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The changes in the configuration of a FENE bead-spring chain in a direct numerical simulation of turbulent channel flow and in some simple rheological flows are examined. Unraveling occurs both in uniaxial and shear flows, but the uniaxial flow is more effective. A vortex with a large rotation rate perpendicular to the principal strain of a uniaxial flow has only a minor retarding effect while a small rotation rate delays the unraveling substantially. In a turbulent flow, the chain unravels the most in the viscous sublayer, to about 90% of its fully extended length. It aligns at a 7° angle with the direction of mean flow. In the buffer zone, it unravels and coils up and takes different orientations at different times. Outside the wall region, the chain assumes a coiled configuration. The unraveling of the chain strongly depends on the relaxation time of the chain normalized with the wall shear rate, λ+. A value of λ+=10 exhibits strong unraveling while very weak unraveling is observed below λ+=1.

FENE-P bead-spring chains unravel in the presence of large enough velocity gradients. In a turbulent flow, this can result in intermittent added stresses and exchanges of energy between the chains and the fluid, whose magnitudes depend on the degree of unravelling and on the orientations of the bead-spring chains. These effects are studied by calculating the average behaviour at different times of an ensemble of chains, contained in a fluid particle that is moving around in a random velocity field obtained from direct numerical simulation of turbulent flow of a Newtonian fluid in a channel. The results are used to evaluate theoretical explanations of drag reduction observed in very dilute solutions of polymers.In regions of the flow in which the energy exchange with the fluid is positive, the possibility arises that turbulence can be produced by mechanisms other than the interaction of Reynolds stresses and the mean velocity gradient field. Of particular interest, from the viewpoint of understanding polymer drag reduction, is the finding that the exchange is negative in velocity fields representative of the wall vortices that are large producers of turbulence. One can, therefore, postulate that polymers cause drag reduction by selectively changing the structures of eddies that produce Reynolds stresses. The intermittent appearance of large added shear stresses is consistent with the experimental finding of a stress deficit, whereby the total local shear stress is greater than the sum of the Reynolds stress and the time-averaged shear stress calculated from the time-averaged velocity gradient and the viscosity of the solvent.

Analytical approximate solutions of Duffing and Van der Pol equations as well as the system of coupled Euler-Bernoulli beams and wave equations are under consideration. To this end, the Adomian Decomposition Method (ADM) and variational iteration method (VIM) have been employed to obtain analytical solutions to these differential equations. The results are compared with accurate numerical computations, which show that ADM is a high performance and accurate method to use for the analytical solution of nonlinear physical problems.