The Koh‐Eringen formulation of nonlinear thermoviscoelasticity is used to study the behavior of a heat‐conducting Rivlin‐Ericksen fluid characterized by a set of constitutive equations involving the coupled dependence of stress and heat flux on temperature gradient and the first two Rivlin‐Ericksen tensors. Slow steady‐state flow of the fluid under a small temperature gradient is considered. With the use of a perturbation technique, the basic equations and boundary conditions are derived for each increasing order of approximation. The first order equations are precisely those of a Newtonian fluid. Succeeding higher order corrections involve linear equations also although the results of preceding lower order approximations are used. The given boundary conditions are satisfied in the first order analysis. Homogeneous boundary conditions are then applicable for higher order corrections. Two specific problems are considered: (1) pressure flow of the fluid between two parallel plates in relative motion and at different constant temperatures; and (2) Couette‐helical flow in the annular region between two coaxial circular cylinders in relative translational velocity parallel to the axis and at two different constant temperatures. Numerical results are plotted for several Eckert numbers and other thermomechanical parameters.