Mathematics,social decisions and the Law†

A number of examples are given on the uses and limitations of mathematical techniques in legal decision making. These examples are of interest in themselves, but they can also provide some insights into the proper role of mathematics in connection with other complex social issues.

     

A self-adjoint “simultaneous crossing of the axis”

By constructing the corresponding Green's function in a trapezoidal domain, we establish the existence of self-adjoint realizations of incorporating boundary conditions of the formu(s, 0)=u(s, T)=0. Such operators correspond to the historically important concept of a simultaneous crossing of the axis for vibrating strings.     

Heat transfer and friction in turbulent vortex flow

Summary This paper presents experimentally measured heat transfer and friction coefficients for air and water flowing through a pipe with several types of inserts designed to induce a swirl in the flow. It was observed that inside-surface heat transfer coefficients in swirling flow can, under favourable conditions, be at least four times as large as heat transfer coefficients at the same mass flow rate in purely axial flow. At the same time the pumping power per unit rate of heat transfer can be reduced. The increase in heat transfer coefficients was found to depend on the degree of swirl and on the density or temperature gradient. However, at comparable Reynolds numbers and swirling motions the heat transfer coefficients for air were found to be smaller than the coefficients for water. The reason for this difference is not definitely known, but the phenomenon is qualitatively compatible with that causing the cooling effect in Ranque-Hilsch vortex tubes. The observed phenomena are analyzed qualitatively and it is shown that they are primarily the result of a centrifugal force which induces a radial inward motion of warmer fluid and a radial outward motion of cooler fluid. The application of vortex flow to boiling heat transfer and other high heat flux systems is discussed briefly.

Nomenclature

Symbols c _p Specific heat at constant pressure, BTU/(lb)(deg F) - D _H Hydraulic diameter, (ft) - D Tube diameter, (ft) - f ₀ Fanning friction factor for axial flow, - f Fanning friction factor for swirling flow, - g Acceleration due to gravity, ft/(sec)² - G Mass velocity, lb/(sec) (sq ft) - h _i Inside surface coefficient of heat transfer, BTU/(hr)(sq ft)(deg F) - k Thermal conductivity, BTU/(hr)(sq ft)(deg F/ft) - L Characteristic length used in Grashof numbers, ft - p Frictional pressure drop in a duct, lbs/sq ft - r Radius of tube, ft - t Temperature potential in Grashof number, deg F - U _i Over-all coefficient of heat transfer based on inside tube area, BTU/(hr)(sq ft)(deg F) - V Axial velocity, ft/sec - Coefficient of thermal expansion, (deg F)^–1 - Absolute viscosity, (lbs)/(ft)(hr) - Density, lbs/(ft)³ - Angular velocity of fluid, rad/sec Dimensionless Parameters Nu ₀ Nusselt Number in axial flow, h _i D _H /k - Nu Nusselt Number in swirling flow, h _i D _H /k - Re Reynolds Number, VD _Hp / - Pr Prandtl Number, c _p /k - j Colburn j-Factor, (Nu/RePr)Pr ^2/3 Member of Technical Staff, Bell Telephone Laboratories, Murray Hill, N. J. formerly Baldwin Research Fellow, Lehigh University.