Instability at the Melting Threshold of Laser Irradiated Silicon as the Underlying Origin of the Ripples Formation

1983 ◽  
Vol 23 ◽  
Author(s):  
M. Combescot ◽  
J. Bok ◽  
C. Benoit A La Guillaume
Keyword(s):  
Steady State ◽  
Penetration Depth ◽  
Steady State Solution ◽  
Critical Value ◽  
Undercooled Liquid ◽  
Strong Decrease ◽  
Pattern Size ◽  
Melting Threshold ◽  
Diffraction Effects ◽  
Previous Idea

ABSTRACTThe increase of reflectivity associated with a strong decrease of the laser penetration depth at the melting threshold of laser irradiated silicon induces a symmetry breaking with formation of a mixture of solid and liquid regions. We present a steady state solution in the case of solid and liquid stripes and we show that the liquid regions are slightly hotter than solid ones in contradiction with the previous idea of an undercooled liquid. The pattern size has to be smaller than a critical value of the order of the laser penetration depth, and can be selected by additional interference and diffraction effects.

scholarly journals ESTIMATES OF BLOW-UP TIME FOR A NON-LOCAL REACTIVE–CONVECTIVE PROBLEM MODELLING OHMIC HEATING OF FOODS

2006 ◽  
Vol 49 (1) ◽  
pp. 215-239 ◽  
Author(s):  
C. V. Nikolopoulos ◽  
D. E. Tzanetis
Keyword(s):  
Steady State ◽  
Initial Data ◽  
Blow Up ◽  
Ohmic Heating ◽  
Steady State Solution ◽  
Critical Value ◽  
Stationary Solutions ◽  
State Solution ◽  
Non Local ◽  
The Difference

AbstractIn this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, $u_t+u_{x}=\lambda f(u)/(\int_{0}^1f(u)\,\mathrm{d} x)^{2}$, together with initial and boundary conditions. It is known that, for $f(s)$, $-f'(s)$ positive and $\int_0^\infty f(s)\,\mathrm{d} s\lt\infty$, there exists a critical value of the parameter $\lambda>0$, say $\lambda^\ast$, such that for $\lambda>\lambda^\ast$ there is no stationary solution and the solution $u(x,t)$ blows up globally in finite time $t^\ast$, while for $\lambda\leq\lambda^\ast$ there exist stationary solutions. Moreover, the solution $u(x,t)$ also blows up for large enough initial data and $\lambda\leq\lambda^\ast$. Thus, estimates for $t^\ast$ were found either for $\lambda$ greater than the critical value $\lambda^\ast$ and fixed initial data $u_0(x)\geq0$, or for $u_0(x)$ greater than the greatest steady-state solution (denoted by $w_2\geq w^*$) and fixed $\lambda\leq\lambda^\ast$. The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given $\lambda$, $\lambda^*$ and $0\lt\lambda-\lambda^*\ll1$, estimates of the following form were found: upper bound $\epsilon+c_1\ln[c_2(\lambda-\lambda^*)^{-1}]$; lower bound $c_3(\lambda-\lambda^*)^{-1/2}$; asymptotic estimate $t^\ast\sim c_4(\lambda-\lambda^\ast)^{-1/2}$ for $f(s)=\mathrm{e}^{-s}$. Moreover, for $0\lt\lambda\leq\lambda^*$ and given initial data $u_0(x)$ greater than the greatest steady-state solution $w_2(x)$, we have upper estimates: either $c_5\ln(c_6A^{-1}_0+1)$ or $\epsilon+c_7\ln(c_8\zeta^{-1})$, where $A_0$, $\zeta$ measure, in some sense, the difference $u_0-w_2$ (if $u_0\to w_2+$, then $A_0,\zeta\to0+$). $c_i\gt0$ are some constants and $0\lt\epsilon\ll1$, $0\ltA_0,\zeta$. Some numerical results are also given.

scholarly journals On the Convergence of the Modified Riccati Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Nicholas Assimakis ◽  
Maria Adam
Keyword(s):  
Kalman Filter ◽  
Steady State ◽  
Target Tracking ◽  
Riccati Equation ◽  
Detection Probability ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Steady State Solution ◽  
Critical Value ◽  
Solution Algorithm

The modified Riccati equation arises in the implementation of Kalman filter in target tracking under measurement uncertainty and it cannot be transformed into an equation of the form of the Riccati equation. An iterative solution algorithm of the modified Riccati equation is proposed. A method is established to decide when the proposed algorithm is faster than the classical one. Both algorithms have the same behavior: if the system is stable, then there exists a steady-state solution, while if the system is unstable, then there exists a critical value of the measurement detection probability, below which both iterative algorithms diverge. It is established that this critical value increases in a logarithmic way as the system becomes more unstable.

scholarly journals On finite-amplitude patterns of convection in a rectangular-planform container

1996 ◽  
Vol 317 ◽  
pp. 111-127 ◽  
Author(s):  
P. G. Daniels ◽  
M. Weinstein
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Numerical Simulations ◽  
Steady State Solution ◽  
Critical Value ◽  
Finite Amplitude ◽  
State Structure ◽  
Amplitude Equations ◽  
Critical Wavelength ◽  
Rayleigh Numbers

This paper considers the development of finite-amplitude patterns of convection in rectangular-planform containers. The horizontal dimensions of the container are assumed to be large compared with the critical wavelength of the motion. An interaction between rolls parallel and perpendicular to the lateral boundaries is modelled by a coupled pair of nonlinear amplitude equations together with appropriate conditions on the four lateral boundaries. At Rayleigh numbers above a critical value a steady-state solution is established with rolls parallel to the shorter lateral sides, consistent with the predictions of linear theory. At a second critical value this solution becomes unstable to cross-rolls near the shorter sides and a new steady state evolves. This consists of the primary roll pattern together with regions near the shorter sides where there is a combination of rolls parallel and perpendicular to the boundary.Analytical and numerical methods are used to describe both the evolution and steady-state structure of the solution, and a comparison is made with the results of full numerical simulations and experiments.

An airfoil theory of bifurcating laminar separation from thin obstacles

1990 ◽  
Vol 216 ◽  
pp. 255-284 ◽  
Author(s):  
C. J. Lee ◽  
H. K. Cheng
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Steady State ◽  
Steady State Solution ◽  
Equation System ◽  
Branch Points ◽  
Laminar Separation ◽  
Kutta Condition ◽  
Numerical Studies ◽  
Steady State Solutions

Global interaction of the boundary layer separating from an obstacle with resulting open/closed wakes is studied for a thin airfoil in a steady flow. Replacing the Kutta condition of the classical theory is the breakaway criterion of the laminar triple-deck interaction (Sychev 1972; Smith 1977), which, together with the assumption of a uniform wake/eddy pressure, leads to a nonlinear equation system for the breakaway location and wake shape. The solutions depend on a Reynolds numberReand an airfoil thickness ratio or incidence τ and, in the domain$Re^{\frac{1}{16}}\tau = O(1)$considered, the separation locations are found to be far removed from the classical Brillouin–Villat point for the breakaway from a smooth shape. Bifurcations of the steady-state solution are found among examples of symmetrical and asymmetrical flows, allowing open and closed wakes, as well as symmetry breaking in an otherwise symmetrical flow. Accordingly, the influence of thickness and incidence, as well as Reynolds number is critical in the vicinity of branch points and cut-off points where steady-state solutions can/must change branches/types. The study suggests a correspondence of this bifurcation feature with the lift hysteresis and other aerodynamic anomalies observed from wind-tunnel and numerical studies in subcritical and high-subcriticalReflows.

Dependency of Unsteady Time-Linearized Flutter Investigations on the Steady State Flow Field

2011 ◽  
Author(s):  
Michael Blocher ◽  
Markus May ◽  
Harald Schoenenborn
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Elastic Stability ◽  
Compressor Cascade ◽  
Steady State Flow ◽  
State Flow ◽  
Flow Parameters ◽  
Pressure Distributions ◽  
German Aerospace ◽  
École Polytechnique

The influence of the steady state flow solution on the aero-elastic stability behaviour of an annular compressor cascade shall be studied in order to determine sensitivities of the aero-dynamic damping with respect to characteristic flow parameters. In this context two different flow regimes — a subsonic and a transonic case — are subject to the analysis. The pressure distributions, steady as well as unsteady, on the blade surface of the NACA3506 profile are compared to experimental data that has been gained by the Institute of Aeroelasticity of the German Aerospace Center (DLR) during several wind tunnel tests at the annular compressor cascade facility RGP-400 of the Ecole Polytechnique Fe´de´rale de Lausanne (EPFL). Whereas a certain robustness of the unsteady CFD results can be stated for the subsonic flow regime, the transonic regime proves to be very sensitive with respect to the steady state solution.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

Observations on the Steady-State Solution of an Extremely Flexible Spinning Disk With a Transverse Load

1983 ◽  
Vol 50 (3) ◽  
pp. 525-530 ◽  
Author(s):  
R. C. Benson
Keyword(s):  
Steady State ◽  
Hybrid System ◽  
Fourth Order ◽  
Steady State Solution ◽  
Transverse Load ◽  
System Of Differential Equations ◽  
Membrane Theory ◽  
Small Disk ◽  
Spinning Disk ◽  
Flexible Disk

The steady deflection of a transversely loaded, extremely flexible, spinning disk is studied. Membrane theory is used to predict the shapes and locations of waves that dominate the response. It is found that waves in disconnected regions are possible. Some results are presented to show how disk stiffness moderates the membrane waves, the most important result being an upper bound on the highest ordered wave of significant amplitude. A hybrid system of differential equations and boundary conditions is developed to replace the pure membrane formulation that is singular, and the full fourth-order plate formulation that is numerically sensitive. The hybrid formulation retains the salient features of the flexible disk response and facilitates calculations for very small disk stiffnesses.

Dynamic Stability of an Impact System Connected With Rock Drilling

1969 ◽  
Vol 36 (4) ◽  
pp. 743-749 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Computer Simulation ◽  
Exact Solution ◽  
Steady State ◽  
Asymptotic Stability ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
External Excitation ◽  
Rock Drilling ◽  
Impact System ◽  
Number Of Cycles

This paper deals with asymptotic stability of an analytically derived, synchronous as well as nonsynchronous, steady-state solution of an impact system which exhibits piecewise linear characteristics connected with rock drilling. The exact solution, which assumes one impact for a given number of cycles of the external excitation, is derived, its asymptotic stability is examined, and ranges of parameters are determined for which asymptotic stability is assured. The theoretically predicted stability or instability is verified by a digital computer simulation.

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