2013

Transcription

1 Lecture 1 3/13/13 University of Washington Department of Chemistry Chemistry 53 Winter Quarter 013 A. Definition of Viscosity Viscosity refers to the resistance of fluids to flow. Consider a flowing liquid constrained between two plates (see Figure. The plates are large. The lower plate is stationary. The upper plate to moved in a plane parallel to the lower plate. It is assumed that an infinitesimally thin layer of liquid sticks to each plate. So a layer of liquid adheres to the lower plate and thus does not move, while a layer of liquid translates with the upper plate. Intervening layers (i.e. lamina of liquid move with velocities that vary as a function of y, the distance from the lower, stationary plate. The velocity gradient v/ y may be thought of as a deformation of the liquid, and is called shear. The shear strain is defined as x dx shear strain = Lim = y 0 y dy (1.1 Shear Stress is defined as the force F pushing the upper plate in the x direction divided by the area A of the upper plate: F σ = shear stress = (1. A In general shear is defined as a force applied to a surface the direction of which is parallel to the surface. The shear is indexed by the direction of the force and the direction of the normal vector to the surface. For the problem above the shear force is in the x direction, and the normal vector is in the y direction so that Fx = σ xyay (1.3 A physical quantity indexed by two directions is called a tensor. Shear is a tensor. Force, which has a single index, is a vector.

2 For many liquids, called Newtonian liquids, the shear stress and strain are related by: Fx d dx d dx x σ xy = = = (1. Ay dt dy dy dt dy Equation 1. states that the stress is proportional to the rate of change of the strain. The constant of proportionality that relates the shear stress to the velocity gradient is the viscosity η: x σxy = η (1.5 dy The viscosity η has units of gm cm -1 sec -1 = dyne s cm - =1 poise=0.1 N s m -. For example, the viscosity of pure water at room temperature is about η=0.01 poise=0.001 N s m - =0.001kg m -1 s -1 The negative sign in 1.5 reflects the fact that as y increases and we get away from the moving plate which applies the shear the velocity decreases. A fluid whose shear stress is proportional to the velocity gradient, and whose viscosity does not change with velocity is said to be Newtonian. If the graph of σ versus the velocity gradient is non-linear the fluid is non-newtonian. Blood is a non-newtonian fluid. Newton s Law of Viscosity: x Fx = η Ay (1.6 dy B. Laminar Flow and Blood Pressure We can learn a lot about viscosity and fluid flow by considering the movement of a viscous fluid through a cylinical tube (see diagram. This study will help us understand problems like the movement of blood through an artery. To treat flow through a cylinder we have to change to cylinical coordinates. The origin is at the center of the tube. The radius of the cylinder is R. the length of the cylinder is L. The velocity goes down as we get closer to the wall of the tube so the shear stress equation has a negative sign in it to reflect this behavior F = η A (1.7 In equation 1.7 A is the surface area of the cylinder of radius r, i.e. A = π rl Now we consider the movement of a lamina of fluid of thickness and a distance r from the wall of the tube. The force of the viscous ag is F = ηa = η( πrl where the length of the cylinder is L.

3 Now a pump applies a pressure P to the fluid and ives it through the cylinder. At steady state the force of the pressure (i.e. (Pπr is exactly opposed by the viscous ag. We imagine a small plate of fluid, radius r and thickness dl. The pressure above it is P and below is P+dP. The net force on this plate is dp( πr = η( πrdl 1 dp or = r η dl The left hand side is integrated from 0 to v and the right hand side is integrated from R (the radius of the tube to position r. The result is the steady state velocity of the lamina of fluid which is R 1 dp 1 dp v= r = ( R r η dl η dl r dp is the pressure op in the pipe per unit length. It is a negative number. The dl dv flux Q = is the volume of liquid that flows through a cross sectional area of dt the cylinder per unit time. Q is given by R π R dp 1 dp Q= v( π r = = 8η dl 8 η / πr dl 0 Q is the rate of flow of fluid through the cylinder and dp/dl is the pressure gradient pushing the fluid along through the cylinder. Because dp/dl is intrinsically negative dp/dl is positive and motions is in the direction opposite to

4 the pressure gradient. The term 8 ηl/ π R is essentially the resistance to the flow. Note the narrower the pipe and/or the more viscous the fluid the greater the resistance to flow. We can put the equation in simpler form if the express the dp P pressure gradient - = in terms of the pressure difference P over a length dl L of pipe L P This equationq =, called Poiseuille s Law, describes the relationship ( 8 ηl/ πr between vessel geometry and pressure that governs blood flow, where P is the pressure op from one point to another point along the length of a blood vessel. The shorter and wider the blood vessel the less resistance to flow. The pressure op from one end to the aorta (R=0.01m to a point 1 cm away (L=0.01 m can be calculated using the fact that blood flows at a rate of 0.08 L/s and the viscosity of blood is 0.0 poise at T=310K. 3 1m ( 8( 0.00Nsm ii ( 0.01m( 8 10 L/ s L ηlq P = = π R ( 3.1( 0.01m 6 = 0.8 N / m = 0.8Pa = 8 10 atm = 0.006mmHg This very small pressure op is expected because the aorta is so large. Blood pressure is measured in terms of systolic pressure, the maximum pressure at the peak of the heart pulse, and the diastolic, the minimum pressure between pulses. In a healthy adult the systolic is 10 mmhg (10 torr=0.158atm and the diastolic is 80 mmhg (=80 torr=0.105atm. The average blood pressure is about 100 mmhg. Note these pressures are expressed as the excess pressure over atmospheric pressure which is 760 mmhg = 1atm. Thus the average systolic and diastolic pressures are actually 880 mmhg=1.158 atm and 80 mmhg=1.105 atm, respectively. As blood enters smaller diameter vessels the pressure ops increase until the venous system is reached, where the blood pressure is barely 10 mmhg. The pressure op is obviously a strong function of R, the radius of the vessel in question. Because of the inverse fourth power dependence, a slight decrease of the blood vessel radius results in a large pressure op and blood flow Q must be maintained by higher blood pressure (hypertension. P The equation Q =. Suppose a liquid flows between two reservoirs ( 8 ηl/ πr whose surfaces are a height h apart. If the liquid flows from the upper to the lower reservoir through a capillary tube of length L the pressure difference due to gravity is P= ρgh. Q is the volume of liquid that flows through the capillary per unit time: Q= V / t. With these definitions: V πr ρgh πr gh = = = η ρ t Aρ t t 8 L 8L V ( η

5 A is called the viscometer constant. It is the same for all liquids for the same viscometer. The easiest way to measure the viscosity of an unknown liquid η 1 is to first measure the viscosity of a known liquid η. Then because A is the same η1 ρ1 t1 for both... = η ρ t C. Viscosity of Macromolecular Solutions: Einstein found that the viscosity of a macromolecular solution can be expressed as a power series on the concentration of the macromolecule: η = η0( 1+ kc 1 + kc + where η is the solution viscosity and η 0 is the viscosity of the pure solvent. Assuming the concentration of macromolecules is small we can retain just the η η0 term linear in concentration: η η0( 1+ kc 1 or = ηsp = kc 1 where η sp is η0 called the specific viscosity. The constant k 1 is of interest because it is dependent on the shape of the η η0 1 ηsp macromolecular solute. This constant is given by = = k1 η0 C C To assure accuracy in the truncation of the power series the viscosity η is ηsp measured as a function of C. The ratio is plotted as a function of C. The C ηsp value of extrapolated to C =0 is called the intrinsic viscosity [η]: C [ η] η = lim sp = k1. C 0 C The intrinsic viscosity is very sensitive to solute shape. Einstein found that k = νv = ν V + Vδ where ν is called a shape factor. For spherical solute ( 1 hyated 1 1 molecules ν=5/. If the molecule is elongated with an axial ratio of P=L/d, the shape factor increases rapidly as a function of P...