High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation
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Journal of Evolution Equations
High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation Peter Constantin , Jiahong Wu, Jiefeng Zhao and Yi Zhu
To Matthias Hieber, with friendship, respect and admiration Abstract. We give a small data global well-posedness result for an incompressible Oldroyd-B model with wave-number dissipation in the equation of stress tensor. The result is uniform in solvent Reynolds numbers and requires only fractional wave number-dependent dissipation (−)β , β ≥ 21 in the added stress.
1. Introduction A class of models of complex fluids is based on an equation for a solvent coupled with a kinetic description of particles suspended in it. In the case of dilute suspensions weakly confined by a Hookean spring potential, a rigorously established exact closure for the moments in the kinetic equation of this Navier–Stokes–Fokker–Planck system yields the Oldroyd-B system [21]. After non-dimensionalization, the coupled OldroydB system is ⎧ 1 ⎪ u = K ∇ · σ, ⎪∂t u + u · ∇u + ∇ p − Re ⎨ (1.1) ∂t σ + u · ∇σ = (∇u)σ + σ (∇u)∗ − W1e (σ − I), ⎪ ⎪ ⎩∇ · u = 0, where σ is the conformation tensor, σ = E(m ⊗ m) with m the end-to-end vector in Rd and E the average with respect to the local distribution, u is the solvent velocity, p is the pressure, Re is the Reynolds number of the solvent, W e is the Weissenberg 1 number, K = γ ReW e and γ is the ratio of solvent viscosity to polymeric viscosity. In the limit of zero Reynolds number, the system (1.1) reduces further and it becomes a nonlinear evolution for σ ∂t σ + u · ∇σ = (∇u)σ + σ (∇u)∗ −
1 (σ − I) We
(1.2)
Mathematics Subject Classification: 35Q30, 35Q35, 35Q92 Keywords: Oldroyd-B, Complex fluid, Weissenberg number, Reynolds number, Global existence, Nonlinear stability, Dissipation.
J. Evol. Equ.
P. Constantin et al.
where u is obtained from σ by solving the Stokes system 1 ∇ · σ, γ We
− u + ∇ p =
∇ · u = 0.
(1.3)
The system (1.2) with (1.3) is an example of an equation which might develop finitetime singularities for large data, even in R2 . The forcing in the right-hand side of (1.3) or in the right-hand side of the momentum equation of (1.1) depends only on the added stress τ = σ − I,
(1.4)
because any multiple of the identity matrix added to σ is balanced by a pressure, even if the factor is a function of space and time. For small added stress, it is known [7] that the system (1.2), (1.3) has global solutions. The problem of global existence of smooth solutions for large data is open and challenging. The large Weissenberg number problem is challenging both numerically and analytically. If we replace the damping term by a wave number-dependent dissipative term, we obtain an equation for the conformation stress ∂t σ + u · ∇σ = (∇u)σ + σ (∇u)∗ − η P(D)(σ − I)
(1.5)
with P(D) being a dissipative differential operator and η a positive number. If a small diffusive term (P(D) = − in (1.5)) is added to the equation for σ coupled with (1.3), then global existence of smooth solutions with arbitrary data has been establis
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