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LECTURE NOTES ON APPLIED MATHEMATICS Methods and Models John K. Hunter Department of Mathematics University of California, Davis June 17, 2009 Copyright c 2009 by John K. HunterContents Lecture 1. Introduction 1 1. Conservation laws 1 2. Constitutive equations 2 3. The KPP equation 3 Lecture 2. Dimensional Analysis, Scaling, and Similarity 11 1. Systems of units 11 2. Scaling 12 3. Nondimensionalization 13 4. Fluid mechanics 13 5. Stokes formula for the drag on a sphere 18 6. Kolmogorov's 1941 theory of turbulence 22 7. Self-similarity 25 8. The porous medium equation 27 9. Continuous symmetries of di erential equations 33 Lecture 3. The Calculus of Variations 43 1. Motion of a particle in a conservative force eld 44 2. The Euler-Lagrange equation 49 3. Newton's problem of minimal resistance 51 4. Constrained variational principles 56 5. Elastic rods 57 6. Buckling and bifurcation theory 61 7. Laplace's equation 69 8. The Euler-Lagrange equation 73 9. The wave equation 76 10. Hamiltonian mechanics 76 11. Poisson brackets 79 12. Rigid body rotations 80 13. Hamiltonian PDEs 86 14. Path integrals 88 Lecture 4. Sturm-Liouville Eigenvalue Problems 95 1. Vibrating strings 96 2. The one-dimensional wave equation 99 3. Quantum mechanics 103 4. The one-dimensional Schr odinger equation 106 5. The Airy equation 116 6. Dispersive wave propagation 118 7. Derivation of the KdV equation for ion-acoustic waves 121 iii 8. Other Sturm-Liouville problems 127 Lecture 5. Stochastic Processes 129 1. Probability 129 2. Stochastic processes 136 3. Brownian motion 141 4. Brownian motion with drift 148 5. The Langevin equation 152 6. The stationary Ornstein-Uhlenbeck process 157 7. Stochastic di erential equations 160 8. Financial models 167 Bibliography 173LECTURE 1 Introduction The source of all great mathematics is the special case, the con- crete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same 1 as a small and concrete special case. We begin by describing a rather general framework for the derivation of PDEs that describe the conservation, or balance, of some quantity. 1. Conservation laws We consider a quantityQ that varies in space,x, and time, t, with density u(x;t), ux q (x;t), and source density  (x;t). For example, ifQ is the mass of a chemical species di using through a stationary medium, we may takeu to be the density,q the mass ux, andf the mass rate per unit volume at which the species is generated. For simplicity, we suppose that u(x;t) is scalar-valued, but exactly the same considerations would apply to a vector-valued density (leading to a system of equa- tions). 1.1. Integral form The conservation ofQ is expressed by the condition that, for any xed spatial region , we have Z Z Z d (1.1) udx = qndS + dx: dt Here, is the boundary of , n is the unit outward normal, and dS denotes integration with respect to surface area. Equation (1.1) is the integral form of conservation ofQ. It states that, for any region , the rate of change of the total amount ofQ in is equal to the rate at whichQ ows into through the boundary plus the rate at whichQ is generated by sources inside . 1.2. Di erential form Bringing the time derivative in (1.1) inside the integral over the xed region , and using the divergence theorem, we may write (1.1) as Z Z u dx = (rq +) dx t 1 P. Halmos. 12 Since this equation holds for arbitrary regions , it follows that, for smooth func- tions, (1.2) u =rq +: t Equation (1.2) is the di erential form of conservation ofQ. When the source term  is nonzero, (1.2) is often called, with more accuracy, a balance law forQ, rather than a conservation law, but we won't insist on this distinction. 2. Constitutive equations The conservation law (1.2) is not a closed equation for the density u. Typically, we supplement it with constitutive equations that relate the ux q and the source density  to u and its derivatives. While the conservation law expresses a gen- eral physical principle, constitutive equations describe the response of a particular system being modeled. Example 1.1. If the ux and source are pointwise functions of the density, q =f(u);  =g(u); then we get a rst-order system of PDEs u +rf(u) =g(u): t 2 For example, in one space dimension, if g(u) = 0 and f(u) = u =2, we get the inviscid Burgers equation   1 2 u + u = 0: t 2 x This equation is a basic model equation for hyperbolic systems of conservation laws, such as the compressible Euler equations for the ow of an inviscid compressible uid 47. Example 1.2. Suppose that the ux is a linear function of the density gradient, (1.3) q =Aru; where A is a second-order tensor, that is a linear map between vectors. It is represented by an nn matrix with respect to a choice of n basis vectors. Then, if  = 0, we get a second order, linear PDE for u(x;t) (1.4) u =r (Aru): t Examples of this constitutive equation include: Fourier's law in heat conduction (heat ux is a linear function of temperature gradient); Fick's law ( ux of solute is a linear function of the concentration gradient); and Darcy's law ( uid velocity in a porous medium is a linear function of the pressure gradient). It is interesting to note how old each of these laws is: Fourier (1822); Fick (1855); Darcy (1855). The conductivity tensor A in (1.3) is usually symmetric and positive-de nite, in which case (1.4) is a parabolic PDE; the corresponding PDE for equilibrium density distributions u(x) is then an elliptic equation r (Aru) = 0: In general, the conductivity tensor may depend uponx in a nonuniform system, and on u in non-linearly di usive systems. While A is almost always symmetric,LECTURE 1. INTRODUCTION 3 it need not be diagonal in an anisotropic system. For example, the heat ux in a crystal lattice or in a composite medium made up of alternating thin layers of copper and asbestos is not necessarily in the same direction as the temperature gradient. For a uniform, isotropic, linear system, we have A = I where  is a positive constant, and then u(x;t) satis es the heat, or di usion, equation u =u: t Equilibrium solutions satisfy Laplace's equation u = 0: 3. The KPP equation In this section, we discuss a speci c example of an equation that arises as a model in population dynamics and genetics. 3.1. Reaction-di usion equations If q =ru and  =f(u) in (1.2), we get a reaction-di usion equation u =u +f(u): t Spatially uniform solutions satisfy the ODE u =f(u); t which is the `reaction' equation. In addition, di usion couples together the solution at di erent points. Such equations arise, for example, as models of spatially nonuniform chemical reactions, and of population dynamics in spatially distributed species. The combined e ects of spatial di usion and nonlinear reaction can lead to the formation of many di erent types of spatial patterns; the spiral waves that occur in Belousov-Zabotinski reactions are one example. One of the simplest reaction-di usion equations is the KPP equation (or Fisher equation) (1.5) u =u +ku(au): t xx Here, , k, a are positive constants; as we will show, they may be set equal to 1 without loss of generality. Equation (1.5) was introduced independently by Fisher 22, and Kolmogorov, Petrovsky, and Piskunov 33 in 1937. It provides a simple model for the dispersion of a spatially distributed species with population densityu(x;t) or, in Fisher's work, for the advance of a favorable allele through a spatially distributed population. 3.2. Maximum principle According to the maximum principle, the solution of (1.5) remains nonnegative if the initial data u (x) =u(x; 0) is non-negative, which is consistent with its use as 0 a model of population or probability. The maximum principle holds because ifu rst crosses from positive to negative values at timet at the pointx , and ifu(x;t) has a nondegenerate minimum atx , 0 0 0 then u (x ;t ) 0. Hence, from (1.5), u (x ;t ) 0, so u cannot evolve forward xx 0 0 t 0 0 in time into the region u 0. A more careful argument is required to deal with degenerate minima, and with boundaries, but the conclusion is the same 18, 42. A similar argument shows that u(x;t) 1 for all t 0 if u (x) 1. 04 Remark 1.3. A forth-order di usion equation, such as u =u +u(1u); t xxxx does not satisfy a maximum principle, and it is possible for positive initial data to evolve into negative values. 3.3. Logistic equation Spatially uniform solutions of (1.5) satisfy the logistic equation (1.6) u =ku(au): t This ODE has two equilibrium solutions at u = 0, u =a. The solution u = 0 corresponds to a complete absence of the species, and kat is unstable. Small disturbances grow initially like u e . The solution u = a 0 corresponds to the maximum population that can be sustained by the available resources. It is globally asymptotically stable, meaning that any solution of (1.6) with a strictly positive initial value approaches a as t1. Thus, the PDE (1.5) describes the evolution of a population that satis es lo- gistic dynamics at each point of space coupled with dispersal into regions of lower population. 3.4. Nondimensionalization Before discussing (1.5) further, we simplify the equation by rescaling the variables to remove the constants. Let  u =Uu;  x =Lx;  t =Tt where U, L, T are arbitrary positive constants. Then 1 1 = ; = :  x Lx  t T t  It follows that u  (x;  t) satis es     T a u  = u  + (kTU)u  u  :  t x x  2 L U Therefore, choosing r 1  (1.7) U =a; T = ; L = ; ka ka and dropping the bars, we nd that u(x;t) satis es (1.8) u =u +u(1u): t xx Thus, in the absence of any other parameters, none of the coecients in (1.5) are essential. If we consider (1.5) on a nite domain of length `, then the problem depends in an essential way on a dimensionless constant R, which we may write as 2 ka` R = :  p We could equivalently use 1=R or R, or some other expression, instead of R. From (1.7), we have R =T =T where T =T is a timescale for solutions of the reaction d r r 2 equation (1.6) to approach the equilibrium value a, and T = ` = is a timescale d for linear di usion to signi cantly in uence the entire length ` of the domain. The qualitative behavior of solutions depends on R.LECTURE 1. INTRODUCTION 5 When dimensionless parameters exist, we have a choice in how we de ne dimen- sionless variables. For example, on a nite domain, we could nondimensionalize as p above, which would give (1.8) on a domain of length R. Alternatively, we might prefer to use the length` of the domain to nondimensionalize lengths. In that case, the nondimensionalized domain has length 1, and the nondimensionalized form of (1.5) is 1 u = u +u (1u): t xx R We get a small, or large, dimensionless di usivity if the di usive timescale is large, or small, respectively, compared with the reaction time scale. Somewhat less obviously, even on in nite domains additional lengthscales may be introduced into a problem by initial data u(x; 0) =u (x): 0 Using the variables (1.7), we get the nondimensionalized initial condition u  (x;  0) =u  (x ); 0 where 1 u  (x ) = u (Lx ): 0 0 a Thus, for example, ifu has a typical amplitudea and varies over a typical length- 0 scale of `, then we may write   x  u (x) =af 0 `  where f is a dimensionless function. Then   p  u  (x ) =f Rx  ; 0 and the evolution of the solution depends upon whether the initial data varies rapidly, slowly, or on the same scale as the reaction-di usion length scale L. 3.5. Traveling waves One of the principal features of the KPP equation is the existence of traveling waves which describe the invasion of an unpopulated region (or a region whose population does not possess the favorable allele) from an adjacent populated region. A traveling wave is a solution of the form (1.9) u(x;t) =f(xct) where c is a constant wave speed. This solution consists of a xed spatial pro le that propagates with velocity c without changing its shape. For de niteness we assume that c 0. The case c 0 can be reduced to this one by a re ection x7x, which transforms a right-moving wave into a left-moving wave. Use of (1.9) in (1.8) implies that f(x) satis es the ODE 00 0 (1.10) f +cf +f(1f) = 0: The equilibria of this ODE are f = 0, f = 1. Note that (1.10) describes the spatial dynamics of traveling waves, whereas (1.6) describes the temporal dynamics of uniform solutions. Although these equations have the same equilibrium solutions, they are di erent ODEs (for example, one6 is second order, and the other rst order) and the stability of their equilibrium solutions means di erent things. The linearization of (1.10) at f = 0 is 00 0 f +cf +f = 0: The characteristic equation of this ODE is 2  +c + 1 = 0 with roots n o p 1 2  = c c 4 : 2 Thus, the equilibriumf = 0 is a stable spiral point if 0c 2, a degenerate stable node if c = 2, and a stable node if 2c1. The linearization of (1.10) at f = 1 is 00 0 f +cf f = 0: The characteristic equation of this ODE is 2  +c 1 = 0 with roots n o p 1 2  = c c + 4 : 2 Thus, the equilibrium f = 1 is a saddlepoint. As we will show next, for any 2c1 there is a unique positive heteroclinic orbit F (x) connecting the unstable saddle point at f = 1 to the stable equilibrium at f = 0, meaning that F (x) 1 as x1; F (x) 0 as x1: These right-moving waves describe the invasion of the state u = 0 by the state u = 1. Re ecting x7x, we get a corresponding family of left-moving traveling waves with1c2. Since the traveling wave ODE (1.10) is autonomous, if F (x) is a solution then so is F (xx ) for any constant x . This solution has the same orbit as F (x), 0 0 and corresponds to a traveling wave of the same velocity that is translated by a constant distance x . 0 There is also a traveling wave solution for 0 c 2 However, in that case the solution becomes negative near 0 since f = 0 is a spiral point. This solution is therefore not relevant to the biological application we have and mind. Moreover, by the maximum principle, it cannot arise from nonnegative initial data. The traveling wave most relevant to the applications considered above is, per- haps, the positive one with the slowest speed (c = 2); this is the one that describes the mechanism of di usion from the populated region into the unpopulated one, followed by logistic growth of the di usive perturbation. The faster waves arise be- cause of the growth of small, but nonzero, pre-existing perturbations of the unstable state u = 0 ahead of the wavefront. The linear instability of the state u = 0 is arguably a defect of the model. If there were a threshold below which a small population died out, then this depen- dence of the wave speed on the decay rate of the initial data would not arise.LECTURE 1. INTRODUCTION 7 3.6. The existence of traveling waves Let us discuss the existence of positive traveling waves in a little more detail. p If c = 5= 6, there is a simple explicit solution for the traveling wave 1: 1 F (x) = :   p 2 x= 6 1 +e Although there is no similar explicit solution for general values of c, we can show the existence of traveling waves by a qualitative argument. 0 Writing (1.10) as a rst order system of ODEs for (f;g), where g =f , we get 0 f =g; (1.11) 0 g =f(1f)cg: For c 2, we choose 0  1 such that   p 1 1 2 + =c; = c c 4 : 2 Then, on the line g = f with 0f 1, the trajectories of the system satisfy 0 dg g f(1f) 1f 1 = =c =c + c + = : 0 df f g 0 Since f 0 for g 0, and dg=df , the trajectories of the ODE enter the triangular region D =f(f;g) : 0f 1; f g 0g: 0 0 Moreover, since g 0 on g = 0 when 0 f 1, and f 0 on f = 1 when g 0, the regionD is positively invariant (meaning that any trajectory that starts in the region remains in the region for all later times). The linearization of the system (1.11) at the xed point (f;g) = (1; 0) is      0 f 0 1 f = : 0 g 1 c g The unstable manifold of (1; 0), with corresponding eigenvalue   p 1 2  = c + c + 4 0; 2 is in the direction   1 r = :  The corresponding trajectory below the f-axis must remain in D, and since D contains no other xed points or limit cycles, it must approach the xed point (0; 0) as x1. Thus, a nonnegative traveling wave connecting f = 1 to f = 0 exists for every c 2. 3.7. The initial value problem Consider the following initial value problem for the KPP equation u =u +u(1u); t xx u(x; 0) =u (x); 0 u(x;t) 1 as x1; u(x;t) 0 as x1:8 Kolmogorov, Petrovsky and Piskunov proved that if 0 u (x) 1 is any initial 0 data that is exactly equal to 1 for all suciently large negativex, and exactly equal to 0 for all suciently large positive x, then the solution approaches the traveling wave with c = 2 as t1. This result is sensitive to a change in the spatial decay rate of the initial data into the unstable state u = 0. Speci cally, suppose that x u (x)Ce 0 as x1, where is some positive constant (and C is nonzero). If  1, then the solution approaches a traveling wave of speed 2; but if 0 1, meaning that the initial data decays more slowly, then the solution approaches a traveling wave of speed 1 c( ) = + : This is the wave speed of the traveling wave solution of (1.10) that decays to f = 0 x at the rate fCe .LECTURE 1. INTRODUCTION 9 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u Figure 1. The phase plane for the KPP traveling wave, showing the heteroclinic orbit connecting (1; 0) to (0; 0) (courtesy of Tim Lewis). u x10 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 x Figure 2. The spatial pro le of the traveling wave. uLECTURE 2 Dimensional Analysis, Scaling, and Similarity 1. Systems of units The numerical value of any quantity in a mathematical model is measured with respect to a system of units (for example, meters in a mechanical model, or dollars in a nancial model). The units used to measure a quantity are arbitrary, and a change in the system of units (for example, from meters to feet) cannot change the model. A crucial property of a quantitative system of units is that the value of a dimensional quantity may be measured as some multiple of a basic unit. Thus, a change in the system of units leads to a rescaling of the quantities it measures, and the ratio of two quantities with the same units does not depend on the particular choice of the system. The independence of a model from the system of units used to measure the quantities that appear in it therefore corresponds to a scale-invariance of the model. Remark 2.1. Sometimes it is convenient to use a logarithmic scale of units instead of a linear scale (such as the Richter scale for earthquake magnitudes, or the stellar magnitude scale for the brightness of stars) but we can convert this to an underlying linear scale. In other cases, qualitative scales are used (such as the Beaufort wind force scale), but these scales (\leaves rustle" or \umbrella use becomes dicult") are not susceptible to a quantitative analysis (unless they are converted in some way into a measurable linear scale). In any event, we will take connection between changes in a system of units and rescaling as a basic premise. A fundamental system of units is a set of independent units from which all other units in the system can be derived. The notion of independent units can be made precise in terms of the rank of a suitable matrix 7, 10 but we won't give the details here. The choice of fundamental units in a particular class of problems is not unique, but, given a fundamental system of units, any other derived unit may be constructed uniquely as a product of powers of the fundamental units. Example 2.2. In mechanical problems, a fundamental set of units is mass, length, time, or M, L, T , respectively, for short. With this fundamental system, velocity 1 2 V = LT and force F = MLT are derived units. We could instead use, say, force F , length L, and time T as a fundamental system of units, and then mass 1 2 M =FL T is a derived unit. Example 2.3. In problems involving heat ow, we may introduce temperature (measured, for example, in Kelvin) as a fundamental unit. The linearity of temper- ature is somewhat peculiar: although the `zeroth law' of thermodynamics ensures that equality of temperature is well de ned, it does not say how temperatures can 1112 be `added.' Nevertheless, empirical temperature scales are de ned, by convention, to be linear scales between two xed points, while thermodynamics temperature is an energy, which is additive. Example 2.4. In problems involving electromagnetism, we may introduce current as a fundamental unit (measured, for example, in Amp eres in the SI system) or charge (measured, for example, in electrostatic units in the cgs system). Unfortu- nately, the ocially endorsed SI system is often less convenient for theoretical work than the cgs system, and both systems remain in use. Not only is the distinction between fundamental and derived units a matter of choice, or convention, the number of fundamental units is also somewhat arbitrary. For example, if dimensional constants are present, we may reduce the number of fundamental units in a given system by setting the dimensional constants equal to xed dimensionless values. Example 2.5. In relativistic mechanics, if we use M, L, T as fundamental units, 8 1 then the speed of light c is a dimensional constant (c = 3 10 ms in SI-units). Instead, we may set c = 1 and use M, T (for example) as fundamental units. This means that we measure lengths in terms of the travel-time of light (one nanosecond being a convenient choice for everyday lengths). 2. Scaling Let (d ;d ;:::;d ) denote a fundamental system of units, such as (M;L;T ) in 1 2 r mechanics, and a a quantity that is measurable with respect to this system. Then the dimension of a, denoted a, is given by 1 2 r (2.1) a =d d :::d 1 2 r for suitable exponents ( ; ;:::; ). 1 2 r Suppose that (a ;a ;:::;a ) denotes all of the dimensional quantities appearing 1 2 n in a particular model, including parameters, dependent variables, and independent variables. We denote the dimension of a by i 1;i 2;i r;i (2.2) a =d d :::d : i 1 2 r The invariance of the model under a change in units d 7 d implies that it j j j is invariant under the scaling transformation 1;i 2;i r;i a   ::: a i = 1;:::;n i i 1 2 r for any  ;::: 0. 1 r Thus, if a =f (a ;:::;a ) 1 n is any relation between quantities in the model with the dimensions in (2.1) and (2.2), then f must have the scaling property that  1;1 2;1 1;n 2;n 1 2 r r;1 r;n   ::: f (a ;:::;a ) =f   ::: a ;:::;  ::: a : 1 n 1 n 1 2 r 1 2 r 1 2 r A particular consequence of this invariance is that any two quantities that are equal must have the same dimension (otherwise a change in units would violate the equality). This fact is often useful in nding the dimension of some quantity.LECTURE 2. DIMENSIONAL ANALYSIS, SCALING, AND SIMILARITY 13 Example 2.6. According to Newton's second law, force = rate of change of momentum with respect to time: Thus, if F denotes the dimension of force and P the dimension of momentum, 2 then F = P=T . Since P = MV = ML=T , we conclude that F = ML=T (or mass acceleration). 3. Nondimensionalization Scale-invariance implies that we can reduce the number of quantities appearing in a problem by introducing dimensionless quantities. Suppose that (a ;:::;a ) are a set of quantities whose dimensions form a fun- 1 r damental system of units. We denote the remaining quantities in the model by (b ;:::;b ), where r +m = n. Then, for suitable exponents ( ;:::; ) deter- 1 m 1;i r;i mined by the dimensions of (a ;:::;a ) and b , the quantity 1 r i b i  = i 1;i r;i a :::a r 1 is dimensionless, meaning that it is invariant under the scaling transformations induced by changes in units. A dimensionless parameter  can typically be interpreted as the ratio of two i quantities of the same dimension appearing in the problem (such as a ratio of lengths, times, di usivities, and so on). In studying a problem, it is crucial to know the magnitude of the dimensionless parameters on which it depends, and whether they are small, large, or roughly of the order one. Any dimensional equation a =f(a ;:::;a ;b ;:::;b ) 1 r 1 m is, after rescaling, equivalent to the dimensionless equation  =f(1;:::; 1;  ;:::;  ): 1 m Thus, the introduction of dimensionless quantities reduces the number of variables in the problem by the number of fundamental units. This fact is called the `Bucking- ham Pi-theorem.' Moreover, any two systems with the same values of dimensionless parameters behave in the same way, up to a rescaling. 4. Fluid mechanics To illustrate the ideas of dimensional analysis, we describe some applications in uid mechanics. Consider the ow of a homogeneous uid with speedU and length scaleL. We restrict our attention to incompressible ows, for which U is much smaller that the speed of sound c in the uid, meaning that the Mach number 0 U M = c 0 1 is small. The sound speed in air at standard conditions is c = 340 ms . The 0 incompressibility assumption is typically reasonable when M 0:2.14 The physical properties of a viscous, incompressible uid depend upon two dimensional parameters, its mass density  and its (dynamic) viscosity . The 0 dimension of the density is M  = : 0 3 L The dimension of the viscosity, which measures the internal friction of the uid, is given by M (2.3)  = : LT To derive this result, we explain how the viscosity arises in the constitutive equation of a Newtonian uid relating the stress and the strain rate. 4.1. The stress tensor The stress, or force per unit area, t exerted across a surface by uid on one side of the surface on uid on the other side is given by t = Tn where T is the Cauchy stress tensor and n is a unit vector to the surface. It is a fundamental result in continuum mechanics, due to Cauchy, that t is a linear function ofn; thus, T is a second-order tensor 25. The sign ofn is chosen, by convention, so that ifn points into uid on one side A of the surface, and away from uid on the other side B, then Tn is the stress exerted by A on B. A reversal of the sign of n gives the equal and opposite stress exerted by B on A. The stress tensor in a Newtonian uid has the form (2.4) T =pI + 2D where p is the uid pressure,  is the dynamic viscosity, I is the identity tensor, and D is the strain-rate tensor  1 D = ru +ru : 2 Thus, D is the symmetric part of the velocity gradientru. In components,   u u i j T =p + + ij ij x x j i where  is the Kronecker-, ij  1 if i =j,  = ij 0 if i6=j. Example 2.7. Newton's original de nition of viscosity (1687) was for shear ows. The velocity of a shear ow with strain rate  is given by u =x e 2 1 th where x = (x ;x ;x ) and e is the unit vector in the i direction. The velocity 1 2 3 i gradient and strain-rate tensors are 0 1 0 1 0  0 0  0 1 A A ru = 0 0 0 ; D =  0 0 : 2 0 0 0 0 0 0LECTURE 2. DIMENSIONAL ANALYSIS, SCALING, AND SIMILARITY 15 The viscous stress t = 2Dn exerted by the uid in x 0 on the uid in x 0 v 2 2 across the surfacex = 0, with unit normaln =e pointing into the regionx 0, 2 2 2 ist =e . (There is also a normal pressure forcet =pe .) Thus, the frictional v 1 p 1 viscous stress exerted by one layer of uid on another is proportional the strain rate  and the viscosity . 4.2. Viscosity The dynamic viscosity is a constant of proportionality that relates the strain-rate to the viscous stress. Stress has the dimension of force/area, so ML 1 M T = = : 2 2 2 T L LT The strain-rate has the dimension of a velocity gradient, or velocity/length, so L 1 1 D = = : T L T Since D has the same dimension as T, we conclude that  has the dimension in (2.3). The kinematic viscosity  of the uid is de ned by   = :  0 It follows from (2.3) that  has the dimension of a di usivity, 2 L  = : T The kinematic viscosity is a di usivity of momentum; viscous e ects lead to the p di usion of momentum in time T over a length scale of the order T . 2 The kinematic viscosity of water at standard conditions is approximately 1 mm =s, meaning that viscous e ects di use uid momentum in one second over a distance of the order 1 mm. The kinematic viscosity of air at standard conditions is approxi- 2 mately 15 mm =s; it is larger than that of water because of the lower density of air. These values are small on every-day scales. For example, the timescale for viscous 6 di usion across room of width 10 m is of the order of 6 10 s, or about 77 days. 4.3. The Reynolds number The dimensional parameters that characterize a uid ow are a typical velocity U and lengthL, the kinematic viscosity, and the uid density . Their dimensions 0 are 2 L L M U = ; L =L;  = ;  = : 0 3 T T L We can form a single independent dimensionless parameter from these dimensional parameters, the Reynolds number UL (2.5) R = :  As long as the assumptions of the original incompressible model apply, the behavior of a ow with similar boundary and initial conditions depends only on its Reynolds number.16 The inertial term in the Navier-Stokes equation has the order of magnitude   2  U 0  uru =O ; 0 L while the viscous term has the order of magnitude   U u =O : 2 L The Reynolds number may therefore be interpreted as a ratio of the magnitudes of the inertial and viscous terms. The Reynolds number spans a large range of values in naturally occurring ows, 20 5 from 10 in the very slow ows of the earth's mantle, to 10 for the motion of 6 10 bacteria in a uid, to 10 for air ow past a car traveling at 60 mph, to 10 in some large-scale geophysical ows. Example 2.8. Consider a sphere of radius L moving through an incompressible uid with constant speed U. A primary quantity of interest is the total drag force D exerted by the uid on the sphere. The drag is a function of the parameters on which the problem depends, meaning that D =f(U;L; ;): 0 2 The drag D has the dimension of force (ML=T ), so dimensional analysis implies that   UL 2 2 D = U L F : 0  Thus, the dimensionless drag D (2.6) =F (R) 2 2  U L 0 is a function of the Reynolds number (2.5), and dimensional analysis reduces the problem of nding a function f of four variables to nding a function F of one variable. The function F (R) has a complicated dependence on R which is dicult to determine theoretically, especially for large values of the Reynolds number. Never- theless, experimental measurements of the drag for a wide variety of values of U, L,  and  agree well with (2.6) 0 4.4. The Navier-Stokes equations The ow of an incompressible homogeneous uid with density  and viscosity is 0 described by the incompressible Navier-Stokes equations,  (u +uru) +rp =u; 0 t (2.7) ru = 0: Here, u (x;t) is the velocity of the uid, and p (x;t) is the pressure. The rst equation is conservation of momentum, and the second equation is conservation of volume. Remark 2.9. It remains an open question whether or not the three-dimensional Navier-Stokes equations, with arbitrary smooth initial data and appropriate bound- ary conditions, have a unique, smooth solution that is de ned for all positive times. This is one of the Clay Institute Millenium Prize Problems.

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