Viscoelastic effects on parallel flow stability, transition to turbulence, and turbulent flow

The classical two dimensional theory of stability of parallel flow is extended to viscoelastic fluids. How the elasticity of the fluid affects the point of stability and determines the point of transition to turbulence is analyzed. In addition the magnification of disturbances is elucidated. This study is based on a viscoelastic constitutive equation which has been successful in predicting the experimental trends of various unsteady high shear rate laminar flows. A viscoelastic stability equation which is an extension to the Orr-Sommerfeld equation for a Newtonian fluid is derived and solved for a flow between parallel plates superimposed by a two-dimensional disturbance. A solution indicates that fluid elasticity minimally shifts the point of instability to lower values of Reynolds\u27 number but to a greater degree than does the second-order/Maxwell stability equation. However, another result shows reduced disturbance magnification for turbulent flow at low wavenumber. The range of values of the disturbance wavenumber for which disturbances grow is diminished at these high Reynolds\u27 numbers and low wavenumber. This may be a trend which offers an explanation to turbulent drag reduction by polymer additives based on viscoelastic properties. It is possible that the reduction in disturbance magnification reduces the turbulence level resulting in a reduction of Reynolds\u27 stresses at the wall