This paper establishes an asymptotic expansion for a second order differential equation with a small diffusion coefficient, which generalizes the configurational probability diffusion equation of the Rigid Dumbbell Model (RDM) of diluted polymer solutions theory for fast shear flows. This is a singular perturbation problem with turning point.
R.C.Ackerberg and R.E.O’Malley, Boundary layer problems exhibiting resonance, Studies in Applied Mathematics49(3) (1970), 277–295. doi:10.1002/sapm1970493277.
2.
R.B.Bird, R.C.Armstrong and O.Hassager, Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics, Wiley, New York, 1987.
3.
R.B.Bird, R.C.Armstrong and O.Hassager, Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theories, Wiley, New York, 1987.
4.
I.S.Ciuperca and L.I.Palade, On the existence and uniqueness of solutions of the configurational probability diffusion equation for the generalized rigid dumbbell polymer model, Dynamics of Partial Differential Equations7 (2010), 245–263. doi:10.4310/DPDE.2010.v7.n3.a3.
5.
S.Cleja-Ţigoiu and V.Ţigoiu, Rheology and Thermodynamics, Part I, Rheology, Editura Universităţii din Bucureşti (1998).
6.
E.M.De Jaeger and F.Jiang, The Theory of Singular Perturbations, Elsevier, Amsterdam, 1996.
7.
M.V.Fedoryuk, Asymptotic methods in analysis, in: Analysis I: Integral Representations and Asymptotic Methods, R.V.Gamkrelidze, ed., Encyclopaedia of Mathematical Sciences, Vol. 13, Springer, NY, 1989, pp. 83–191. doi:10.1007/978-3-642-61310-4_2.
8.
A.Georgescu, Asymptotic Treatment of Differential Equations, R.J.Koops and K.W.Morton, eds, Applied Mathematics and Mathematical Computation, Vol. 9, Springer, Berlin, 1995.
9.
C.-Y.Jung and R.Temam, Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point, Journal of Mathematical Physics48 (2007), 065301. doi:10.1063/1.2347899.
10.
C.-Y.Jung and R.Temam, Singularly perturbed problems with a turning point: The non-compatible case, Analysis and Applications12(3) (2014), 293–321. doi:10.1142/S0219530513500279.
11.
K.Niijima, On the behavior of solutions of a singularly perturbed boundary value problem with a turning point, SIAM Journal of Mathematical Analysis9(2) (1978), 298–311. doi:10.1137/0509021.
12.
F.W.J.Olver, Sufficient conditions for Ackerberg–O’Malley resonance, SIAM Journal of Mathematical Analysis9(2) (1978), 328–355. doi:10.1137/0509023.
13.
H.-C.Öttinger, A note on rigid dumbbell solutions at high shear rates, Journal of Rheology32(2) (1988), 135–143.
14.
W.E.Stewart and J.P.Sorensen, Hydrodynamic interactions effects in rigid dumbbell suspensions. II. Computations for steady shear flow, Journal of Rheology16(1) (1972), 1–13. doi:10.1122/1.549275.
15.
W.Wasow, Linear Turning Point Theory, Applied Mathematical Sciences, Vol. 54, Springer, New York, 1985.
16.
R.Wong and H.Yang, On an internal boundary layer problem, Journal of Computational and Applied Mathematics144(1–2) (2002), 301–323. doi:10.1016/S0377-0427(01)00569-6.
17.
H.Yang, On a singular perturbation problem with two second-order turning points, Journal of Computational and Applied Mathematics190(1–2) (2006), 287–303. doi:10.1016/j.cam.2005.01.040.
