Motion in Viscous Liquids Simplified derivations of the Stokes and Einstein equations Max A. Lauffer The Biophysical Laboratory. Department of Biological Sciences, University of Pittsburgh. Pittsburgh, PA 15260 Newton's Law of Flow Students at intermediate levels are frequently introduced to Stokes's law for the frictional resistance of a sphere moving in a viscous fluid and, perhaps, to the Einstein equation for the viscosity of a liquid suspension of rigid spheres. These equations, and others more complicated for nonspherical or flexible particles, are widely used in biophysics, in biochemistry, and in polymer chemistry. I t would be highly desirable for the student to have some understanding of the physical basis underlying these formulations. Textbooks frequently simply present these equations in final form; some start with eqns. 1and 2 expressed in vector notation, ,,VZii= V P
(1)
v.ir=o
(2)
where v is the vector differential operator, q the viscosity coefficient, P the pressure and ii the velocity of fluid, a vector, and then merely state that the solution under appropriate boundary conditions is Stokes's law or the Einstein equation (I). Equation 1is obtained by summing all of the forces per unit volume which could operate on a flowing fluid. When external forces are absent, when inertial forces are made negligible by restricting consideration to steady low velocities and when the medium is incompressible, only the terms of eqn. (1)remain. Equation (2) is the equation of continuity for an incompressible medium. These equations have been solved by the method of vector analysis. However, for all except those who, with an appropriate background in physics and vector mathematics, pursue their studies to an advanced level, the manner of arriving a t the final results must remain a mystery. The purpose of this communication is to show how some of these imoortant final eauations can be ohtained from Newton's law of flow, using no mathematical language beyond elementary differential and integral calculus. The liquid is treated as a continuous incompressiblemedium, just as is done in vector analysis by satisfying eqn. (2), and the suspended spheres are considered as rigid solids large enough to be macroscopic, that is, large enough so that by contrast the molecular discontinuities of the solvent can be ignored. I t is assumed thoughout that there is no slippage between liquids and solid surfaces. Newton's law of flow, like all other laws of science, is a generalization which adequately describes the behavior of matter under appropriate circumstances. Imagine two parallel planes of area, A, one a distance, z , above the other, with a liquid of viscosity, q, hetween them, as illustrated in Figure
Figure 1. A diagram illustrating Newton's law of flow. The space between the two parallel planes of Area A, zcm apart, is filled with liquid of viscosity caefficient, 7. The lower plane is held stationary while a force F, applied in the direction of the arrow, on the upper plane causes it to move with velocity, u, in the same direction. Intervening liquid exhibits a velocity gradient, duldz.
250
Journal of Chemical Education
1.If the lower plane is held stationary and a horizontal force, F. is aonlied to the uoner. in the steadv state it will move in the dire'ction of the f&e at a constant Gelocity, u. The liquid between the two olanes can be considered to be an infinite numlwr of infinitesimally thin sherts all parallel to the twu danei. When there is n o s l i m ~ a ~between e the planes and the adjacent sheets of liquid, the sheet next to the lower plane will have a velocity of 0 and that next to the upper plane will have a velocity of u in the direction of F. If the velocity is low enough to avoid turbulence, the flow of the intervening sheets of liquid will be plane laminar. Starting with the sheet in contact with the lower stationary plane and proceeding upward, each successive sheet will have a slightly higher velocity than the preceding one, until the sheet adjacent to the upper plane is reached. A uniform velocity gradient, duldz, also called the rate of shear, will be established in the liquid between the two planes. Newton's law of flow states simply that for such a system, F = qAduldz
(3)
The dimensions of q are mll-It-'. Equation (3) is in reality the definition of the coefficient of viscosity, q. I t is alaw because equations derived from it, such as eqn. (5) for the volume rate of flow through acapillary tuhe and eqn. (6) for the torque exerted by a rotating fluid on a stationarv cvlinder. accuratelv describe the behavior of solae fluids. ~ ; a & ~ l eofsuch s fluids are liquids like water and other common laboratow solvents and solutions of small molecules and ions. Such fluids are known as Newtonian fluids. However, other fluids, some pure liquids and also solutions of extremely anisodimensional particles like tobacco mosaic virus, do not flow in accord with equations derived from eqn. (3). They are said to be non-Newtonian. ~~~~~
~
~~
Flow of a Viscous Liquid through a Capillary Tube The simplest application of Newton's law of flow is the derivation of Poiseuille's law for the flow of a viscous liquid through a capillary tube. This derivation is frequently given in textbooks (1,2). Consider a capillary tuhe of length, L, and radius, rl, filled with a liquid with viscosity coefficient, q (Figure 2). When pressure, P, is applied to a core of liquid of
Figure 2. A diagram illustratingflow of liquid of viscosity coelficient. q, through a tube of length. L, and radius, r,, when pressure. P, isappliedtoane end. The symbol, r, is the radius of a ''plug" of liquid extending the entire length of the tube.
radius, r, in the tube, the force causing the plug to flow is nr2P. When steady flow is achieved, this force will be exactly balanced by a frictional force given by Newton's law, eqn. (3), equal to 2nrLqduldr. This statement is in reality an adaptation of eon. (1) . . to the volume of liauid renresented bv the cylindrical core of radius, r , and length, i.Thus, T r i p + 2nqLrduldr = 0,du = -(P/2qL)rdr and u = -Pr2/4qL C. When there is no slippage a t the capillary wall, the value of u is 0 a t r = rl and the constant, C, equals Pr12/4qL. Thus, ~~
~
~
~.
+
u = -(r12 - r2) 4qL
(4)
Equation (4) shows that the velocity is greatest when r is 0 a t the center of the tube and 0 a t the wall where r is rl. The flow can be thought of as the movement of concentric hollow cylinders. each havine a radius with a articular value of r and a thickness, dr, more or less resembling a nest of cork borers. The cvlinder with the smallest radius moves alone- the tuhe with the maximum velocity and each larger hollow cylinder moves more slowly with the one next to the capillary wall having a value of u equal to 0.The distance that each hollow cylinder moves in 1see is u and the volume of liquid associated with it is u.2nrdr. When the value of u from eqn. (4) is introduced, the total volume of liquid flowing through the tuhe in 1sec, V, is given by TP n (r12r - 19dr = ?rPr14 (5)
v =2qL
Figure 3. Cross section of two concentric cylinders of length, L. The radius of me inmr cylinder is r, and mat of the outer is e. The space between die cylinders is filled wim liquid with viscosity coefficient. q. When the outer cylinder rotates with angular velocity, Q, radians per second and the inner one is held stationary. liquid at radius, r, will rotate with angular velocity, w.
811L
0
Equation (5), Poiseuille's law, is the basis for interpreting the results of measurements made in all kinds of capillary viscometers (14). Differentiation of eqn. (4) yields duldr = -PrI2qL. This illustrates the interrelatedness of pressure and velocity gradients specified in vector languageby eqn. (1).~urthermore, the differential equation shows that pressure in a fluid is relieved by viscous flow. The work involved in maintaining such flow is converted into heat. Motion of a Liquld Between Two Concentric Cylinders: Outer Rotating and Inner Stationary (4)
Imagine two concentric cylinders of length, L, the inner with radius, rl, and the outer with radius, r2, illustrated in cross section in Fieure 3. Let the mace between be filled with a liquid with a ;;iscosirs coeffirie~r.7. Rotate the outer cylinder V radian3 per second and hold the with an anmlar V L ~ I I C ~ Iof!! inner cyliider stationary. Assume Newtonian flow and no slippage a t the walls of the cylinders; neglect end effects. At the steady state, an infinite number of cylindrical shells of length, L, and thickness, dr, with radii, r , ranging from r l to r2, will rotate with angular velocities, w, varying between 0 a t r l and R a t rz. The condition for the steady state is that the accelerating moment (torque) on the outer surface of each shell equals the decelerating moment on the inner surface. Thus, the moment, M, is constant for all values of r.' As shown by Figure 4, the rate of shear, replacing duldz in eqn. (31, is r dwldr. Thus, from eqn. (31,
A = 2arL
M = rF Therefore,
=4aqLw+C and - - M 12
When r = rl, 0 = 0. Hence
Figure 4. Diagram explaining the rate of shear in liquid in an apparatus like that illustrated in Fiaure 3. When liouid at radius. r. rotates withanaular velocitv. w. raa ansper second. on one secona 11 w I rnovetnmughanarc equallo rd Wnen them r no snear. lhq.lo at rad ds, r r dr, wll move thrown an arc of length ( r T o r ) IW). Wnen there s shear. 1 qua a! rad us, r r o r , w;ll rotate wth ang-lar velocity, w do, and will movean additional distance. ( r + do dw in one second. The rate of shear will be ( r dr) doidr which approaches rdwtdr as dr approaches zero.
+
+
and
M rl
-M =4~qLw r
W h e n r = rz, w = R. ---=
4 a q ~ ~ (6) r12 ra2 Equation (6) shows how the moment, M, on the inner cylinder of a rotating cylinder viscometer varies with Rand q. M can be evaluated from the angle through which the inner cylinder turns when supported by a torsion wire, thus permitting the calculation of q. This equation is used to interpret data obtained with a Couette viscometer (3,4).
From elementary mechanics, the condition of equilibrium for a simple lever with a force Ft perpendicular to the lever at a distance r z from the fulcrum and with a farce F1 opposite in direction at r l is Far2 = -F1rl, or the moment or torque at r , and ra are equal but opposite. When this is applied to a rotating hollow cylinder of liquid being accelerated on its outer surface at r z and retarded on its inner surface at rr, the steady state condition is likewise that the moment on the outer surfaceis equal to but opposite to that on the inner surface. In this case, however, the retarding moment or torque on the inner surface of the hollow cylinder between r l and ra is equal to hut opposite to the accelerating moment on the liquid inside the hollaw cylinder at rl. Thus, the accelerating moments at rl and at raare the same at the steady state. In genera1,Mis a constant, independent of r.
Volume 58
Number 3
March 1981
25 1
Motion of a Liquid between Two Concentric Cylinders: Inner Rotating and Outer Stationary When the inner cylinder is rotated with angular velocity, Cl, and the outer is held stationary, duldz is replaced by -r dwldr. By steps like those in the previous section, one obtains; Mlr2) = 4avLw C. When r = rz, w = 0. Therefore, C = Ml(rz2)Also, when r = rl, o = Cl.
+
M I-= 4 a q L q 2
(8)
n
Equation (8) is the expression for the friction coefficient,
