Appendix Hyperbolic Equations With Two Independent Variables

Transcription

1 Appendix Hyperbolic Equations With Two Independent Variables 1. SECOND ORDER SCALAR EQUATIONS Consider the general quasi-linear equation (A.l) where a, b and c are functions of x, y, ~. ~x boundary data on an open curve; that is, and ~, with Cauchy y (A.2) where s is a parameter. Note that by integration along C, ~ = ~ 0 (s) is prescribed except for a constant. If this constant is specified then a necessary condition for (A.l) and (A.2) to define a well-posed boundary value problem for ~ is that by differentiating along r the derivatives of all orders may be calculated. Differentiating once ~ = x ~ + y'~ ~ox o~xx o~xy' ~ = x ~ + y'~ ~Oy O~xy O~yy' (A. 3) so that a necessary condition is that there is a unique solution of equations (A.l) and (A.3) for ~xx' ~ and ~ xy. This implies that yy (A.4) and it may be shown that this condition is also sufficient to ensure the existence of derivatives of all orders if the coefficients and boundary values are infinitely differentiable. We define a characteristic to be a curve in the x,y plane along which Cauchy data does not uniquely define the second derivatives of ~ 132

2 Appendix 133 Hence given x(t), y(t), ~x(t), ~y(t) on a characteristic, we have + X = x~ XX + y~xy' ~Y = x~xy + Y~yy' (A.S) which together with (A.l) must lead to a non-unique solution for the second derivatives ~, ~ and ~ Thus equations (A.l) and (A.S) u xy yy are linearly dependent, and this implies that a 2b c a f c x y 0 0 x!x 0 0 x y 0 ~y y which reduces to and (A.6) (A.7) In the semi-linear case, when a, b and c are functions of X and only, (A.6) determines the characteristics, which are real if b2 ~ ac, with two real characteristic directions at a point if b2 > ac. In the quasi-linear case when a,b,c also depend on ~. ~, ~, this is still X y a necessary condition for two real characteristic directions to exist at a point, but the directions cannot in general be evaluated without a knowledge of the solution ~. If two real characteristic directions exist at a point the equation is said to be locally hyperbolic, and if this is true at all points in the domain D on one side of the open boundary defined by (A.2), then the equation is hyperbolic in D and the Cauchy-Kowaleski theorem gives the existence of a solution ~ in some neighborhood of the boundary. (See Garabedian [12].) Note that these conditions require that the boundary curve does not touch a characteristic of either family. For a hyperbolic equation, the characteristics also define the "region of influence" of the initial data. Equation (A.7) shows that there is a relation between the derivatives of ~ and f along a characteristic, X y and it can be shown (see Courant & Hilbert, Vol. II [8]) that the solution at P depends on the initial data between A and B, where AP and BP are the positive and negative characteristics through P as shown in Figure A.l. Conversely, Cauchy data given on AB solution in the region APB defines a unique bounded by the characteristics through A y

3 134 APPENDIX negative characteristic Figure A.l and B. For two dimensional supersonic flow, the characteristics are the Mach lines and the "region of influence" of a body may be identified with the part of the fluid which is not in the "zone of silence" described on p. 64. In a number of examples it is possible to integrate (A.7) explicitly along a characteristic of either family and define two functions, called Riemann invariants, which are constant along characteristics of the appropriate family. The existence of these invariants allows solutions to be constructed for boundary value problems of a special class, called simple wave solutions, of which the two dimensional steady Prandtl-Meyer flow of a gas described in Chapter IV is an example. If we apply the theory to equation (4.19), then the characteristic directions are defined by ( 2 2) 2. ( 2 2). 2 a -u y + 2uv xy + a -v x = 0, which reduces to equation (4.24), namely ~ = -uv±a[u 2 +v 2 -a 2 l~ dx 2 a -u The characteristics exist if u 2 +v 2 > a2, that is if the flow is supersonic. On a characteristic, (A.7) gives ( 2 2).. ( 2 2). a -u yu + a -v xv = 0, which is relation (4.25).

4 Appendix 135 The Riemann invariants are complicated functions in terms of u and v, but rewritten in terms of ~ and e, where sin ~ = a/(u 2 +v 2 )~ and tan e = v/u, they are e ± f(~). as in equations (4.27) and (4.28). In this example the characteristics are physically identified with the Mach lines of the flow. 2. FIRST ORDER SYSTEMS OF EQUATIONS Consider the first order system Au + Bu = c, X y (A.8) where u Emn and A and B are n x n matrices. The system is quasilinear if the entries and components of A, B and c are functions of x, y and u, and linear if they are functions of x and y alone. If Cauchy boundary data u = u 0 (s) on x = x 0 (s), y = y 0 (s) is given, then a necessary condition for a well posed problem is that the partial derivatives of u may all be uniquely determined on the boundary. Hence, from (A.B) evaluated on the boundary and the relation u' = x'u + y'u 0 0 X 0 y' we require a unique solution components of ux and uy. of 2n simultaneous equations for the 2n This implies that the 2n x 2n determinant of the coefficients of u and u is non-zero, and the condition can be simplified to Jsx0 - Ay0 X y J 1 o. This corresponds to the condition (A.4) of the previous section and in the case n = 2 will be a quadratic function of x0 and y 0 as in (A.4). Indeed if the coefficients in (A.l) are independent of ~. it may easily be written as a first order system by the substitution u 1 = ~x' u2 = ~y As in the previous section we may define characteristics as curves in the x,y plane along which Cauchy data does not uniquely define the first derivatives of u. However, a more direct approach, which is easily shown to be equivalent, is to define a characteristic (x(t),y(t)) by I sx - AY"I = o. A system is hyperbolic at a point if there exist n tions, that is, if (A.9) has n real roots for A eigenvalue A we can define a left eigenvector z Then, using (A.B), T z c, (A.9) characteristic direcy/i. For such an such that z (B-AA) 0. T

5 136 APPENDIX and hence along a characteristic (A.lO) is an ordinary differential equation for u. If this equation is integrable, then the resultant function which is constant along the characteristic is called a Riemann invariant. A simple example is that of the shallow water theory equations described in Chapter III.l, where variables t and x replace x and y. For equations {3.3) and (3.4), u = (~). A = (~ ~). The roots of lu~a hyperbolic for n > U;AI = 0. The nz 1 ; z 2 /:in = 0, so that from (A.lO) B = (~ ~). and c 0. 0 are A = u ± lin so that the system is T left eigenvector z = (z 1,z 2 ) 0 TA du ( r=)du du r= dn = z Tx = n,±>"&n Tx = n dx ± >'&n dx satisfies This is integrable and gives Riemann invariants u ± 2/:in, as obtained by elementary manipulation in (3.5). A second example is the one dimensional flow of a gas described in Chapter IV.3 and governed by equations (1.6) and (1.8). In this case u = (~). A = (~ ~) and 8 = [: a~/p] An example with n = 3 is the two dimensional steady flow of an isentropic gas discussed in Chapter IV.4, which is described in the previous section in the homentropic case, when it may be reduced to a second order scalar equation. case the equations are (4.16), (4.17) and (4.18), and In the general 0 0 u and 8 It is easily verified that in the supersonic case there are three families of characteristics, the two Mach lines and the streamlines. However, only one Riemann invariant exists if the flow is not homentropic, and that is along the streamlines where 2 2 J a 2 d (u +v ) + T = const.,

6 Appendix 137 which is Bernoulli's equation. One property of the characteristics of (A.l) and (A.B) is that they are curves across which discontinuities in the second derivatives of ~ or the first derivatives of u may occur. Thus a physical interpretation of a characteristic in terms of gas dynamics is that of a weak shock wave. For a linear second order equation or system the characteristics are also curves across which discontinuities in V~ or u may occur. Thus in the linearized theory of Chapter VI, shock waves lie along characteristics. However, for non-linear problems shock waves do not lie along characteristics, and Riemann invariants are discontinuous at shock waves.

7 References [1] Batchelor, G. K. An Introduction to Fluid Dynamics. CUP, (2] Becker, E. Gas Pynamics. Academic Press, (3] Birkhoff, G. and Zarantonello, E. H. Jets, Wakes & Cavities. Academic Press, (4] Carrier, G. F. and Pearson, C. E. Ordinary Differential Equations. Blaisdell, [5] Chester, C. R. Techniques in Partial Differential Equations. McGraw-Hill, [6] Cole, J. Perturbation Methods in Applied Mathematics. Blaisdell, [7] Copson, E. T. Asymptotic Expansions. CUP, [8] Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. II. Interscience, [9] Erdelyi, A. Higher Transcendental Functions, Vol. II. McGraw-Hill, [10] Erdelyi, A. Tables of Integral Transforms, Vol. I. McGraw-Hill, [11] Gakhov, S. D. Boundary Value Problems. Pergamon, [12] Garabedian, P. R. Partial Differential Equations. Wiley, [13] Greenspan, H. The Theory of Rotating Fluids. CUP, [14] Hayes, W. D. and Probstein, R. F. Hypersonic Flow Theory. Academic Press, [15] Howarth, L. (Ed.) Modern Developments in Fluid Dynamics, High Speed Flow. OUP, 1953, Vol. I. [16] Landau, L. D. and Lifschitz, E. M. Fluid Mechanics. Pergamon, [17] Liepmann, H. W. and Roshko, A. Elements of Gas Dynamics. Wiley, [18] Milne Thomson, L. M. Theoretical Hydrodynamics. Macmillan,

8 References 139 [19] Murray, J. D. Asymptotic Analysis. OUP, [20] Officer, C. B. Introduction to the Theory of Sound Transmissions. McGraw Hill, Rosenhead, L. (Ed.) Laminar Boundary Layers. OUP, [21] [22] [23] [24] [25] Schiffer, M. Article F, Handbuch der Physik IX. Springer-Verlag, Schlichting, H. Boundary Layer Theory. McGraw-Hill, Stoker, J. J. Water Waves. Interscience, Van Dyke, M. D. Perturbation Methods in Fluid Mechanics. Parabolic Press, [26] Woods, L. C. The Theory of Subsonic Plane Flow. CUP, [27] Yih, C. S. Pynamics of Nonhomogeneous Fluids. Macmillan, 1965.

9 Index Acoustic equation, 101 waves, 76 Acoustics, lolff. Aerofoil, 105, 115, Angular momentum, conservation of, 4, 20 Aperture, flow through, 23, 24 Asymptotic expansion, 18, 19 Axisymmetric flow, subsonic, supersonic, 109, 110 Barotropic flow, 13, 15, 20, 21, 61 Batchelor, G. K., 1, 16, 119 Beach, disturbance propagating onto, 53 Becker, E., 80, 85 Bernoulli's equation for compressible flow, 13 for homentropic flow, 15 for irrotational incompressible flow, 9, 40, 119, 120, 128 for isentropic flow, 137 for steady flow, 62, 63, 70, 71, 74, 76 for steady, isentropic flow, 15 for steady, two-dimensional, rotational flow, 12 linearized form of, 100, 103 Birkhoff, G., 120 Bjerknes' form of Kelvin's theorem, 21 Blasius' theorem, 117, 118 Blast wave, 89ff., 115 Bluff body, inviscid flow past, 24' 119 Bore, 52, 83, 84 Boundary condition, at fluid boundary, 2, 6 layer, 9, 17, 19, 119 separation, 24 Bow wave, 36 Burger's equation, 88, 98 Capillary wave, 34 Carrier, G. F., 32 Cauchy data for partial differential equation, 132, 133, 135 equation (for vorticity), 7 integral, 124 principal value, 125, 131 residue theorem, 125 Cauchy-Kowaleski theorem, 133 Centred simple wave, 46, 47, 49 Channel of slowly varying cross section, flow in, 39, 42 Chaplygin's equation, 76, 78 Characteristics, 44, 50, fan of, 46, 74 for Chaplygin's equation, 78 for dam break problem, 45, 46 for flow between reservoirs, 49, 58 for piston problem, compressive, 69 compressive, impulsive velocity, 48 expansive, 68, 69 expansive, impulsive velocity, 46, 47 for Prandtl-Meyer flow, 73, 134 for two-dimensional steady flow, 71,72 Charpit's equations, 33 Chester, C. R., 33 Chord line of wing,

10 Index 141 Circular wave, 35 Circulation, 7, 8, 104, 115, 118, 119' 127 Cnoidal wave, 57, 58 Cole, J., 115 Compressible flow, 13, 6lff. linearized theory, 99ff. Conductivity, 5, 85 Conformal transformation, 117, 119, 120, 123, 130 Conservation law, 82 Conservative force, 6, 8 Continuity equation (see also Mass, conservation of), 13, 39, 40, 44, 61, 65 integral form of, 81 for two-dimensional steady flow, 90 linearized, 99 Continuum hypothesis, 1, 2, 3 Contraction ratio, of a twodimensional jet, 121 Convection, of vorticity, 9, 100 Convective derivative, 2 of Jacobian, 2 effects, vs. rotational effects, 18 heat conduction equation, 5 Copson, E. T., 30 Corner continuous convex, supersonic flow past, 72 sharp concave, flow past, 92 sharp convex, flow past, 73, 78 sharp, incompressible flow past, 119 Courant, R., 44, 82, 133 Critical speed of stream with free surface, 29 of sound, 63, 83, 92 Crocco's theorem, 16, 70, 96 Cylinder porous, incompressible flow past, 20, 21 two-dimensional steady incompressible flow past, 21 Dam break problem, 45, 46 Deep water gravity waves on, 27, 28, 30 Diffusion equation, 88 Dilatation, 2 Dimensional analysis, 17ff. Discontinuous solution (see also Bore, Hydraulic jump, Shock) as zero-viscosity limit of continuous solution, 85 Discontinuous solution (cont.) for flow between reservoirs, 49, 58 for piston problem, compressive case, 48, 79 for shallow water equations, Dispersion, 27ff. relation, 27, 33, 41 for interfacial wave, 35, 41 for three-dimensional Stokes wave, 36 including surface tension, 33 Dispersive wave, 27 Dissipation, 17 Dissociation, 61, 85 Doublets, line distribution of, 109 Drag in inviscid, incompressible flow without circulation, 104 in subsonic flow without circulation, 104 on slender bodies, 110, 115 on thin wings in supersonic flow, 106 wave, 106 Eikonal equation, 102 Ekman number, 19 Energy Elliptic partial differential equation, 9, 65, 112 conservation of, 4, 5, SO, 52, for incompressible flow, 5 for viscous flow, 17 integral form, 82, 90 flux, for gravity wave, 28 kinetic, 19, 20 rate of change in inviscid, compressible fluid, 5 rate of transmission, for small amplitude waves on deep water, 28 stored, for gravity waves, 28 Enthalpy, 19 Entropy, 14 jump, across shock, 83 across weak shock, 94, 97 nondecreasing, 17, 84, 85 Equation of state, for perfect gas, 14 Equilibrium, thermodynamic, 14 Erdelyi, A., 108, 109 Eulerian flow description, 1 Euler's equation, 4, 17, 62 in rotating reference frame, 10 linearized, 99

11 142 INDEX Fan of characteristics, 46, 74 Flat plate, two-dimensional flow past, 9, 117, 122, 123 Flow separation, 117, 119, 128 Force, conservative, 6, 8 Fourier transform, 29, 108 Free boundary, 23, 25 Free streamline flow, 136 Froude number, 18, 40 Gakhov, S. D., 124, 125 Garabedian, P. R., 133 Geometrical optics, 102 Gravity waves, 25ff. non-dimensional parameters of, 26, 54 non-dimensional variables of, 54 Greenspan, H., 11 Group direction, 36 velocity, 29ff. for Stokes waves, 34, 41 for small amplitude waves on deep water, 31 Hayes, W. D., 115 Helicity, 21 Helmholtz' equation, 7, 102 Hilbert, D., 44, 82, 133 Hilbert problem, 124, 131 Hodograph plane, 24, 75ff., 92, 119 transformation, 24 Homentropic flow, 15, 61, 63, 68, 70, 83, 96, 114, 136 Howarth, L., 90 Hydraulic jump, 48, SOff., 80 Hyperbolic partial differential equation, 44, 65, 71, 73, 112, 132ff. Hypersonic flow, 100 two-dimensional, steady, Ideal gas, 14, 62 form of internal energy for, 5 Incompressible fluid, 4 Incompressible, inviscid flow in horizontal channel of slowly varying cross section, 39 non-dimensional equations of, 18 two-dimensional, past porous circular cylinder, 20, 21 past thin wing, 103 steady, past cylinder, 21 Incompressible, inviscid, irrotational flow, equations of, 8 kinetic energy for, 20 Inertial effects vs. viscous effects, 19 Inertial wave, 22 Inhomogeneous fluid, 12, 13 Interface, wave at, 35, 41 Internal wave, 22 Inviscid flow equations of, 3ff. past bluff body, 119 past blunt body, 24 past sharp, concave corner, 92 through orifice, 119 turning sharp corner, 119 Irrotational flow, 7, 8, 100, 144 two-dimensional, steady, compressible, 70, 74 Isentropic flow, 15, 96 speed of sound in, 62 two-dimensional, steady, 136 Jacobian, 2 Jet of fluid emerging from hole, 23, 24 impinging on fixed wall, 23, 24 two-dimensional, 120, 130 Jump conditions, SO Kelvin's theorem, 8, 10, 118 Bjerknes' form of, 21 Kinematic viscosity, 17 Kinetic energy for incompressible, inviscid, irrotational flow, 20 of fluid flowing past sphere, 20 rate of change of, for inviscid, compressible flow, 19 Korteweg-deVries equation, 56, 57, 88 three-dimensional, 60 Kutta condition, 119, 123, 126, 127 Kutta-Joukowski formula for lift, 115 Lagrangian flow description, 1 Landau, L. D., 90, 102, 112 Laplace equation, 9 transform, 109, 115 Laval nozzle, 65, 66 Legendre transformation, 75 Liepmann, H. W., 14, 80, 109, 112 Lifschitz, E. M., 90, 102, 112 Lift, 105, 106 on flat plate in two-dimensional, steady, incompressible, inviscid flow, 118 on slender body, 110 on thin wing in separated flow, 129 Line source of sound, 101

12 I;ndex 143 Mach angle, 64, 71, 72 cone, 64, 101, 110 line, 64, 71, 72, 73, 78, 94, 102, 104, 106, 110, 112, 134, 136 approximating weak shock, 93 number, 19, 63-65, 66, 85 Mass conservation of (see also Continuity equation), 3, 4, 44, so, 51, 83 for incompressible fluids, 4 integral form of, 81 virtual, for sphere moving through liquid, 20 Meteorological problem, 10, 18 Method of stationary phase, 30 Milne-Thomson, L. M., 117, 120 Mixed type partial differential equation, 112 Momentum, conservation of, 4, SO, 52, 83 for viscous flow, 16 integral form, 82, 90 Murray, J. D., 108 Navier-Stokes equation, 16 non-dimensional form of, 19 Newtonian approximation in two-dimensional steady hypersonic flow, 114 fluid, 16 Newton's equations, 4 Non-dimensional parameters, 18, 19, 63 for gravity waves, 26, 54 variables, 17, 18 for gravity waves, 54 for shallow water theory, 43 for sound waves, 63 Nozzle flow through, 6Sff., 77 Laval, 65, 66 Nusselt number, 19 Oceanographic problems, 10 Officer, C. B., 102 One-dimensional, unsteady, compressible flow, 68ff., 80, 114, 136 viscous, 96 Optics, geometrical, 119 Orifice, flow through, 119 Particle path, 1, 11 for Stokes wave, 28, 41 Pearson, C. E., 32 Perfect gas equation of state for, 14 normal shock in, 97 Phase function, 33 speed, 31 Piston problem compressive, 69, 77, 79 impulsive velocity, 47, 48, 53, 96 expansive, 63, 69, 77 impulsive velocity, 46, 47, 69, 73, 77 Plane wave, 101 Plemelj formulae, 125, 126, 127, 129 Point source, 36 of sound, 101 Poisson's equation, 12 Polar, shock, 92, 93, 97 Porous cylinder, two-dimensional incompressible inviscid flow past, 20, 21 Potential, velocity, 8 complex, 24, 136, 145, 150 Prandtl number, 19 Prandtl-Glauert similarity rule, 104 Prand tl-meyer expansion, 73 flow, 72, 73, 134 Pressure coefficient, 104 in steady rotational flow, 12 reduced, 10 Principal value integral, 124 Probstein, R. F., 115 Progressive wave, 27 Radiation, 5, 61, 85 condition, 38, 101 Rankine-Hugoniot shock relations, 83, 84, 85, 91, 97, 113 Rate of strain tensor, 16 Reduced pressure, 10 Region of influence, 64, 73, 133, 134 of slender, axisymmetric body in supersonic flow, 110 Relaxation, 85 Reservoirs, flow between, 49, 58 Resonance, 29, 38 Reynold's number, 19 Riemann invariant, 44, SO, 59, 134, 135, 136, 137 mapping theorem, 119 Riemann-Hilbert problem, 128, 130 Rosenhead, L., 17 Roshko, A., 14, 80, 109, 112 Rossby number, 18, 21, 22

13 144 INDEX Rotational effects, vs. convective effects, 18 Rotational incompressible flow equations of, 18 examples of, loff., 59 Schiffer, M., 76 Schlichting, H., 110 Schwartz-Christoffel theorem, 120 Separation, boundary layer, 24 flow, 117, 119, 128 Shallow water theory, 26, 43ff., SO, 52-56, 73, 83, 136 analogy to one-dimensional compressible flow, 68 small amplitude theory, 55 three-dimensional, 60 two-dimensional, over bottom of constant slope, 59 variable depth, 55 Shear flow, 11, 12 layer, 9, 17 Ship travelling on deep water, 36 Shock, 61, 69, 75, 77, 78, 79ff. as continuous transition in viscous gas, 99 cone, 96, 98 curved, 94, 96 detached, 96 in flows past bodies, in nozzle, 66 intersecting, 97 normal, 79ff., 97 oblique, 90ff., 97 one-dimensional unsteady, 80 polar, 92, 93, 97 relations, SO, 90 (see also Rankine-Hugoniot shock relations) speed of, 83 spherical, 89 strong, 86, 89, 97, 115 unsteady structure of, 86ff. weak, 86, 87, 93, 94, 97, 99, 104, 112, 114, 137 width of, 84, 87 Similar solution, 18, 115 Similarity argument, 63, 69, 89, 90 relation, for subsonic flow past thin wing, 104 for transonic flow past thin wing, 112 Similarity (cont.) rule, for subsonic or supersonic flow past slender body, 110 Prandtl-Glauert, 104 Simple wave, 44, 56, 68, 73, 76, 134 centred, 46, 47, 49 generalized, 59 Singular integral, 124, 125, 126, 131 perturbation, 19 Slender body theory, 106ff. Slip flow, 97 velocity, 9 Small disturbance theory, 26ff., 54 in shallow water, 55 Solitary wave, 57, 58 Sound line source of, 101 point source of, 101 speed of, 62, 63, 69 critical, 63, 83, 92 stagnation, 62, 101 waves, 6lff. Sources, line distribution of, 109, 125 Spherical wave, 101 Stagnation speed of sound, 62, 101 Standing wave forced, 38 in rectangular tank, 38, 42 on stream, 41 State of a fluid, 1 thermodynamic, 14 Stoker, J. J., 36 Stokes theorem, 81 wave, 26, 54 group and phase velocity,of, 31, 41 particle path for, 28, 41 three-dimensional, 35 with surface tension, 33 Stratified flow, 13, 22 Stream function, 11, 21 tube, 7 Streamline, 11, 71, 136 as jet boundary, 120 Streamlined body, 9 Stretching of variables, 103 Stress tensor, 16 symmetry of, 20 Stress-strain relation, 16 Subcritical flow of stream with free surface, 29, 40, 53

14 Index 145 Subsonic flow, 64, 65 axisymmetric, in nozzle, 66 past thin wing, 103 Supercriti.;al flow of stream with free surface, 40, 53 Supersonic flow, 64, 65, 71, 72, 73, 78 axisymmetric, 107, 109, 110 in nozzle, 66 past body, past cone, 98, 115 past continuous convex corner, 72 past sharp corner, 73, 78 past thin wing, 104 Surface tension, 33, 34, 41 effect on surface disturbance, 34 Taylor column, 21 Thermodynamic equilibrium, 14 state, 14 Thermodynamics, 13ff. first law of, 14 second law of, 14 Thin wing separated flow past, 128ff. subsonic flow past, 103, 104 incompressible, attached, 123ff., 130 supersonic flow past, theory, 102ff. two-dimensional steady flow past, incompressible, 103 transonic, 116 Thrust on flat plate in two-dimensional separated flow, 123 Tidal wave, 39ff., 55 Transonic flow, 100, 111, 112 two-dimensional, steady, past thin wing, 116 Transport theorem, 3, 19, 20 Travelling wave, 27 forced, surface, 28 on surface of fluid in rotational motion, 59 solution of Korteweg-deVries equation, 57 Two-dimensional steady flow, 70ff., 75 hypersonic, Newtonian approximation in, 114 incompressible, inviscid (complex variable methods), 117ff. incompressible, past thin wing, 103 Two-dimensional steady flow (cont.) isentropic, 136 past flat plate, 9 transonic, past thin wing, 116 Van Dyke, M. D., 19, 111 Velocity potential, 8, 13, 123 complex, 24, 117, 127 Virtual mass, of sphere moving through liquid, 20 Viscosity, 46, 73, 85 coefficients of, 16 kinematic, 17 Viscous effects, vs. inertial effects, 19 flow, 16ff. one-dimensional, unsteady, 113 fluid, forces in, 16 forces, work done by, 17 Vortex line, 20 shedding, 119 starting, 119 Vorticity, 6, 7, 9, 10, 11, 13, 16, 70 convection of, 9, 100 distribution, 127 flux, 7 in rotating reference frame, 10 over stream tube, 7 shed behind flat plate in twodimensional incompressible inviscid flow, 119 Wake, 9, 122, 128 in inviscid flow past a blunt body, 24 Wave acoustic, 76, 101 amplitude, 27 at an interface, 35 blast, 89ff., 115 bow, 36 breaking, 53, 58 capillary, 34 circular, 35 cnoidal, 57, 58 dispersive, 27 drag, 106 equation, 39, 62, 101 gravity, 25ff. inertial, 22 internal, 22 shock (see Shock) simple, 44, 56, 59, 68, 73, 76, 134 centred, 46, 47, 49 solitary, 57, 58

15 146 INDEX Wave (cont.) standing forced, 38 in rectangular tank, 38, 42 on stream, 41 Stokes (see Stokes wave) tidal, 39ff., 55, 56 travelling, 27, 57, 59 forced, 28 Weak solution, of flow equations, 81, 84 Well-posed boundary value problem, 132' 135 Wind tunnel, 66 WKB method, 32, 102 three-dimensional, 36 Woods, L. C., 130 Work done by viscous forces, 17 Yih, c. s.' 13 Zarantonello, E. H., 120 Zone of silence, 45, 46, 64, 134

16 Applied Mathematical Sciences 1. John: Partial Differential Equations, 4th ed. (cloth) 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. (cloth) 4.. Percus: Combinatorial Methods. 5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-Linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 3rd ed. (cloth) 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I. 19. Marsden/McCracken: The Hopi Bifurcation and its Applications. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. (cloth) 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications. 26. Kushner/Ciark: Stochastic Approximation Methods for Constrained and Unconstrained Systems. 27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. Ill. 34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. (cloth) 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghii/Kallen: Dynamic Meterology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion. (cloth) 39. Piccinini/Stampacchia/Vidossich: Ordinary Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. (cloth) 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. 43. Ockendon/Tayler: lnviscid Fluid Flows.