ON A CYLINDRICAL BENDING OF A PRISMATIC SHELL WITH TWO

CUSPED EDGES UNDER THE ACTION OF AN INCOMPRESSIBLE VISCOUS FLUID^{1}

N. Chinchaladze I.Vekua Institute of Applied Mathematics I.Javakhishvili Tbilisi State University

This paper deals with the proof of a uniqueness theorem of a problem of the interaction of the incompressible viscous fluid and of a thin elastic cusped plate of variable thickness (i.e., prismatic shell).

Let the projection of a prismatic shell be an infinite strip

Ω = {(x_{1}, x_{2}) : −∞ < x_{1} < ∞, 0 ≤ x_{2} ≤ ℓ},

and let the shell thickness be given by the equation

2h(x_{2}) = h_{0}x_{2}^{α/3} (ℓ − x_{2})^{β/3} , h_{0}, α, β = const, h_{0} > 0, α, β ≥ 0.

The edges x_{2} = 0 and x_{2} = ℓ are cusped if α > 0 and β > 0, respectively.

The geometry of the elastic solid along thickness is taken into account in the coefficients of the following bending equation [1]

\frac{∂^{2}}{∂x_{2}^{2}} [x_{2}^{2}(ℓ − x_{2})^{β} \frac{∂^{2}w(x_{2}, t)}{∂x_{2}^{2}}] = −\frac{2ρ^{s}h(x_{2})}{D_{0}} \frac{∂^{2}w(x_{2}, t)}{∂t^{2}} + \frac{q(x_{2}, t)}{D_{0}}, (1)

where D(x_{2}) = D_{0}x_{2}^{α}(ℓ − x_{2})^{β}, D_{0} = const > 0, is a flexural rigidity of the plate, ρ^{s} is a density of solid, w(x_{2}, t) ∈ C^{4}(Ω) ∩ C^{2}(t > 0) ∩ C^{1}(t ≥ 0) is a deflection of the plate, q(x_{2}, t) ∈ C(Ω̄ × {t ≥ 0}) is an intensivity of a lateral load.

On ∂Ω the admissible boundary conditions of the plate theory should be given [1,2].

Let a flow of the viscous and incompressible fluid be independent of x_{1} and parallel to 0x_{2}x_{3} (i.e., v_{1} ≡ 0) with the following conditions at infinity

\begin{matrix}v_{i}(x_{2}, x_{3}, t)|_{x_{3}→∞} = v_{i∞}(x_{2}, t), p(x_{2}, x_{3}, t)|_{x_{3}→∞} = p_{∞}(x_{2}, t), \\ v_{i∞}(x_{2}, t)|_{x_{3}→−∞} = O(1), p_{∞}(x_{2}, t)|_{x_{3}→−∞} = O(1), \\ v_{i∞}(x_{2}, t)|_{x_{2}^{2}→∞} = O(1), p_{∞}(x_{2}, t)|_{x_{2}^{2}→∞} = O(1), i = 2, 3, \end{matrix}  (2)

here v := (v_{2}, v_{3}) ∈ C^{2}(Ω^{f}) ∩ C^{1}(t > 0) ∩ C(t ≥ 0) is a velocity vector of the fluid, p(x_{2}, x_{3}, t) ∈ C^{2}(Ω^{f}) ∩ C(t ≥ 0) is a pressure, v_{2∞}(x_{2}, t), v_{3∞}(x_{2}, t), and p_{∞}(x_{2}, t) ∈ C({] − ∞, 0[∩]ℓ, +∞[} × {t ≥ 0}) are given functions.

Because of the incompressibility

div v(x_{2}, x_{3}, t) = 0, (x_{2}, x_{3}) ∈ Ω^{f}, t ≥ 0,  (3)

1This research was supported by a grant of Georgian President for the university

and (see [5], p.5)

σ_{ij}(x_{2}, x_{3}, t) = −pδ_{ij} + μ (\frac{∂v_{i}}{∂x_{j}} + \frac{∂v_{j}}{∂x_{i}} ) ,  (4)

where σ_{ij} is a stress tensor, μ is a coefficient of velosity, δ_{ij} is Kroneker delta. Further, if the plate is thin, we can assume that: − the fluid occupies the whole space R^{3} but the middle plane Ω of the

plate, i.e., Ω^{f} = R^{3}\Ω.

− values of a normal component of stress tensor, σ_{n3}^{f} (x_{2}, h^{(±)}(x_{2}), t) in the

fluid part Ω^{f} are transferred at appropriate points of Ω from corresponding sides, i.e.,

q(x_{2}, 0, t) = σ_{33}^{f}(x_{2}, 0_{+}, t) − σ_{33}^{f}(x_{2}, 0_{−}, t), x_{2} ∈ [0, ℓ], t ≥ 0;  (5)

− transmission conditions for v_{i}(x_{2}, x_{3}, t) (i = 2, 3) have the forms as follows (see [3], [4])

v_{2}(x_{2}, 0, t) = 0, v_{3}(x_{2}, 0, t) = \frac{∂w(x_{2}, t)}{∂t}, x_{2} ∈ [0, ℓ], t ≥ 0.  (6)

From (3) and (4) a normal component of the stress tensor is equal to

σ_{33}^{f}(x_{2}, x_{3}, t) = −p(x_{2}, x_{3}, t) − 2μ\frac{∂v_{2}(x_{2}, x_{3}, t)}{∂x_{2}}.  (7)

By virtue of (6), from (7) we get

σ_{33}^{f}(x_{2}, 0_{±}, t) = −p(x_{2}, 0_{±}, t).

Hence, (5) has the following form

−p^{+}(x_{2}, t) + p^{−}(x_{2}, t) = q(x_{2}, t), x_{2} ∈ [0, ℓ], t ≥ 0.  (8)

Let the motion of the fluid flow be sufficiently slow i.e., v_{i} and v_{i,j} (i, j = 2, 3) be so small that linearization of Navier-Stokes equations (see [4,5,6,7]) be admissible. Hence,

\frac{∂v_{i}}{∂t} = −\frac{1}{ρ^{f}} gradp + νΔv_{i} + F_{i}(x_{2}, x_{3}, t), ρ^{f}, ν = const, i = 2, 3, (9_{i})

where ρ^{f} is a density of fluid, ν = μ/ρ, F := (F_{2}, F_{3}) is a volume force. Let the initial conditions be

\begin{matrix}v_{i}(x_{2}, x_{3}, 0) = v_{i}^{0}(x_{2}, x_{3}), i = 2, 3, (x_{2}, x_{3}) ∈ Ω^{f}, \\ w(x_{2}, 0) = w^{0}(x_{2}), \frac{∂w(x_{2}, t)}{∂t} = v_{3}^{0}(x_{2}, 0), x_{2} ∈ [0, ℓ]. \end{matrix}  (10)

Theorem 1. The solution of the initial transmission boundary value problem (1), (3), (9_{i}) (i = 2, 3), (10), (6), (8), (2) is unique.

Proof. Let the difference of two admissible solutions of the problem under consideration be v_{i}(x_{2}, x_{3}, t) (i = 2, 3), p(x_{2}, x_{3}, t), q(x_{2}, t), w(x_{2}, t).

For these functions we get equations (1), (3), and

\frac{∂v_{i}}{∂t} = −\frac{1}{ρ^{f}} gradp + νΔv_{i}, i = 2, 3,  (11_{i}),

under the following conditions at infinity

p(x_{2}, x_{3}, t) = 0 when x_{3} → ∞; p(x_{2}, x_{3}, t) = O(1) when x_{2}^{2} + x_{3}^{2} → ∞, (12_{1})
v_{i}(x_{2}, x_{3}, t) = 0 when x_{3} → ∞;
v_{i}(x_{2}, x_{3}, t) = O(1) when x_{2}^{2} + x_{3}^{2} → ∞, i = 2, 3,  (12_{i})

initial conditions

\begin{matrix}v_{i}(x_{2}, x_{3}, 0) = 0 (i = 2, 3), p(x_{2}, x_{3}, 0) = 0, (x_{2}, x_{3}) ∈ Ω^{f}, \\ q(x_{2}, 0) = 0, w(x_{2}, 0) = 0 w,_{t} (x_{2}, 0) = 0, x_{2} ∈ [0, ℓ], \end{matrix}  (13)

and transmission conditions (6), (8).

After differentiation of (11_{i}) with respect to x_{i} and termwise summation, in virtue of (3), we obtain that p(x_{2}, x_{3}, t) is a harmonic function, i.e.,

Δp(x_{2}, x_{3}, t) = 0,  (14)

where Δ = \frac{∂^{2}}{∂x_{2}^{2}} + \frac{∂^{2}}{∂x_{3}^{2}} is a Laplace operator.

The solution of the problem (14), (8), (12_{1}), has the following form (see [8])

p(x_{2}, x_{3}, t) = −\frac{1}{2ωπ} ∫_{0}^{ℓ} \frac{q(ξ, t)x_{3}}{(ξ − x_{2})^{2} + x_{3}^{2}} dξ;

v_{i}(x_{2}, x_{3}, t) (i = 2, 3) can be written as follows

v_{i}(x_{2}, x_{3}, t) = −\frac{1}{ρ^{f}} ∫_{0}^{t} p,_{i} (x_{2}, x_{3}, τ)dτ + ϕ_{i}(x_{2}, x_{3}, t),  (15_{i})

where the first term is a particular solution of (11_{i}), and ϕ_{i}(x_{2}, x_{3}, t) (i = 2, 3), taking into account (11_{i}), (13), (3), (12), (6), are solutions of the problems

ϕ_{i,t} − νΔϕ_{i} = 0,

ϕ_{i}(x_{2}, x_{3}, 0) = 0; ϕ_{i}|_{x_{3}→∞} = 0, ϕ_{i}|_{x_{2}^{2}+x_{3}^{2}→∞} = O(1),  (16_{i})

ϕ_{2}(x_{2}, 0, t) = 0, ϕ_{3}(x_{2}, 0, t) = w,_{t} (x_{2}, t) + \frac{1}{ρ^{f}} ∫_{0}^{t} p,_{3} (x_{2}, 0, τ)dτ, x_{2} ∈ [0, ℓ],

ϕ_{2,2}(x_{2}, x_{3}, t) + ϕ_{3,3}(x_{2}, x_{3}, t) = 0.  (17)

After differentiating (15_{i}) with respect to x_{j} (i, j = 2, 3, i ≠ j) and using (14) it is easy to show that there exists the function ψ(x_{2}, x_{3}, t) given by the expression

ψ,_{i} (x_{2}, x_{3}, t) = v_{i}(x_{2}, x_{3}, t) − ϕ_{i}(x_{2}, x_{3}, t), i = 2, 3.  (18_{i})

By virtue of (11_{i}) and (17), we can find ψ(x_{2}, x_{3}, t) from the the following problem

Δψ = 0, ψ,_{2} (x_{2}, 0, t) = 0, ψ,_{2} |_{x_{2}^{2}+x_{3}^{2}→∞} = O(1).  (19)

(19) means (see [8]) that ψ,_{2} (x_{2}, x_{3}, t) = 0. Therefore ψ,_{33} (x_{2}, x_{3}, t) ≡ 0, and

ψ(x_{2}, x_{3}, t) = c_{1}(t)x_{3} + c_{2}(t).

Taking into account (18_{3}), (12_{3}), and (16_{3}), we get

ψ,_{3} = c_{1}(t) = v_{3} − ψ_{3} = 0, when x_{3} → ∞.

Finally, we obtain ψ,_{3} ≡ 0,

v_{i}(x_{2}, x_{3}, t) − ϕ_{i}(x_{2}, x_{3}, t) = 0, i = 2, 3.  (20)

By virtue of (15_{i}), (20), and p(x_{2}, x_{3}, t) is a function only of t. On the other hand (see (12_{i})) this function is equal to zero at infinity, i.e.,

p(x_{2}, x_{3}, t) = A(t) = 0.  (21)

Further, the transmission condition (8) and equation (1) mean that

q(x_{2}, t) = 0, w(x_{2}, t) = 0.  (22)

In view of (21), (22), from (11_{i}) and (6) we obtain

v_{i,t}(x_{2}, x_{3}, t) − νΔv_{i}(x_{2}, x_{3}, t) = 0, i = 2, 3.  (23_{i})

v_{i}(x_{2}, 0, t) = 0, x_{2} ∈ [0, ℓ], i = 2, 3.  (24)

It is easy to show that the problem (3), (23_{i}), (24), (13), (12) has only trivial solution, i.e., v_{i} ≡ 0, i = 2, 3.

30 Bulletin of TICMI

R e f e r e n c e s

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[2] Jaiani, G.V., Elastic Bodies with Non-smooth Boundaries-Cusped Plates and Shells. ZAMM, 76, Suppl. 2, 1996, 117-120.

[3] Chinchaladze N., Jaiani G., On a Cylindrical Bending of a Prismatic Shell with Two cusped Edges Under Action of an Ideal Fluid. Bulletin of TICMI, 2, 1998, 30-34. (Web-site:http//www.viam.hepi.edu.ge/Others/TICMI)

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[8] Muskhelishvili N., Singular Integral Equations. Noordhoff, 1953.