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Vector Calculus Applicationsž 1. Introduction 2. the Heat Equation

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Vector Calculus Applicationsž 1. Introduction 2. the Heat Equation Vector Calculus Applications 1. Introduction The divergence and Stokes’ theorems (and their related results) supply fundamental tools which can be used to derive equations which can be used to model a number of physical situations. Essentially, these theorems provide a mathematical language with which to express physical laws such as conservation of mass, momentum and energy. The resulting equations are some of the most fundamental and useful in engineering and applied science. In the following sections the derivation of some of these equations will be outlined. The goal is to show how vector calculus is used in applications. Generally speaking, the equations are derived by first using a conservation law in integral form, and then converting the integral form to a differential equation form using the divergence theorem, Stokes’ theorem, and vector identities. The differential equation forms tend to be easier to work with, particularly if one is interested in solving such equations, either analytically or numerically. 2. The Heat Equation Consider a solid material occupying a region of space V . The region has a boundary surface, which we shall designate as S. Suppose the solid has a density and a heat capacity c. If the temperature of the solid at any point in V is T.r; t/, where r x{ y| zkO is the position vector (so that T depends upon x, y, z and t), then theE total heatE eergyD O containedC O C in the solid is • cT dV : V Heat energy can get in or out of the region V by flowing across the boundary S, or it can be generated inside V . Let’s suppose that the heat flux vector is called q. This vector measures rate of energy flow past a point per unit area. Thus, if we have a small elementE of surface area dS with outward unit normal vector n, the rate of energy flow outward through this element of surface area is q n dS. Integrating over theO entire surface, the total rate of energy flow (i.e., the flux) out of the regionE O is therefore — q n dS : E O This is shown schematically in Fig. 1. c W. L. Kath, 2004, 2005, 2006, 2010, 2011, 2015. E-mail: [email protected] Version 1.3, February 2015 1 Winter 2015 Vector calculus applications Multivariable Calculus z q n V S y x Figure 1: Schematic diagram indicating the region V , the boundary surface S, the normal to the surface n, and the heat flux vector q. O E Let’s suppose that the rate at which heat energy per unit volume is being generated is Q. Then, the total rate at which heat is being generated inside V is • Q dV : V Conservation of energy then requires that the rate at which the energy within the region V changes and the rate at which energy crosses the boundary of region must balance, i.e., d • — • cT dV q n dS Q dV : dt V C E O D V The signs of the terms must appear in this manner; assuming that Q 0 for the moment, if the flux of energy out of the region is positive, then the total energy inside theD region must be decreasing, hence its derivative will be negative. Similarly, if the flux of heat through the boundary is zero, but heat is being generated inside V , then the temperature must be increasing. To simplify this, we first take the time derivative inside the volume integral: d • • @ cT dV .cT / dV : dt V D V @t We can do this because the region V over which the integral is being taken is independent of time; V is the same region for all time. Thus, at any fixed time, integration over V just gives a single number, i.e., the energy inside V at that time. The energy will change with time because T (and possibly and c) changes with time. Note that the time derivative must be turned into 2 Winter 2015 Vector calculus applications Multivariable Calculus a partial derivative when it is moved inside the volume integral, however, because the functions being integrated depend upon both position and time. Similarly, we apply the divergence theorem to the flux integral: — • q n dS q dV : E E O D V r E The result is • @ • • .cT / dV q dV Q dV 0 ; E V @t C V r E V D or, equivalently, • Ä @ .cT / q Q dV 0 : E V @t C r E D It would be nice to say at this point that the quantity in brackets is zero; this would allow us to get rid of the integral. This would be true, but the right argument is needed, because an integral can be zero either because the integrand is zero or because it is equally positive and negative, and when integrated there is perfect cancellation. In the present case, however, there is one more fact: the integral must be zero for arbitrary volumes V . This forces the integrand to be zero everywhere. The proof is by contradiction. First, assume that the integrand is not zero everywhere. Pick a point at which it is positive, and then choose a region V that is entirely within the region where it’s positive. Since we are integrating a quantity that is entirely positive, the resulting integral must be zero. This can’t be true, however, since we know the integral must be zero. Therefore, we must have @ .cT / q Q 0 : @t C rE E D This is the partial differential equation form of the fundamental principle of energy conservation for heat transfer. More can be done if one has an explicit result for the heat flux q in terms of the temperature. In many cases, observations show that the heat flux is given by Fick’sE law, q k T; E D rE i.e., the heat transfer is proportional to the negative of the temperature gradient. Thus, heat tends to flow from hotter regions to colder regions. While this is true in most cases, it is not always true; in particular, if phase transitions are possible it can be violated. If we use Fick’s law in the equation for conservation of heat energy, it becomes @ Á .cT / k T Q: @t D rE rE C 3 Winter 2015 Vector calculus applications Multivariable Calculus Finally, if , c and k are all constant, and Q 0, this equation simplifies to the heat equation D @T 2T; (1) @t D r where k=c is the thermal diffusivity. D A special case of this equation results if the object is in thermal equilibrium, i.e., the temperature doesn’t change with time: 2T 0 : (2) r D This is known as Laplace’s equation. 3. Fluid mechanics 3.1. Conservation of mass Conservation laws are also the basis for the equations of fluid mechanics. Let be the fluid density and v be the fluid velocity. Then the mass in a region V is given by E • dV ; V and the rate at which mass leaves the region through its boundary S is — v n dS E O I see Fig. 2 for a diagram of the situation. Here v is the mass flux vector; this has units of kg=m3 m=sec kg=m2sec, i.e., mass per unit area perE second. D The rate of change of the mass in V must be balanced by the rate at which mass leaves through S, d • — dV v n dS 0 : dt V C E O D Pulling the time derivative inside the V integral, and converting the surface integral into a volume integral using the divergence theorem, we have • Ä@ v dV 0 : E V @t C r E D 4 Winter 2015 Vector calculus applications Multivariable Calculus v n V S Figure 2: Schematic diagram indicating the region V , the boundary surface S, the normal to the surface n, the fluid velocity vector field v, and the particle paths (dashed lines). O E As before, because the region V is arbitrary, we must have the terms between the brackets be identically zero, which gives @ v 0 : (3) @t C rE E D This is the partial differential equation form of conservation of mass. Equation 3 can be simplified somewhat by using the vector identity .v/ v v rE E D rE E C rE E D v v, which gives E rE C rE E @ v v 0 : (4) @t CE rE C rE E D The combination @=@t v occurs so much in fluid mechanics that it is given a special notation, CE rE @ D v : @t CE rE Á Dt This is called the material or substantial derivative, and physically it represents taking the time derivative ‘following the fluid’, i.e., taking the time derivative of something transported along by the fluid. One can see this by considering .x; t/ not at a fixed position x, but instead .x.t/; t/, where dx=dt v; because the position x movesE at the fluid velocity vE, this is the densityE of a specific fluidE element.DE Taking the time derivativeE using the chain rule, E d dx @ @ dx @ D .x; t/ E E v : dt E D rE dt C @t D @t C dt rE D @t CE rE D Dt 5 Winter 2015 Vector calculus applications Multivariable Calculus In any event, with this notation we have D v 0 : (5) Dt C rE E D This has a nice physical interpretation: if v > 0, so that the fluid velocity is locally expanding, rE E then D=Dt < 0, i.e., the local density is decreasing.
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