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THE CALCULATION OF THE FRONTOGENETIC FUNCTION

1–2

Abdurakhimov B.F.

State University Tashkent, Uzbekistan

Abstract The development of methods of diagnosis and forecasting of atmospheric fronts is one of priority problems of atmospheric dynamics. It is known that in overwhelming majority of cases the diagnosis of fronts is determined by the synoptic analysis. In this paper we develop some quantitative methods of diagnosis of fronts.

Development of methods of the diagnosis and the forecast of atmospheric fronts is one of prioritary problems of atmospheric dynamics. It is known, that in overwhelming majority of cases the diagnostics of fronts is determined by the synoptic analysis. In the present work we develop quantitative methods of the diagnosis of fronts. One parameter on which it is possible to judge the presence of the atmospheric front is the frontogenetic function, representing the total derivative with respect to time from a horizontal gradient of temperature.

Using the thermodynamics equation, the frontogenetic function can be written as

F = dtd |∇θ|= Convergence + deformation + a bend + inflows of heat where convergence = - 2|∇θ|12x2y)(Ux+Vy)

deformation = - −2|∇θ|1 [(θ2x−θy2)(Ux−Vy) + 2θxθy(Uy+Vx)]

bend = - |∇θ|θzxWxyWy) , Inflows of heat = |∇θ|1 ∇θ∇H

where ∇ −horizontal gradient, H - inflows of heat owing to evaporation and condensation, θ- temperature, u, v, w - components of speed of a wind.

To positive values F the process of amplification of horizontal gradients of temperature, i.e. frontogenesis, while to negative values, a negative - frontolise correspond.

As at calculation of the frontogenetic function it is necessary to calculate derivatives from nonlinear terms, vertical speeds, sources and drains high smoothness requirements on the initial data must be imposed. Indeed, a series of calculations which have been carried out on GARP data, has shown high sensitivity of the frontogenetic function on the possible noise in given mea- surements (fig. 1a,b). In addition the noise level quite often appears to be equal in intensity to the level of a useful signal. In order to suppress the noise the median filtration, which is one of methods of nonlinear processing signal

1

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as used. Median filtration keeps sharp differences in fields whereas the usual linear filter smoothes these differences.

Fig. 1 Frontogenetic function without using median filter (a,b). Below after filtration.

GARP dataset H500 and H1000 for 00h. 5 January 1979. forF >0, —-F <0. Fronts position were obtained by using the synoptic analysis.

A series of experiments with GARP data by using the median filter has shown that the best results are obtained with the filter with the aperture 5x5, in case when the field of temperature is exposed to a filtration only. Median filter substantially suppresses noise, allocates a useful signal, leaving constant its site.

Position of areas of positive values of F and change of their intensity can be connected with certain sites of fronts rather easily. The greatest positive values of F are marked along a zone of the Arctic front while at the top of an internal wave begun the occlusion of the cyclone. Concurrence of maxima of positive values of F to the position of frontal sections and tops of waves on them on 12 has forecasts value. Areas of negative values F coincide either with areas of divergence of anticyclones on a surface 500GPa, or with the position of the anticyclones crosspieces - saddles on the ground.

In order to study the structure of the front, the vertical cuts of an atmo- sphere along two parallels 42,50and 47,50n.l., for 00h. January 5, 1979, GARP data were executed. They cross the cold fronts under an ungle of almost 900, but on different distances from the top of a wave. On vertical cuts the fol- lowing parameters were analyzed: temperature, horizontal speed of the wind, relative humidity, vertical speed, convergence, deformation, a bend and total frontogenetic function.

Making separate comparison on the frontogenetic function shows that a bend isentropic surface is sometimes more than its other components. The contribution of heat inflows of in the total frontogenetic function appeared insignificant.

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SINGULAR PERTURBATION FOR A PROBLEM FROM HYDRAULICS

3–6

Narcisa C. Apreutesei

Department of Mathematics Technical University ”Gh. Asachi”, Iasi, Romania [email protected]

Abstract Of concern is the comparison of the solution (u, v) to the nonlinear b.v.p.

(P) below with the solution (uε, vε) of its elliptic regularization (Pε). An asymptotic approximation for (uε, vε) is constructed using the boundary layer method of Vishik-Lyusternik. The order of this approximation is found.

2000MSC: 35J25, 35L20, 41A60

Keywords: boundary function method, regularly perturbed problem, sin- gularly perturbed problem.

1. INTRODUCTION

Consider problem (P) (arising in hydraulics) given by

ut+vx+αu(x, t) =f(x, t), (x, t)∈Q= (0,1)×(0, T)

vt+ux+βv(x, t) =g(x, t), (x, t)∈Q= (0,1)×(0, T), (S) u(0, t) =u(1, t) = 0, 0< t < T, (BC) u(x,0) =u0(x), v(x,0) =v0(x), 0< x <1 (C) and its elliptic regularization (Pε)

εuεtt−uεt =vxε+αuε(x, t)−f(x, t), (x, t)∈Q

εvttε −vtε=uεx+βvε(x, t)−g(x, t), (x, t)∈Q, (Sε) uε(0, t) =uε(1, t) = 0, 0< t < T, (BCε) uε(x,0) =u0(x), vε(x,0) =v0(x), 0< x <1

uε(x, T) =u1(x), vε(x, T) =v1(x), 0< x <1. (Cε) Problem (P) governs the unsteady fluid flow (water-hammer) with nonlin- ear pipe friction. It is also a model for the fluid flow through a tree-structured system of transmission pipelines [4].In problem (P), udenotes the instanta- neous discharge at a point andv is the elevation of hydraulic gradeline.

We construct an asymptotic approximation (Yε, Zε) of (uε, vε) with the aid of the solution of problem (P) and find the order of this approximation in the spacesL2(Q) andC [0, T] ;L2(0,1)

. 3

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This approximation of problem (P) with its elliptic regularization is stronger than those from [3] and from other related papers. In [2],an elliptic regular- ization is also used to approximate the solution of the heat equation.

2. THE ASYMPTOTIC APPROXIMATION

Denote K = L2(0,1) and H = L2(0,1)2. It is known that, under some specific hypotheses, problems (P) and (Pε) have unique strong solutions (u, v) inW1,∞(0, T;H) ([4]) and (uε, vε)∈W2,2(0, T;H),respectively [1].

Our purpose is to investigate the behavior of the solution (uε, vε) of (Pε) as ε→ 0.Remark that (u, v) does not generally satisfy the boundary condition (Cε), therefore at least in some neighborhood (ρ, T] of the final point t = T, functions (uε, vε) and (u, v) are not close to each other. Thus (Pε) is a singularly perturbed problem and the set L = (0,1)×(ρ, T] is a boundary layer. We shall prove that uε−u and vε−v converge to zero in C([0, ρ];K) and construct an asymptotic approximation for (uε, vε) which is valid in the whole domain Q. Using the boundary function method of Vishik -Lyusternik [5],[6],we are looking for an asymptotic approximation (Yε, Zε) of (uε, vε) of the form

Yε(x, t) =u(x, t) + Π (x, τ), Zε(x, t) =v(x, t) + Φ (x, τ), (1) where τ = (T−t)/ε is the boundary layer variable and Π,Φ are boundary layer functions. They satisfy Π (x,∞) = Φ (x,∞) = 0,0< x <1.We find

Π (x, τ) =e−τ[u1(x)−u(x, T)], Φ (x, τ) =e−τ[v1(x)−v(x, T)]. (2) Since in the equations which define the remainder, the second derivativesutt, vtt arise, we need high order regularity results for (P).Suppose that:

(H1)α, β ≥0;

(H2)f, g∈W2,∞(0, T;K) ;

(H3) f(.,0), g(.,0)∈ H1(0,1), u0, v0 ∈ H2(0,1), f(0,0) = f(1,0) = 0, u0(0) =u0(1) = 0, v0 (0) =v0(1) = 0.

Theorem 2.1. If(H1)−(H3)hold, then(u, v)∈W2,∞(0, T;H)andux, vx ∈ L(0, T;K).

Proof. One differentiates formally problem (P) with respect tot.Denoting U =ut, V =vt,one obtains the equation

Ut+Vx+αU(x, t) =ft(x, t), (x, t)∈Q= (0,1)×(0, T) Vt+Ux+βV (x, t) =gt(x, t), (x, t)∈Q= (0,1)×(0, T), with the boundary conditions

U(0, t) =U(1, t) = 0, 0< t < T,

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and

U(x,0) =U0(x), V (x,0) =V0(x), 0< x <1,

whereU0(x) =f(x,0)−αu0(x)−v0 (x) andV0(x) =g(x,0)−βv0(x)−u0(x), forx∈(0,1).

One shows that this problem admits a unique strong solution (U, V) ∈ W1,∞(0, T;H). LetS(t), t≥0 be the nonlinear semigroup generated by the linear operator A:D(A)⊆H×H→H×H,

D(A) ={(U, V)∈H, U(0, t) =U(1, t) = 0, 0< t < T}, A(U, V) = (αU, βV).

With the aid of this semigroup and of the constant variation formula, we express (U, V) and show that it coincides with the derivatives (ut, vt) of the solution of (P).Thus we conclude the proof.

Now we are interested to find the order ofuε−Yε, vε−Zε,where (Yε, Zε) is given in (1).The main result is the following.

Theorem 2.2. If u1 ∈ H01(0,1), v1 ∈ H1(0,1), then under assumptions (H1)−(H3), for sufficiently small ε >0,the pair (Yε, Zε) is an asymptotic approximation of (uε, vε) in the entire domainQ and we get the estimates:

|uε−u|L2(Q)=o ε1/2

, |vε−v|L2(Q)=o ε1/2

, (3)

|uε−u−Π|C([0,T];K) =o ε1/4

, |vε−v−Φ|C([0,T];K)=o ε1/4

. (4) Conclusions.a) If u(., T) =u1, v(., T) =v1, then (u, v)is an asymptotic approximation of (uε, vε) in the entire domain Q. Problem (Pε) is regularly perturbed and estimates (4)hold with Π = Φ = 0.

b) Otherwise, for every ρ ∈ (0, T), (u, v) is an asymptotic approximation of (uε, vε) in Q1 = (0,1)×(0, ρ], (Pε) is singularly perturbed and we have

|uε−u|C([0,ρ];K)=o ε1/4

, |vε−v|C([0,ρ];K) =o ε1/4

. (5)

References

[1] A.R. Aftabizadeh, N.H. Pavel, Boundary value problems for second order differential equations and a convex problem of Bolza, Diff. Integral Eqns.2(1989), 495-509.

[2] N. Apreutesei, An asymptotic analysis for a semilinear elliptic equation, Comm. in Applied Analysis,7, 1 (2003), 67-77.

[3] L. Barbu, G. Morosanu, Asymptotic analysis of the telegraph equations with non-local boundary conditions,PanAmerican Math. J.,8(1998), no.4, 13-22.

[4] V. Hokkanen, G. Morosanu, Functional methods in differential equations, Chap- man&Hall/CRC, Boca Raton, 2002.

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[5] A. Vasilieva, V. Butuzov, L. Kalachev, The boundary function method for singular perturbation problems, SIAM, Philadelphia, 1995.

[6] M. Vishik, L. Lyusternik, Regular degeneration and boundary layer for linear differ- ential equations with small parameter multiplying the highest derivatives,Usp. Mat.

Nauk, 12 (1957), 3-122 (in Russian), Amer. Math. Soc. Transl., Ser. 2, 20 (1962), 239-364.

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GEOMETRIC DYNAMICS OF CUTTING MACHINES

7–10

Dumitru B˘al˘a

,,Gheorghe Anghel” University, Drobeta Turnu Severin dumitru [email protected]

The kinematic scheme of the teeth-wheel cutting machine, Romanian pro- duction FD-320A is represented in fig. 1.

Fig. 1.

In it three branches can be distinguished: the branch from the pointME to the point A; the branch from the point A to the tool screw milling; the branch the point A to the piece. Shortly, these branches will be named: uncommon branch, common branch (of the tool) and working surface branch.

In this paper we apply the Udri¸ste method [6] in order to obtain further information regarding the description of the cutting machine behavior during the threading. In order to increase the threading capacity of the teeth wheel cutting machine FD-320A it is necessary to diminish as much as possible the vibrations that appear during the cutting process. The increase of the cutting capacity and of the quality of the working surfaces determines the increasing of the production capacity. In order, to study the vibrational movement of the elastic system presented in fig. 2d from [5] we use the Riemann manifold (R6, δij).

7

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Fig. 2.

The vibrations of the elastic system are modeled by the solutions of the dif- ferential kinematic system

8

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:

˙ x1=x2,

˙

x2=Cme

Jme

CS(x3x1) +CP(x5x1) Cme+CS+CP

,

˙ x3=x4,

˙ x4=CS

JS

Cme(x1x3) +CP(x5x3) Cme+CS+CP

,

˙ x5=x6,

˙ x6=CP

JP

Cme(x1x5) +CS(x3x5) Cme+CS+CP

,

(1)

where the constants Jme, CS, CP represent the inertia moment and two constants of elasticity respectively. The index selection took into account the electrical engine, tool and piece. Let us build the Lagrange extension by Udri¸ste method. To this aim we introduce the,,cutting machine” vector field which has the following six components

8

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:

X1(x1, x2, x3, x4, x5, x6) =x2, X2(x1, x2, x3, x4, x5, x6) = Cme

Jme

CS(x3x1) +CP(x5x1) Cme+CS+CP

, X3(x1, x2, x3, x4, x5, x6) =x4,

X4(x1, x2, x3, x4, x5, x6) = CS

JS

Cme(x1x3) +CP(x5x3) Cme+CS+CP

, X5(x1, x2, x3, x4, x5, x6) =x6,

X6(x1, x2, x3, x4, x5, x6) = CP

JP

Cme(x1x5) +CS(x3x5) Cme+CS+CP

,

(2)

whereX= (X1, X2, X3, X4, X5, X6) andx= (x1, x2, x3, x4, x5, x6).

The autonomous system (1) has the form dxi

dt =Xi(x1, x2, x3, x4, x5, x6), i= 1, 6. (3)

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The equilibria of (1) are x = (a,0, a,0, a,0), where a is a constant. The function

f = 1 2

X6

i=1

Xi2 (4)

represents the energy density associated with the vector field ,,cutting ma- chine” and the Euclidian structure δij). The geometric dynamics associated with the elastic system is described by the second order differential system

d2xi

dt2 = ∂f

∂xi+X

j

∂Xi

∂xj −∂Xj

∂xi dxj

dt , i, j= 1, 6, (5) which proves to be an Euler-Lagrange extension. In other words, the La- grangian L = 1

2 P6 i=1

dxi

dt −Xi 2

or L = 1 2

P6 i=1

dxi

dt 2

− P6

i=1

Xidxi

dt +f determines the system (5) (which has 6 degrees of freedom. The trajec- tories of (5) contain also the solutions of the system (1). This system de- scribes a pregeodesic movement in a gyroscopical field forces. More precisely,

P6 i=1

∂Xi

∂xj −∂Xj

∂xi dxj

dt is the gyroscopical force and ∂f

∂xi represents a conser- vative force. In order to see if the trajectories are pregeodesic we build the associated Hamiltonian H = 1

2 P6 i=1

dxi

dt 2

−f, and the geometric structure formed of Riemann metricsgij = (H+f)δij,and of the nonlinear connection Nji = Γijk−Fji,where Γijk is the Riemann connection induced bygij metrics.

The skqewsymmetric matrix ofFji elements

FjiihFjh, (6)

Fij = ∂Xj

∂xi − ∂Xi

∂xj (7)

corresponds to the rotor in the case of three dimmensional space.

The solutions of the system (5) (computations using (1),(2),(4) were made) are horizontal pregeodesics on the Riemann-Jacobi-Lagrange manifold (R6 Omega, gij, Nji), where Ω is the set of equilibrium points, and Nji is a non- linear connection.

Theorem 0.3. Any nonconstant trajectory of the dynamic system associated with (5), which has the total energy H (constant), is a reparameterized hor- izontal pregeodesic of the Riemann-Jacobi-Lagrange manifold (R6 Ω, gij = (H+f)δij, Nji = Γijk+Fji, i, j = 1, 6).

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Because the new detaching of the splint maintains the vibration process, the phenomenon is called a regenerative vibration. It is known that the delay is the central idea of the regenerative effect. In analyzing the nonlinear model of the vibrating tool machine, mathematically described by differential equations with a delay [2] the following stages are followed: 1) the analysis of the linear part of the system

X(t) =˙ L(p)X(t) +R(p)X(t−τ) +F(X(t), X(t−τ), p), (8) whereL(p) =

0 1

−(1 +p) −2ξ

, R(p) =

0 0 p 0

,

F(X(t), X(t−τ), p) = 3p 10

0

(x1(t)−x1(t−τ))2−(x1(t)−x1(t−τ))3

; 2) the analysis of the roots of the characteristic equation and Hopf bifurca- tion; 3) the determination of the generalized eigensubspaces associated with the system (6) at the Hopf bifurcation point; 4) the determination of the cen- tre manifold at the bifurcation point and the associated limit cycle; 5) the study of the orbit of the system (6) and the specification of the invariants of the limit cycle. Starting with the simplified scheme of the dynamical system machine-tool-piece-contrivance-tool during the cutting process, the stability of the cutting machine functioning is studied [1]. Actually, the aim is the lifting of the stability charts. They are graphical representations which have on the abscissa the n revolutions of the machine and on the ordinal the w value of the threading depth. In order to calculate the revolution of the cutting and of the threading depth a computation algorithm was formulated too.

References

[1] B˘al˘a, D., Sila¸s, Gh.,Probleme privind stabilitatea dinamicii ¸si echilibrului unor sisteme mecanice, Ed. S¸coala Mehedint¸iului, Dr. Tr. Severin 2001.

[2] B˘al˘a, D.,Metode geometrice ˆın studiul mi¸sc˘arilor sistemelor vibrante ¸si vibropercutante, Tez˘a de doctorat, Timi¸soara, 2004.

[3] Mircea, G., Neamt¸u, M., Opri¸s, D.,Sisteme dinamice din economie, mecanic˘a, biologie descrise prin ecuat¸ii diferent¸iale cu argument ˆıntˆarziat, Ed. Mirton, Timi¸soara, 2003.

[4] Mircea, G., Neamt¸u, M., Opri¸s, D.,Bifurcat¸ie Hopf pentru sisteme dinamice cu argu- ment ˆıntˆarziat ¸si aplicatii, Ed. Mirton, Timi¸soara, 2004.

[5] Moraru, V., Ispas, C., Rusu, S¸t.,Vibrat¸iile ¸si stabilitatea ma¸sinilor unelte, Ed. Tehnic˘a, Bucure¸sti, 1982.

[6] Udri¸ste, C., Postolache, M.,Atlas of magnetic geometric dynamics, Geometry Balkan Press, Bucharest, 2001.

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CONTINUITY OF CHARACTERISTICS OF A THIN LAYER FLOW DRIVEN BY A SURFACE TENSION GRADIENT

11–16

Emilia Borsa, Adelina Georgescu

University of Oradea, University of Pite¸sti

Abstract The flow of a thin layer down an inclined rigid impermeable plane driven simultaneously by the gravity and a surface tension gradient is considered.

Geometric and mechanical reasons suggested us the continuity of the depth of the layer, the pressure and the volume flux of the fluid. By using suitable asymptotic expansions it was shown that, indeed, this is the case.

2000 MSC: 76D08

Keywords: viscous flow, thin-layer approximation, surface tension gradi- ent

1. PROBLEM FORMULATION

Consider the flow of a thin liquid layer and use a lubrication (thin-layer) approximation. The liquid is a Newtonian viscous fluid, of constant densityρ and coefficient of dynamic viscosityµ. The thin layer has a steady flow, down a plane inclined at an angleα, 0≤α≤πto the horizontal. The flow is driven simultaneously by gravity and a surface tension gradient,∂γ/∂x >0.

With respect to the Cartesian coordinates system Oxyz as indicated in fig.1,

Fig.1. The geometry of the thin layer.

the fluid velocity has the form u=u(y, z)i. Introduce the notation: p is the pressure in the fluid , pa is the atmospheric pressure, g is the gravitational acceleration, γ is the reference value of surface tension, β is the angle of the three-phase contact line (it is assumed constant, such thatβ < π/2), 2ais the

11

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width of the layer, hm is the maximum depth of the liquid (hm =h(0)). A constant value of β means that any contact angle hysteresis is ignored. We neglected the viscosity of the air above the liquid and we took the surface curbature to be h′′ = d2h/dy2, because h2 << h. Denote by z = h(y) the free-surface profile. Then, the Navier-Stokes equations are

0 =−px+ρgsinα+µ(uyy+uzz),0 =−py,0 =−pz−ρgcosα. (1) In the thin-layer theory, [1], [5] these equations reduce to

0 =−px+ρgsinα+µuzz,0 =−py,0 =−pz−ρgcosα. (2) and are to be integrated subject to the boundary conditions

u= 0, onz= 0; (3)

p−pa=−γh′′, γuz=∂γ/∂xon z=h(y). (4) The origin of the stress τ could be very diverse. We are interested in the case whenτ comes from a local variation of surface, i.eτ =∂γ/∂x.

Impose the following contact conditions

h= 0, h=±tanβ, aty=±a. (5)

The angle of the layer and the horizontal straightline belongs to the follow- ing three cases : i) α ∈ (0, π/2), ii)α = π/2 , iii)α ∈ (π/2, π) . Taking into account the closed form expressions (Section 2) for the velocity profile, free surface profile, pressure and nondimensional volum flux, in all these cases, in Section 3 their continuity atα=π/2 is shown.

2. CLOSED FORM SOLUTIONS

In [2] we obtained the following results. The velocity of the fluidu(y, z) is u(y, z) = ρgsinα

2γ (−z2+ 2zh) + 1 µ·∂γ

∂x·z. (6)

The free surface profide z=h(y) is given by h(y) = a·tanβ

B ·coshB−coshBξ

sinhB , α∈(0, π/2) (7) h(y) = a·tanβ

B ·B·1−ξ2

2 , α=π/2 (8)

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h(y) = a·tanβ

B ·cosBξ−cosB

sinB , α∈(π/2, π) (9)

whereB =a(ρg|cosα|γ )1/2, forα 6=π/2;B = 0, forα=π/2,ξ =y/a∈[−1,1] . The pressure of the fluid p=p(z) has the expression

p(z) =pa−ρgzcosα+tanβp

ρgγ|cosα|



cothB, α∈(0, π/2) B−1, α=π/2 cotB, α∈(π/2, π)

(10) The nondimensional volume flux reads

Q=F(B) +9 2

∂γ

∂x 1 B

1

tanβ|tanα|G(B), (11) where

F(B) = 15Bcoth3B−15coth2B−9BcothB+ 4, α∈(0, π/2) (12) F(B) = 12

35B4, α=π/2 (13)

F(B) =−15Bcot3B+ 15cot2B−9BcotB+ 4, α∈(π/2, π) (14) and

G(B) = 3Bcoth2B−3cothB−B, α∈(0, π/2) (15) G(B) = 4

15B3, α=π/2 (16)

G(B) = 3Bcot2B−3cotB+B, α∈(π/2, π). (17) Remark 2.1. Rigorously speaking, for the caseα=π/2, these formulae cotain nondeterminations of the form0/0. It is easy to see that the elimination of this nondetermination proceeds immediately. However, we kept the above forms for the sake of symmetry. On the other hand, this kind of expressions are currently used in the litterature on thin layers.

3. CONTINUITY OF

H

,

P

AND

Q

AT

α= π/2 Some of the above expressions are sufficiently complicated to allow us to decide ifh,pand Qare continuous or not atα=π/2 . The asymptotic study carried out in this section shows that they are continuous, indeed.

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3.1. CONTINUITY OF H(Y)

For the sake of simplity, we consider the following functions f1(x) = coshx−coshxξ

xsinhx , f2(x) = cosxξ−cosxxsinx ,where B:=x. The asymptotic expan- sion of f1(x) and f2(x),as x→0, are

f1(x)

1 +x2!2 +...

1 +x22!ξ2+...

x

x

1!+x3!3 +...

=

x2

1 2!ξ2!2

+...

x2+x3!4 +... =

1−ξ2 2!

1 +x3!2 +... 1ξ2 2 ;

f2(x)

1x22!ξ2+x44!ξ4 +...

1x2!2+x4!4...

x

x

1!x3!3 +x5!5 ...

=

x2

1 2!ξ2!2

+...

x2x3!4 +... 1ξ2 2 .

Here we used [6] and the asymptotic expansions cosx ∼ 1− x2!2 + x4!4... and sinx∼ 1!xx3!3+x5!5−...,as x→0.In fact, 1−ξ22 is the asymptotic representation of f1(x) and f2(x),as x→0 and it was obtained by performing the asymptotic expansion of the involved sines and cosines. The same results can by obtained by using the lHospital rule, namely

x→0limf1(x) = lim

x→0

sinhx−ξsinhxξ sinhx+xcoshx = lim

x→0

coshx−ξ2coshxξ

xsinhx+ 2coshx = 1−ξ2 2 ;

x→0limf2(x) = lim

x→0

sinx−ξsinxξ sinx+xcosx = lim

x→0

cosx−ξ2cosxξ

−xsinx+ 2cosx = 1−ξ2 2 . Therefore

αրπ/2lim h(y, α) =h(y, π/2) (18)

and

αցπ/2lim h(y, α) =h(y, π/2). (19)

Here α is the asymptotic variable.The relations (18), (19) and (8) show that h(y,α) is continuous atα=π/2.

3.2. CONTINUITY OF P

Consider the following functions g1(x) =xcothxand g2(x) =xcotx, where B:=x. Using the asymptotic expansions [5] cothx ∼ 1x + x3x453 + 2x9455 −...

and cotx ∼ 1xx3x4532x9455 −..., as x→ 0, we have g1(x) ∼ x(1x + x3

x3

45 + 2x9455 −...) = 1 + ..., g2(x) ∼ x(1xx3x4532x9455 −...) = 1 + ..., and, by means of the l Hospital rule, we get lim

x→0g1(x) = lim

x→0

coshx+xsinhx coshx = 1;

(15)

x→0limg2(x) = lim

x→0 x

tanx = lim

x→0cos2x= 1,whence

αրπ/2lim p(y, α) =p(y, π/2), (20)

αցπ/2lim p(y, α) =p(y, π/2). (21)

The relation (20), (21) and (10) show that p(y,α) is continuous atα=π/2.

3.3. CONTINUITY OF

Q

The following asymptotic representations F(B) ∼ 1235B4 , G(B) ∼ 154B3 , as B → 0 (i.e. as α → π/2) hold. Indeed, consider the functions F1(x) =

15xcoth3x−15xcoth2x−9xcothx+4

x4 ,F2(x) = −15xcot3x+15cotx4 2x−9xcotx+4,

G1(x) = 3xcoth2x−3cothx−xx3 ,G2(x) = 3cot2x−3cotx+xx3 , and take into account that coth2x∼ x12 +x92 +...+232x452 +4x94542x1354 +..., cot2x∼ x12 +x92 +...−23

2x2

454x9454 +2x1354 +...

Remark that the first nonvanishing term of the asymptotic expansion of F1, F2, G1, G2 is obtained if the first four terms in the asymptotic expansion of coth2x and cot2x are considered. We have

F1(x) ∼ 5· 1x(1x + x3x453 + 2x9455 +...)·[3x(

1

x2+x92+...+232x452+4x94542x1354+...)

x3

3(1x+x3x453+2x9455−...)−x

x3 ] + −4x(1xx3x

3

45+2x9455+...)+4

x4 =

= (x52+535x452+10x9454+...)(15494536x2+...)+4548x9452+...= 49+45418936 = 1235, F2(x) ∼ 5· 1x(1xx3x4532x9455 −...)·[−3x(

1

x2+x92+...−232x4524x9454+2x1354+...)

x3 +

3(1xx3x4532x9455−...)−x

x3 ] + −4x(1xx3x

3

452x9455+...)+4

x4 =

= (x52535x45210x9454−...)(−45994530x24539456 x2...) +454 +8x9452+...= 1235, G1(x)∼ x32(x922x452 +4x94542x1354 +...)− x33(1x +x3x453 +2x9455 +...)−x12 =

1

3456 +12x94526x1352 +4536x9452 +...= 13453 = 154 ,

G2(x)∼ x32(x922x4524x9454 +2x1354 +...)− x33(1xx3x4532x9455 −...)−x12 =

1

345612x94526x1352x34 +x12 +453 +6x9452 +...+x12 = 13453 = 154 and , by the lHopital rules, we get lim

x0F1(x) =

x→0lim

15xcoshx(1 +sinh2x)15sinhx(1 +sinh2x)9xcoshxsinh2x+ 4sinh3x

x4sinh3x =

= lim

x0

15xsinhx27coshxsinh2x+ 6xsinh3x+ 12cosh2xsinhx 4x3sinh3x+ 3x4sinh2xcoshx =

= lim

x0

36sinh2x+ 36xcoshxsinhx

12x2sinh2x+ 20x3sinhxcoshx+ 3x4+ 6x4sinh2x=

(16)

= lim

x0

36(x+ 2xsh2xshxchx)

24xsinh2x+ 84x2sinhxcoshx+ 32x3+ 64x3sinh2x+ 12x4sinhxcoshx =

= lim

x→0

144xsinhxcoshx

180x2+ 24sinh2x+ 216xsinhxcoshx+ 360x2sinh2x+ 12x4+ 176x3sinhxcoshx+ 24x4cosh2x =

= lim

x→0

144xcoshx

180sinhxx + 24sinhxx + 216coshx+ 360xsinhx+ 12x3sinhxx + 176x2sinhxcoshx+ 24x3coshx =

=1235 ;

x→0limF2(x) = lim

x→0

15xcos3x+15sinxcos2x9xcosxsin2x+4sin3x

x4sin3x = lim

x→0

27xcos2x27sinxcosx+9xsin2x

7x5 =

xlim0

−36xcosx+36sinx 35x3 = lim

x0 36xsinx

35·3x2 =1235;

xlim0G1(x) = lim

x0

4xsinhxcoshx−4sinh2x

3x2sinh2x+2x3sinhxcoshx = lim

x0

4xsinhx

6xsinhx+3xcosh2x+6xcosh2x+2x3sinhx =

x→0lim

4

6+3coshx·sinhxx +6coshx·sinhxx +2x2 =154 ,

xlim0G2(x) = lim

x0

3xcos2x3sinxcosx+xsin2x

x3·sin2x = lim

x0

4xcosx+4sinx 5x3 = lim

x0 4xsinx

15x2 = 154 ;

whence

αրπ/2lim Q(y, α) =Q(y, π/2) (22)

and

αցπ/2lim Q(y, α) =Q(y, π/2). (23)

The relations (22), (23) and (11)-(17) show that Q(y, α) is continuous at α=π/2 .

Consequently, by direct asymptotic expansions and by the lHopital rule, we proved that the following characteristics h(y, α) ,p(y, α) and Q(y, α) of the flow of the thin layer are continuous atα=π/2 .

References

[1] Acheson, D.J.,Elementary fluid dynamics, Oxford University Press, 1990,238–259.

[2] Borsa, E., Viscous fluid flows driven by surface tension gradients, Series of Applied and Industrial Mathematics,17, Ed. University of Pitesti, 2004.(Romanian)

[3] Georgescu, A., Asymptotic treatment of differential equations, Chapman & Hall, London, 1995. (first published by Ed. Tehn., Bucharest, 1989).

[4] Chifu, E., Gheorghiu, C.I.and Stan, I., Surface mobility of surfactant solution.

Numerical analysis for the Marangoni and gravity flow in a thin liquid layer of triangular section, Rev. Roumaine Chim.,29, (1984), 31–42.

[5] Ockendon, H., Ockendon, J.R.,Viscous flows, Cambridge University Press, 1995.

[6] D.J., RIjic, I.M., Grandstein, I.S., Tables of integralls, sums, series and products, Ed. Tehn., Bucharest, 1955. (Romanian)

[7] Tipei, N.,Theory of lubrication, Stanford University Press, 1962.

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DISTRIBUTION OF DYNAMIC PERTURBATION TO THE TWO-COMPONENT ELASTIC MEDIUM

17–23

Ghenadie Bulgac, Ion Naval

Abstract The present paper is devoted to the solution of the non-stationary self-similar problems a linearly-elastic waves in the quarter-space, filled with a two-compo- nent medium, under the action of impulse external force of the boundary. We explain the idea of Smirnov-Sobolev method of functionally invariant solutions, using as an example the problem of the shock of quarter of elastic space on rigid fixed barrier.

Keywords: linearly-elastic waves, two-component medium, Smirnov-Sobolev method, func- tionally invariant solutions, self-similar solution.

Herein we explain the idea of the Smirnov-Sobolev method of functionally invariant solutions, using as an example the problem of shock of a quarter of elastic space on a rigid fixed barrier. The material is considered to be homogeneous and isotropic, composed of two rigid media. It is supposed that the medium moves uniformly with velocityv0, parallel to thexOy plane, and at an instant t=0 it strikes at a rigid fixed barrier in the plane xOz. The xOz plane is considered to be free of strains. When t= 0, on the plane xOz, shearing stresses are equal to zero, i.e. there is the possibility of sliding.

It is necessary to study the flat ondular movement that satisfies, forx≥ 0, y≥ 0, the equations of motion. In Ui, Vi (i= 1,2) these equations read

Ai12Ui

∂ x2 +Ai122Ui

∂ y2 + (Ai1Ai2)∂ x∂ y2Vi +Bi12U3i

∂ x2 +Bi22U3i

∂ y2 + + (Bi1Bi2)

2V3−i

∂ x∂y ==ρ1i2U1

∂ t2 +ρi22U2

∂ t2 (1)ib ∂U∂ t1 ∂U∂ t2

, Ai12Vi

∂ x2 +Ai122Vi

∂ y2 + (Ai1Ai2)∂ x∂ y2Ui +Bi12V3−i

∂ x2 +Bi22V3−i

∂ y2 + + (Bi1Bi2)

2U3−i

∂ x∂y ==ρ1i2V1

∂ t2 +ρi22V2

∂ t2 (1)ib ∂V∂ t1 ∂V∂ t2,

(1)

where Ai1i+ 2µi+ (−1)i ρ3−iρα2; Ai2i−λ5; Bi12+i+ 2µ3+

ρiα2

ρ (−1)i;Bi253; (i= 1, 2).

The solution of the system (1) should satisfy the initial conditions U1 = 0, ∂U1

∂ t =−V0, U2 = 0, ∂U2

∂ t =−V0,

V1= 0, ∂V1

∂ t = 0, V2 = 0, ∂V2

∂t = 0 (2)

17

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