## Asymptotic Methods in Fluid Mechanics: Survey and Recent by Herbert Steinrück (auth.), Herbert Steinrück (eds.)

By Herbert Steinrück (auth.), Herbert Steinrück (eds.)

A survey of asymptotic tools in fluid mechanics and functions is given together with excessive Reynolds quantity flows (interacting boundary layers, marginal separation, turbulence asymptotics) and occasional Reynolds quantity flows to illustrate of hybrid equipment, waves as an instance of exponential asymptotics and a number of scales tools in meteorology.

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Nonlinear problems of this type arise in many applications, including steady-state combustion theory where F (w) is an exponential function (cf. Ward et al. (1993)). The primary diﬀerence between the linear problem (1) and the unperturbed problem corresponding to (53) is that, depending on the form of the nonlinearity F (w), the unperturbed problem may have no solution, a unique solution, or multiple solutions. We shall assume that the unperturbed problem has at least one solution, and we will focus on determining how a speciﬁc solution to this problem is perturbed by the presence of the subdomain Ωε .

Consider the steady-state convection-diﬀusion equation for T (X), with X = (X1 , X2 ) posed outside two circular disks Ωj for j = 1, 2 of a common radius a, and with a center-to-center separation 2L between the two disks: κ T = U · ∇T , X ∈ R2 \ ∪j=1 Ωj , X ∈ ∂Ωj , j = 1, 2 , T = Tj , |X| → ∞ . T = T∞ , (102a) (102b) (102c) Here κ > 0 is constant, Tj for j = 1, 2 and T∞ are constants, and U = U(X) is a given bounded ﬂow ﬁeld with U(X) → (U∞ , 0) as |X| → ∞, where U∞ is constant. We introduce the dimensionless variables x, u(x), and w(x) by x = X/γ , T = T∞ w , γ ≡ κ/U∞ .

The constants Aj for j = 1, . . , N and χ satisfy the N + 1 dimensional linear algebraic system deﬁned by (50) and (46). To implement the hybrid method, we must compute the Neumann Green’s function GN and its regular part RN . 3) of Kolokolnikov et al. (2005)) and for a rectangle. In particular, upon representing points as complex numbers, we obtain for the unit disk that GN (x; ξ) = 1 2π − log |x − ξ| − log x|ξ| − ξ 1 3 + (|x|2 + |ξ|2 ) − |ξ| 2 4 , (52a) 1 RN (ξ; ξ) = 2π ξ 3 − log ξ|ξ| − + |ξ|2 − |ξ| 4 .