Dynamics-governing PDEs

Although it is desirable to be able to capture the full dynamics of both ion transport and solvent transport [Asaka and Oguro (2000) Tadokoro et al. (2000)], such an attempt typically does not lead to analytical solutions. Therefore, we follow Nemat-Nasser and Li (2000) and focus on the dynamics of cations only. Let D, E, f , and p denote the electric displacement, electric field, electric potential, and charge density, respectively. The following field equations hold  

Since the thickness of an IPMC is much smaller than its length or width, we can assume that, inside the polymer, D, E, and J are all restricted to the thickness direction (a -direction), and will drop the boldface notation for 

Substituting (4.6) into the original ion flux equation (4.4) and using = —E, one can rewrite J as 

Assuming KgVE C C E (see [NemaLNasser (2002)] for justiflcation), the nonlinear term involving VE E in (4.7) is dropped, resulting in 

One of the boundary conditions to be used is that the ion flux at the polymer/metal interface is zero [Farinholt (2005)], which results in 

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The projection-based model order reduction algorithm begins with a spatial discretization of the governing PDEs to attain the dynamic system equations as Eq. 11. Specifically, here, X(t) is the state vector of unknowns (a function of time) on the discrete nodes, n is the total number of nodes A is formulated by the numerical discretization Z defines the functions of boundary conditions and source terms and B relates the input function to each state X. Equation 11 can be recast into the frequency domain in terms of transfer function T(s). T(s) then is expanded as a Taylor series at s = 0 yielding... 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

A detailed presentation of computational fluid dynamics applied to combustion is beyond the scope of this work. The CFD is concerned with the numerical solution of Partial Differential Equahons, (PDEs) governing the transport of mass, momentum, energy, chemical species in moving fluids. 

In this section we present a nonlinear circuit model for IPMC actuators. A key component of the circuit is the nonlinear capacitance, derived based on the original PDEs governing the ion dynamics. In addition, the circuit includes ion diffusion resistance [Bonomo et al. (2007) Porfiri (2008)], pseu-... 

The strong form of a hyperbolic PDE, that is, the governing equation, the boundary conditions (BC), the displacement-strain relation (DS) and the initial conditions (IC) are given as follows (for example, the equation of classical elasto-dynamics for a Hookean solid with mass density p and elasticity constant tensor D see, e.g., Gurtin 1972 Selvadurai 2000b) ... 

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