One-parameter equations in describing average-velocity profiles for turbulent flows of weak polymer solutions

Introducing a weak polymer solution into a flowing liquid reduces the resistance coefficient, thickens the viscous sublayer, and increases the velocity gradients. To describe the average-velocity profile in a turbulent flow, one can use one-parameter equations in which the adjustable constants in the logarithmic distributions for the average velocities are expressed in terms of the thickness of the viscous sublayer in the transitional region and the turbulent core, that thickness being $2/\kappa$ where $\kappa$ is the Karman parameter.