Вычислительная математика - ИСЭМ СО РАН

CONTINUOUS SOLUTIONS OF BOUNDARY LAYER PROBLEM IN MELT
SPINNING PROCESS MODELING 1
Aliona Dreglea
SI GASIS, Irkutsk
e-mail: adreglea@gmail.com
Abstract. The continuous solutions for BVP of third order nonlinear differential equations appears
in mathematical model of the melt spinning process. The existence theorem is proved for such BVP.
Key words: melt spinning manufacturing process control, ODE, BVP, fluid mechanics.
Let us consider the following problem
x ′′′ (t) + M(x(t), t)x ′′ (t) = 0, α < t < β, (1)
x(α) = a, x ′ (α) = b, x(β) = c. (2)
BVP (1)-(2) appears in the mathematical model of the melt spinning process (see [1] for details). Some
problems of the boundary layer type from hydrodynamics [1] [2], [5] can be formulated as problem (1),
(2).
Let us introduce the following condition:
A) Let function M(x, t) be defined and continuous in the domain
D = {x, t| |x| |a| + |b| |β| + |c − a − bβ| , α t β} ,
{
}
m = min M(x, t), M = max max M(x, t), 0 .
x,t∈D x,t∈D
The existence of the solution in the partial case M(x, t) = x, α = 0, β = 1 is proved (see [8] page
412 − 413).
Here we consider problem (1)-(2) in general case.
Theorem: Let condition A) be satisfied. Then BVP (1), (2) has the solution in the class C (3)
[α,β] .
Proof. The following nonlinear integral equation
is equivalent to the problem (1),(2).
Let us introduce the following notation
∫ t
η∫
α α
x(t) = a + bt + (c − a − bβ)
β∫ η∫
F x (η) =
∫ η
α
α α
e − σ∫
α
M(x(s),s)ds
dσdη
σ∫
. (3)
e − M(x(s),s)ds
α dσdη
e − σ∫
α
M(x(s),s)ds
dσ, α < η < β. (4)
1 Partially supported by Klüber Lubrication KG (Germany) and Dublin Institute of Technology (Ireland).
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