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for λ = λ1 > α/β. It follows from (3.8), (3.17) and (3.18) that the limiting point (x∗, y∗) lies in the first quadrant with ∗∗ y = λ1x > 0. (3.19) The function x is monotonic increasing by (3.13), and x(t) is bounded above by x∗. Hence x(t) has a limiting value, say x∞, as t → ∞, and there holds limt→∞ x(t) = x∞ ≤ x∗. It is easy to show (by contradiction) that there must hold x∞ = x∗. Indeed if there were to hold x∞ < x∗, then the segment 0∞ of the hyperbola (3.7)–(3.8) extending between x = x0 and x = x∞ < x∗ is uniformly bounded away from the lines y = λ1x and y = λ2x, and the right side of the differential equation satisfies a bound (y − λ1x)(y − λ2x) ≥ δ > 0 (3.20) for a fixed positive constant δ, uniformly on a domain con- taining the hyperbolic segment 0∞ but excluding the point (x∗, y∗). It would follow then from the differential equation (3.4) and the bound (3.20) that x(t1) = x∞ for some finite t1 satisfying t0 < t1 ≤ t0 + ε · (x∞ − x0)/δ and also dx(t)/dt > 0 at t = t1. This would contradict the fact that x∞ is the limiting value of x(t) as t → ∞, so x∞ < x∗ is not possible. A similar argument applies to y(t), and this proves limt→∞ x(t) = x∗ and limt→∞ y(t) = y∗. Collecting these results, we have proved the following the- orem based on the stated assumptions (1.4), (1.7), (2.2), (2.3), (2.12) and (2.13). Theorem 1 The dominant (zero torque) equilibrium line y = λ1x is an attractor for the wedge region (3.6) in the first quadrant for the momentum-energy system (3.4)–(3.8). For initial states (x0, y0) in the wedge (3.6), the solution state (x(t), y(t)) satisfies Even though the solution never arrives (in an exact mathemat- ical sense) at the limiting state (3.17) in any finite time, we show in Section 4 that the system moves quickly to a very close proximity of the limiting state when the scaled inertia ε is small. Hence, in a practical sense, the state of the system can be considered to arrive essentially (or effectively) at the equilibrium limiting state in a short time that is proportional to the scaled inertia ε. 4 The Liquid Hammer When we wish to emphasize the dependence of the solution on the initial state (x0, y0) for the problem (3.4)–(3.8), we de- note the solution functions as x(t) = x(t; x0, y0) and y(t) = y(t; x0, y0). (4.1) The following theorem, which gives estimates on the differ- encesx(t;x0,y0)−x∗ andy(t;x0,y0)−y∗ betweenthesolu- tion functions and the coordinates of the corresponding limit- ing equilibrium point (x∗, y∗) of (3.17), is proved using tech- niques from perturbation theory (cf. SMITH [1985]). Theorem 2 Assume that the initial state (x0, y0) lies in the wedge region (3.6), and assume that the system parameters ε, α, β, γ , λ1 and λ2 satisfy the natural inequalites (1.4), (1.7), (2.2), (2.3), (2.12), and (2.13). Then there are fixed positive constantsκ>0,ξ>0andη>0notdependingonεbutde- pending on the initial state and on the other parameters of the problem excluding ε, so that the solution functions (4.1) for the energy-momentum initial value problem (3.4)–(3.8) satisfy the estimates y(t)−λ1x(t)>0 and x(t),y(t)≥0 forall t ≥t0, (3.21) (4.2) where the limiting values x∗ = x∗(x0,y0) and y∗ = so the state of the system remains in the wedge region (1.9). The solution moves steadily along the hyperbola (3.7)–(3.8) toward the limiting state (3.17) located at the point of inter- section of the hyperbola and the attracting line y = λ1x. The solution approaches the limiting point (x∗, y∗) asymptotically along the hyperbola (3.7) with y∗(x0, y0) are given by (3.17), and the quantity y0 − λ1x0 is positive. Proof. Subtract λ1 times the angular momentum equation (3.4) from the linear momentum equation (3.14) and find lim (x(t), y(t)) = (x∗, y∗), t→∞ with (3.22) β(α−γ)y(t)+2αγx(t) |x(t; x0, y0) − x∗| |y(t;x0,y0)−y∗| fort≥t0 ≤ ≤ ξ · |y0 − λ1x0| · e−κ(t−t0)/ε η · |y0 − λ1x0| · e−κ(t−t0)/ε d ε y−λ1x =−A(t) y−λ1x (4.3) dt but the state of the system never arrives at this limiting point in any finite time. A(t): = λ1−2j+2β2y(t)+β(γ−α)x(t) 6 y(t) − λ2x(t) , (4.4)PDF Image | Transient Characteristics of Radial Outflow Turbine Generators
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