A transform fault is modeled as a vertical cut through an elastic layer (schizosphere) of thickness H overlying a viscous substrate (plastosphere). We consider a steady transform motion accommodated in the schizosphere wholly by slip on the fault and in the plastosphere, insofar as possible, by viscous flow. For the case where the viscosity in the plastosphere is strain rate dependent but independent of temperature, the velocity solution in the plastosphere is Vθ/π, where V is the slip rate on the fault in the schizosphere and r and θ are the cylindrical coordinates with the origin at the bottom of the fault. The viscous stress is singular at the bottom of the fault (r = 0) and exceeds the brittle (frictional) strength for r r0. Equating the brittle strength to the viscous stress defines the brittle--ductile boundary r0 in the plastosphere as a function of H and viscosity. The additional condition that r0 must be small allows the viscosity to be estimated from H. For small r, the temperature-independent solution is a valid approximation to the temperature-dependent solution, and the relation between viscosity and H should remain valid. From the temperature-independent model, we estimate that self-heating due to dissipation in the plastosphere for reasonable Earth parameters is less than ~20¿C.