Dirichlet boundary condition diffusion modeling

The mass diffusivity coefficient of isobutane blowing agent from LDPE foam was found using a onedimensional diffusion model of two concentric cylinders with Dirichlet boundary conditions. An average mass diffusivity coefficient was used to calculate the mass of isobutane remaining in the foam for different boundary conditions. The influence of temperature and additives on diffusion was also examined. The use of the mass diffusivity coefficient in assessing the flammability of PE foam in the post-extrusion period is discussed. 2 refs. USA... 

Every partial differential equation needs an initial value or guess for numerical solver to start computing the equations. On the other hand, boundary conditions are specific for each conservation equation, described in Section 6.2. The variable in the continuity equation and momentum equations is the velocity vector, the variable in the energy equation is the temperature vector, and the variable in the species equation is the concentration vector. Therefore, appropriate velocity, temperature, and concentration values, which represent real-world values, need to be prescribed on each computational boundary, such as inlet, outlet, or wall at time zero. The prescribed values on boundaries are called boundary conditions. Each boundary condition needs to be prescribed on a node or line for 2D system or on a plane for 3D system. In general, there are several types of boundary conditions where the Dirichlet and Neumann boundary conditions are the most widely used in CFD and multiphysics applications. The Dirichlet boundary condition specifies the value on a specific boundary, such as velocity, temperature, or concentration. On the contrary, the Neumann boundary condition specifies the derivative on a specific boundary, such as heat flux or diffusion flux. Once the appropriate boundary conditions are prescribed to all boundaries on the 2D or 3D model, the set of the conservation equations is closed and the computational model can be executed. 

In most experimental runs, the volume and feeding flows of the two CSTRs at both ends of the Couette reactor were large enough for their internal state not to be significantly influenced by the dynamics inside the Couette reactor [33]. This corresponds mathematically to imposing Dirichlet boundary conditions to our model reaction-diffusion system (3). In most of the simulations... 

Fig. 5. A perspective plot of the spatio-temporal variation of the variable u x t) as computed with the reaction-diffusion system (3), with Dirichlet boundary conditions (4), the slow manifold (6) and the model parameters iii = -1.5, e = 0.01. (a) Stationary single-front pattern (u() = 1.1,D = 0.1,q = 0.01) (b) periodically oscillating single-front pattern (uo = 1.1, D = 0.045, a = 0.01) (c) periodic alternation of a single-front and a three-front pattern (u[) = 1.1, D = 0.01, a = 0.01) (d) stationary three-front pattern (uo = 0.5, D = 0.08, a = 0.2) (e) periodically oscillating three-front pattern (uq = 0.5, D = 0.06, a = 0.2) (f) periodic alternation of a single-front and a three-front pattern (uo = 0.5, D = 0.02, a = 0.2).

Fig. 8. Diffusion-induced chaos obtained when integrating the reaction-diffusion system (3) with Dirichlet boundary conditions (4) and the slow manifold (7) 6 = 10 ). The model parameters are e = 0.01, D = 0.05, a = A, uo = 2.5, u = -3.1. (a) Spatio-temporal variation of the variable u x t) coded as in Figure 6 (b) phase portrait (c) Poincar map (d) ID map.

Fig. 9. Spatio-temporal pattern forming phenomena in the reaction-diffusion system (3) with symmetric Dirichlet boundary conditions and the slow manifold (8). The model parameters are e = 0.01, a = 0.01, uo = ui = —2. (a) D = 0.0322560 (7,7 oscillating pattern confined to the lower branch (b) D = 0.0322550 Cr crisis-induced intermittent bursting (c) D = 0.0322307 homoclinic intermittent bursting (d) D = 0.0322400 PJ periodic bursting,...

In this section, we study the existence and stability of the stationary solutions of our reaction-diffusion model (3). Our goal is not to give a full description of all the possible cases, but rather to emphasize some general properties which are common to the class of reaction-diffusion systems given by a set of equations similar to Equation (3) with Dirichlet boundary conditions [104]. 

Fig. 16. Comparison between the exact stationary single-front solution (n(a ),t (a )) (full line) and the external approximation of the same solution (U x),V x)) (dashed line) of the reaction-diffusion model (3) with Dirichlet boundary conditions the piecewise linear slow manifold is given by Equation (7) with 6 = 0. (a) The functions u x) and U x) note that u and U differ significantly in the inner region only (b) phase portrait in the (n, v) plane by definition, the points (U x),V[x)) are located on the slow manifold (dashed line) (c) comparison between the exact solution (full line) and fhe function U[x) + i(0 (dashed line) (d) comparison between the exact solution v x) (full line) and the external function V x) (dashed line).

Fig. 18. Periodically oscillating two-front solution (breathing pattern) of the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6) and the model parameters e — 0.01, D = 0.03, a = 0.2, uo = ui = —1.5. (aj Unstable stationary profile u(x) (b) spatio-temporal variation of u(x, t) using the same coding as in Figure 6 (c) real part and (d) imaginary part of the u component of the critical Hopf eigenmode.

Fig. 21. Direct simulations of the reaction-diffusion system (3) with Dirichlet boundary conditions versus Hopf normal form predictions. z = (- u/Re is the amplitude of the Hopf critical mode, (a) Determination of the critical exponent (3 = 1/2 (b) measurement of Re K, the dashed line corresponds to the normal form predictions on the critical surface (fi =0) Re AC = —0.7301. Model parameters f u) = y e = 0.01, Z Hopf = 1.537 x 10 , a = 0.001, V ) = 1.5, v = —1.5.

The applicable mathematical form of the flux equation is model dependent. For example, the first type model consists of differential equations (DEs). They are developed to yield concentration profiles in the sediment layers as well as the flux. These DEs typically use Equation 4.1 as a boundary condition. The solutions to these DEs require one or more of the following boundary condition categories the Dirichlet condition, the Neuman condition, or a third condition. The first two types are the most common these require mathematical functions containing gradients of the dependent variable (i.e., Cw) as well as functions of the dependent variable itself. For these diffusive-type fluxes, the transport parameter is a diffusion coefficient such as Dg. Several other transport parameters are commonly used and represent diffusion in air and the biodiffusion or bioturbation of soil/sediment particles. 

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