Euler equations, instabilities

Euler equations, instabilities 516 Euler-Lagrange equations... 

Prior knowledge of the local sound speeds is not required when beginning a simulation. If compression occurs at state A, then > c. If compression stops at state B, then , +c, > v. Note that as a consequence of the instability at point A of Eq. (12), runaway expansion on the tensile strain side of state A is also a valid solution of the steady state Euler equations. Such an expansion solution may have physical significance if there exists a larger volume where... 

Example 8.8 Explore conservation of mass, stability, and instability when the convective diffusion equation is solved using the method of lines combined with Euler s method. 

A high degree of accuracy is not called for in many calculations of the evolution of environmental properties because the mathematical description of the environment by a reasonably small number of equations involves an approximation quite independent of any approximation in the equations solution. Figure 2-3 shows how the accuracy of the reverse Euler method degrades as the time step is increased, but it also shows the stability of the method. Even a time step of 40 years, nearly five times larger than the residence time of 8.64 years, yields a solution that behaves like the true solution. In contrast, Figure 2-2 shows the instability of the direct Euler method a time step as small as 10 years introduces oscillations that are not a property of the true solution. 

For elastomers, a bifurcation due to the nonlinearities is observed with the emergence of a sub-harmonic mode, qH (Fig. 8.8). This bifurcation clearly corresponds to a period-doubling sequence similar to what is observed for Faraday instability of shaken fluid layers, except that this bifurcation refers to time and not space [20]. For the Faraday instability, the period-doubling is usually described with a nonlinear parametric oscillator model. Interestingly, the Euler-Lagrange equation describing the minimization of energy for the compressed sheet on elastomer bears also some resemblance with a parametric oscillator [14]. 

Numerical Solution of Steady State Equations.The simple Euler method is used in numerical integration. Reasonable results are obtained for a minimum step size of 50 cm. Step sizes over 250 cm. causes instability. 

The very short time constant in the second equation requires a very small h to follow both y and z in time. Problems of instability can easily arise. To solve stiff problems, the most typical approach is to use an implicit method, which is known to exhibit excellent stability properties (recall the backward Euler method). Special software packages are available for solving stiff systems. Fortunately, many of the straightforward chemical engineering problems enconntered in practice do not yield stiff systems, but when difficulties arise, stiffness might well be the culprit. 

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