This paper continues the theme of the opening Part I by analysing a number of problems designed to illustrate some particular aspects of the general theory. Three are concerned with thermal stresses and the last applies inter alia the theorems on maxima and minima to find lower and upper bounds to the St Venant torsional stiffness of a thin solid section.

The general equations of motion were solved numerically for the laminar isothermal flow of Newtonian fluids from a large tube of circular cross section through an abrupt contraction into a coaxial tube of smaller diameter and through the flow‐development region of the smaller tube. The ratio of the diameter of the large tube to that of the smaller tube was varied from one to eight (the latter in one case). Solutions were obtained for the case where the larger tube is real, with no slip at the wall, and for the case where it is a frictionless “stream” tube. The results are presented as charts giving excess pressure losses attributable to contracted and developing flow in terms of equivalent smaller‐tube diameters as functions of the tube‐contraction ratio and the Reynolds number, which was varied from 0.01 to as high as 500 in one case. Both radial‐ and axial‐velocity profiles are presented. The computed results are shown to be in satisfactory agreement with some experimental data. The results are presented in a manner convenient for use in the design of equipment in which contracted Newtonian flow occurs, such as fiber spinnerettes and heat exchangers, and in the analysis of experimental data for contracted flow.

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Temperature and velocity profiles and pressure losses were computed for laminar, temperature-dependent Newtonian flow from a stream tube through an abrupt contraction into and through the entrance region of a smaller coaxial tube, in which the fluid was cooled or heated at constant wall temperature. The equations of motion and energy, including axial diffusion and viscous dissipation, were solved numerically for diameter ratios of one and two, a practical temperature range, and N p e and N R~ up to 100. Entrance temperatures and velocities are far from uniform, and pressure losses are greater than those computed using simplified equations and uniform entrance temperatures and velocities. E. B. CHRISTIANSEN and S. J. KELSEY Department of Chemical EngineeringThe University of Utuh Salt Lake City, Utah 841 12 SCOPEThe purpose of the presently reported study was to provide more accurate means for predicting pressure losses and the nature of the velocity and temperature fields at the entrance of and in the tubes of such equipment as shell-and-tube heat exchangers, chemical and nuclear reactors, and similar devices. Improved means for predicting such pressure losses and temperature and velocity fields should provide a basis for more effective and economical heat-exchanger design. Also, the nature of these temperature and velocity fields has an important influence on the progress of a chemical reaction and is important in the design and analysis of data from tubular chemical reactors.Recent as well as earlier theoretical and experimental studies of flow in a tube accompanied by heat exchange with the tube walls have demonstrated that the dependence of viscosity on temperature can importantly affect pressure losses and heat transfer. In some recent numerical solutions of the equations of motion and energy in which the dependence of viscosity on temperature is accounted for, it was assumed that flow at the tube entrance is parabolic and that radial convection, inertial terms, axial diffusion, and viscous dissipation are negligible. Some improved numerical solutions account for temperature-dependent viscosity, radial flow, and inertial effects but are restricted to uniform tube-entrance velocities and temperatures. Although these solutions approximate some situations, it is known that entrance velocity profiles are not flat in laminar, isothermal flow, particularly in the flow geometry of concern here, and that axial diffusion and viscous dissipation may have important effects. The effects of temperature-dependent flow, combined with those of axial diffusion and viscous dissipation, for the case of interest here have not, to our knowledge, been clearly defined in previous reports.In the presently reported study, flow from the header into the tubes of a shell-and-tube heat exchanger or reactor was approximated by flow from a larger stream tube (a stream tube has a frictionless wall) through an abrupt contraction into and through the flow-and temperaturedevelopment region of a smaller coaxial tube. The wall of the small tube...