A numerical solution was developed for steady state heat transfer and pressure drop through circular tubes from the coupled equations of motion and energy. The energy equation was integrated numerically by use of finite difference approximations to obtain local temperatures. Axial conduction, radial convection and viscous dissapation were neglected. The equation of motion was integrated analytically. Radial acid angular velocities were taken as zero. From a one-dimensional momentum balance in the axial direction by assumption of constant pressure drop over each small tube length, newtonian behavior and constant density throughout the radial layers, an integral expression was obtained for the axial velocity as a function of viscosity. The viscosity was considered solely as a function of temperature. Each axial step in the nemerical solution began by calculating the viscosity profile from the temperature profile by means of an empirical relationship based on an experimental fluid viscosity parameter. The fluid was assumed to enter the tube ar a uniform temperature. The velocity profile was then calculated from the viscosity profile by use, of the integrated equation of motion. The temperature profile was then extrapolated a short distance down the tube by the explicit solution of a forward step finite difference form of the energy equation. Temperature and velocity gradients were measured at the wall. For a given experimental fluid viscosity parameter the method was demonstrated on a digital computer with a series of inlet fluid temperatures and a constant tube wall temperature for small temperature driving forces. Numerically calculated values of friction factor were lower than experimentally obtained values, but approached the theoretical value predicted by newtonian flow as isothermal flow was approached. Numerically calculated values of the local Nusselt ne.nber agreed closely with the analytical solution of Yamagata for constant fluid core temperature and constant fluid density.