The problem of the movement of a fluid in an elastohydrodynamic contact is examined in cases of high pressures, high displacement velocities, and low characteristic times, taking into account the nonlinear properties of the fluid — dependence of viscosity on pressure, temperature, and displacement velocity. Simple asymptotic formulas are obtained. The problem is solved numerically in the one-dimensional case. A comparison is made with an experiment on the measurement of the frictional force in an elastohydrodynamic contact.