MIT 3.016 Fall 2005 � W.C Carter Lecture 20 c125 Nov. 09 2005 : Lecture 20: Linear Homogeneous and Heterogeneous ODEs Reading: Kreyszig Sections: 1.4 (pp:19–22) , 1.5 (pp:25–31) , 1.6 (pp:33–38) § § §Ordinary Differential Equations from Physical Models In engineering and physics, modeling physical phenomena is the means by which techno-logical and natural phenomena are understood and predicted. A model is an abstraction of a physical system, often with simplifying assumptions, into a mathematical framework. Every model should be verifiable by an experiment that, to the greatest extent possible, satisfies the approximations that were used to obtain the model. In the context of mo deling, differential equations app ear frequently. Learning how to model new and interesting systems is a learned skill—it is best to learn by following a few examples. Grain growth provides some interesting modeling examples that result in first-order ODES. Grain Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In materials science and engineering, a grain usually refers a single element in an ensemble cqkcq that comprises a polycrystal. In a single phase polycrystal, a grain is a contiguous region of material with the same crystallographic orientation. It is separated from other grains by grain boundaries where the crystallographic orientation changes abruptly. A grain boundary contributes extra free energy to the entire system that is proportional to the grain boundary area. Thus, if the boundary can move to reduce the free energy it will. Consider simple, uniformly curved, isolated two- and three-dimensional grains.c� 126 a a three-ΔG2D = 2 and ΔG3D = 42γv = M γ κ where M κ is vcurvature. vn∆A =vn∆tdsdsMIT 3.016 Fall 2005 W.C Carter Lecture 20 Figure 20-1: Illustration of two-dimensional isolated circular grain and dimensional isolated spherical grain. Because there is an extra energy in the system πRγgb πRgb, there is a driving force to reduce the radius of the grain. A simple model for grain growth is that the velocity (normal to itself) of the grain boundary is gb gb gb gb is the grain boundary mobility and the mean curvature of the boundary. The normal velocity gb is towards the center of A relevant question is “how fast will a grain change its size assuming that grain boundary migration velocity is proportional to curvature?” For the two-dimensional case, the rate of change of area can be formulated by considering the following illustration. Figure 20-2: A segment of a grain boundary moving with normal velocity vn will move a distance vnΔt in a short time Δt. The motion will result in a change of area −ΔA for the shrinking grain. Each segment, ds, of boundary contributes to the loss of area by ΔA = −vnΔtds.MIT 3.016 Fall 2005 � W.C Carter Lecture 20 c127 Because for a circle, the curvature is the same at each location on the grain boundary, the curvature is uniform and vn = Mgbκgbγgb = Mgbγgb/R. Thus dA 1 = −Mgbγgb 2πR = −2πMgbγgb (20-1)dt R Thus, the area of a circular grain changes at a constant rate, the rate of change of radius is: dA dπR2 dR = = 2πR = −2πMgbγgb (20-2)dt dt dt which is a first-order, separable ODE with solution: R2(t) − R2(t = 0) = −2Mgbγgbt (20-3) For a spherical grain, the change in volume ΔV due to the motion of a surface patch dS in a time Δt is ΔV = vnΔt dS. The curvature of a sphere is � 11 �κsphere = + (20-4)R R Therefore the velocity of the interface is vn = 2Mgbγgb/R. The rate of change of volume due to the contributions of each surface patch is dV 2 = −Mgbγgb 4πR2 = −8πMgbγgbR == −4(6π2)1/3MgbγgbV 1/3 (20-5)dt R which can be separated and integrated: V 2/3(t) − V 2/3(t = 0) = −constant1t (20-6) or R2(t) − R2(t = 0) = −constant2t (20-7) which is the same functional form as derived for two-dimensions. The problem (and result) is more interesting if the grain doesn’t have uniform curvature.MIT 3.016 Fall 2005 � W.C Carter Lecture 20 c128 Figure 20-3: For a two-dimensional grain with non-uniform curvature, the local normal velocity (assumed to be proportional to local curvature) varies along the grain boundary. Because the motion is in the direction of the center of curvature, the velocity can b e such that its motion increases the area of the interior grain for some regions of grain boundary and decreases the area in other regions. However, it can still be shown that, even for an irregularly shap ed two-dimensional grain, A(t) − A(t = 0) = −(const)t.MIT 3.016 Fall 2005 � W.C Carter Lecture 20 c129 Integrating Factors, Exact Forms Exact Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In classical thermodynamics for simple fluids, expressions such as cqkcq dU =T dS − P dV � ∂U � � ∂U � = dS + dV (20-8)∂S ∂V V S =δq + δw represent the differential form of the combined first and second laws of thermodynamics. If dU = 0, meaning that the differential Eq. 20-8 is evaluated on a surface for which internal energy is constant, U (S, V ) = const, then the above equation becomes a differential form � ∂U � � ∂U �0 = dS + dV (20-9)∂S ∂V V S This equation expresses a relation between changes in S and changes in V that are necessary to remain on the surface U (S, V ) = const. Suppose the situation is turned around and you are given the first-order ODE dy M (x, y) = (20-10)dx − N (x, y) which can be written as the differential form 0 = M (x, y)dx + N (x, y)dy (20-11) Is there a function U (x, y) = const or, equivalently, is it possible to find a curve represented by U (x, y) = const?MIT 3.016 Fall 2005 � W.C Carter Lecture 20 c130 If such a curve exists then it depends only on one parameter, such as arc-length, and on that curve dU(x, y) = 0. The answer is, “Yes, such a function U(x, y) = const exists if an only if M(x, y) and N(x, y) satisfy the Maxwell relations” ∂M(x, y) ∂N(x, y) = (20-12)∂y ∂x Then if Eq. 20-12 holds, the differential form Eq. 20-11 is called an exact differential and a U exists such that dU = 0 = M(x, y)dx + N(x, y)dy. Integrating Factors and Thermodynamics . . . . . . . . . . . . . . . …