problem 6.4 in the attached
CHAPTER 5 Scalar and Vector Fields 5.1. Introduction The preceding two chapters introduced you to systems in which the dynamic variables are functions of time and one spatial variable. Examples included heat conduction in a thin bar of metal, diffusion of dopant in a semiconductor and pollutants in a river, standing waves on a vibrating string, and traveling waves on a transmission line. We will now expand the spatial dependence from one, to two and three dimensions. This complicates the analysis, of course, but also permits us to study more interesting and realistic systems, including heat conduction in structures that are not long and thin, fluid flow around obstacles, such as boat hulls and airplane wings, the vibrations of membranes like drumheads or the eardrum, and electromagnetic waves. 5.2. Fields and their visualization A field is a function of two or three spatial coordinates. If the field is described by a single value at each point in space (e.g., temperature), it is a scalar field. If the field is described by two or more values at each point in space (e.g., heat flow), it is a vector field. Fields may be static (constant in time) or dynamic (time varying). Table 5.1 gives a few examples of scalar and vector fields. Table 5.1: Examples of Scalar and Vector Fields Scalar Fields mass distribution (solid bodies ) moment of inertia) pressure distribution (atmospheric pressure cells) elevation of the earth’s surface (topographic map) black and white image display Vector Fields wind/flow pattern (hurricanes; aerodynamic object) magnetic field surrounding a dipole (geomagnetism; bar magnet) planetary gravitational forces (satellite orbital dynamics) electric field around a charged body (electrostatic printers) electromagnetic fields surrounding power lines, antennas, microwave ovens color image display (red, green, blue) 224 Scalar and Vector Fields 5.2.1. Scalar fields In this book we will denote a general scalar field as ˆ.x; y; ´; t/ or ˆ.r; t /. Despite the fact that scalar fields are among the simplest, they can still be quite difficult to visualize if all you have is a mathematical formula. It helpful to have ways of representing fields graphically. If a scalar field is not time-varying and a function of only two spatial coordinates, ˆ.x; y; ´; t/ D h.x; y/, where h for example can represent height above sea level at different locations .x; y/, we can make a three dimensional plot of h vs x and y (Figure 5.1). A function h maps an input −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0 20 40 60 80 y x h(x,y) Figure 5.1: Scalar field ˆ D h.x; y/ representing height above sea level. array of points .x; y/ into an output array of function values h.x; y/:264 .x1; y1/ � � � .xNx ; y1/::: : : : ::: .x1; yNy / � � � .xNx ; yNy / 375 7�! 264 h.x1; y1/ � � � h.xNx ; y1/::: : : : ::: h.x1; yNy / � � � h.xNx ; yNy / 375 In Matlab, an array of .x; y/ points is actually represented by two arrays of coordinates, one for the x values and one for the y values, X D 264x1 � � � xNx::: : : : ::: x1 � � � xNx 375 ; Y D 264 y1 � � � y1::: : : : ::: yNy � � � yNy 375 The pair of matrix elements .Xmn; Ymn/ is the coordinate pair .xn; ym/. Matlab supplies a function meshgrid to create these X and Y arrays. For example, for equally-spaced x and y coordinates, with x; y 2 Œ�3; 3, meshgrid is called like this: Npts = 51; xmax = 3; [X, Y] = meshgrid(linspace(-xmax, xmax, Npts)); 5.2 Fields and their visualization 225 The arrays X and Y are each 51 � 51. The X and Y arrays are then passed to the function that computes h, i.e., h = f(X,Y). Then, to make the three-dimensional height plot, the mesh function is called, mesh(X,Y,h); The surf function makes a three-dimensional plot of a different form (try it). Additional Matlab commands adjust the orientation of the axes and attach labels. We can also use contour lines (level surfaces) to provide complete information about the field. They are constructed in the following way (Figure 5.2)1 � Cut across the mountain with a plane at some height h � Identify the curve generated at the intersection of the mountain and the cutting plane � Project this curve onto the xy-plane The projection on the xy-plane is the contour line. By repeating these steps for an arbitrary h(x,y) h+dh h 0 x y Figure 5.2: Contour lines are constructed by slicing the scalar field at different heights, then projecting the intersections onto a plane. number of heights h we generate a set of contour lines that both give the footprint of the mountain as well as a complete description of how the mountain height changes with location. The closer the contour lines are together, the steeper the mountain is at that location. Matlab provides the contour function to mechanize the process. 1McQuistan, Scalar and Vector Fields: A Physical Interpretation, [page]. 226 Scalar and Vector Fields [c,hndl] = contour(X,Y,h,10); % Make 10 contours clabel(c,hndl); % Label the contours with values in c vector Additional Matlab commands adjust and label the axes. The contour lines for the mountain in Figure 5.1 are shown in Figure 5.3. 8. 13 34 4 8.13344 8.1 334 4 8.13344 8.13344 8.13344 16.2669 16.2 669 16.2669 16.2669 16.2669 24.4003 24.4 003 24.4003 24.4003 32.5338 32.5 338 40.667240.6672 48.8007 48.8007 56.9341 65.0675 73.201 81.3344 x y −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Figure 5.3: Scalar field representing height above sea level (Figure 5.1), represented by contour lines. Each contour is labeled with the height, or value of the function. Yet another useful way to visualize a two dimensional scalar field is an array of square picture elements, or pixels, where each pixel is assigned a color based on its value, e.g., blue for 0 and red for 100, with the other colors spaced out between these extremes. This method is called pseudocolor (Figure 5.4). Matlab provides the pcolor command to generate a pseudocolor representation. To plot our function in pseudocolor, with overlaid contour lines and a color bar to indicate relative amplitude values, pcolor(X,Y,h); shading interp hold on contour(X,Y,h,10,’k’) % ’k’ causes the contour lines to all be black hold off colorbar EastOutside Additional Matlab commands adjust and label the axes. In some ways the contour lines in Figure 5.3 and the combination of pseudocolor and contour lines in Figure 5.4 have more information about the field than the more realistic looking mountain in Figure 5.1, since we cannot look behind the mountain (unless, of course, the graphical display permits interactive rotation of the mountain plot—Matlab has this capability). 5.2 Fields and their visualization 227 Figure 5.4: Scalar field representing height above sea level (Figure 5.1), rendered in pseudocolor with contour lines overlaid. 5.2.2. Vector fields A vector field in three dimensions is determined by three numbers at each time instant and spatial location. In cartesian coordinates, these numbers are the field’s components relative to three mutually orthogonal unit basis vectors fOx; Oy; Ozg. We will always use the circumflex O to denote a unit vector. For example, a three-dimensional heat field is represented in cartesian coordinates by an x component qx , a y component qy , and a ´ component q´: q.x; y; ´; t/ D qx.x; y; ´; t/OxC qy.x; y; ´; t/ OyC q´.x; y; ´; t/Oz : In general, each component may depend on all three coordinates and time. A vector field may also depend on only two coordinates, e.g., a planar field q.x; y; t/ D qx.x; y; t/OxC qy.x; y; t/ Oy : The electric, magnetic, and gravitational fields familiar from your introductory physics courses are vector fields. In fluid mechanics, the average behavior of a large number (1023/ml) of fluid “particles” is also described by a fluid velocity field, u D ux Ox C uy Oy C u´ Oz.2 Each component of the velocity field is a continuous function of time and all three spatial coordinates, e.g., ux.x; y; ´; t/. In a steady fluid flow the velocity field does not change with time, @u=@t D 0; the x compo- nent of a steady flow is ux.x; y; ´/. This does not mean, however, that a steady velocity field is constant. Although it does not waver at any particular point in space, it may vary from point to point, and a particle may be accelerated as it moves under the influence of the field from one point to another (picture a leaf floating down a narrowing stream). For simplicity, we always assume steady flow when we talk about velocity fields. 2The traditional notation for the velocity field components in fluid mechanics is u; v;w rather than ux ; uy ; u´. 228 Scalar and Vector Fields Example 5.1 (Simple fluid flows). A few simple examples of steady fluid flows are � u D U0 Ox: Steady flow in the Cx direction � u D U0 �y a � Ox: A flow in the x direction whose speed varies with respect to y. � u D U0 � 1 �R2=a2 � Oz, R D p x2 C y2: A flow down a pipe of radius a having a parabolic velocity profile that is zero at the walls of the pipe and U0 in the center of the pipe. � If you fix a sensor at a location .x; y; ´/ in space and observe the velocities of particles passing through that point, you are measuring the velocity field there. Moving the sensor around allows you to map the magnitude and direction of the velocity field in space. An alternative is to tag particles and observe their positions as functions of time, e.g., drifting buoys in the ocean, and single-particle tracking in biofluid flows. Figure 4.14 in the previous chapter showed both pictures: vectors for the velocity field u and ellipses for the particle trajectories .�; �/. In steady flow, particles entrained in a velocity field follow curves that are tangent to the velocity field vectors. These curves are called streamlines of the flow. With enough data, one description can be derived