Boundary conditions Dirichlet type

Let us consider a steady 2-D BVP on a rectangular domain involving only diffusion and a position-dependent somce term, —V (p = f(r), the Poisson equation. We require the solution to be zero on all boundaries, a Dirichlet-type boundary condition. Thus, the BVP is... 

Typically velocity components along the inlet are given as essential (also called Dirichlet)-type boundary conditions. For example, for a flow entering the domain shown in Figure 3.3 they can be given as... 

Typically the exit velocity in a flow domain is unknown and hence the prescription of Dirichlet-type boundary conditions at the outlet is not possible. However, at the outlet of sufficiently long domains fully developed flow conditions may be imposed. In the example considered here these can be written as... 

We formulate boundary conditions in the two-dimensional theory of plates and shells. Denote by u = U,w), U = ui,U2), horizontal and vertical displacements at the boundary T of the mid-surface fl c R. Then the horizontal displacements U may satisfy the Dirichlet-type conditions... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

Thus, in practice, the potential distribution within the electrolyte is obtained by solving Laplace s equation subject to a time-dependent, Dirichlet-type boundary condition at the end of the double layer of the WE, a given value of (j> at the end of the double layer of the CE and zero-flux or periodic boundary conditions at all other domain boundaries. Knowing the potential distribution, the electric field at the WE can be calculated, and the temporal evolution of the double layer potential is obtained by integrating Eq. (11) in time, which results in changed boundary conditions (b.c.) at the WE. 

In the treatment of explicit and implicit difference methods, we have used Dirichlet type boundary conditions, for the sake of simplicity, which specify the values of the solution on the boundaries. A more general type of boundary condition can be defined in the form of a linear combination of the solution and its derivative. Considering in particular the left boundary, such a mixed boundary condition can be written ... 

For (3 = 0 this is a Dirichlet type condition, while for a = 0 it is a Neumann type condition, a, (3, and y may, eventually, be functions of time. A practical example for a mixed boundary condition is the evaporation condition ... 

Irving Langmuir (1908) first replaced the assumption of no axial mixing of the PFR model with finite axial mixing and the accompanying Dirichlet boundary condition ((Cy) = Cyyn at x = 0) by a flux-type boundary condition... 

The Dirichlet-type boundary conditions assume that a solution of known concentration (C0) is applied at the soil top surface for a given duration tp. This solute pulse-type input is assumed to be followed by a solute-free solution application at the soil top surface ... 

We first consider Jeffreys-type models, namely system (7) with e < 1, which is complemented with the Dirichlet boundary condition... 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

Figures 1 is an example for (non-dimensionalized) thermal stresses (von Mises stress) under steady-state heat conduction in a rectangular body defined over (a ,j/),0 < x < 1,0 < y < 1 subject to the Dirichlet type homogeneous boundary condition (T = 0 and Ui — 0) on the boundary with a uniform internal heat generation [5]. The thermal conductivity, elastic modulus and thermal expansion coefficient are all assumed to vary in the form of ko -f kix + k2y where fc, s are constants. For the purpose of illustrations, the values of all the material properties are taken to be unity.

This important relation holds for all the states for the Dirichlet problem (see e.g. [5], Sect. XIII.15). However, the analogous relation is wrong for the other types of boundaries [29]. For example, the constant function in a sphere (that is, the wavefunction r/r(r) = const) satisfies the Neumann boundary conditions and has a zero value for the kinetic energy. Hence the energy functional is the mean value for the potential, that is, it equals —3/(2R) for the hydrogen atom. When the sphere radius R goes to zero, the energy value decreases, in contrast to Equation (2.4). 

It should also be noted that for general type boundary conditions 3y = ptp, the state order is similar to that defined by Dirichlet boundaries. One may note [29] that the functions Ekp R) for a given angular momentum have no common points for p p, and for any finite R value... 

Two types of boundary conditions at the cylinder boundary r = a will be considered (1) Dirichlet and (2) Neumann. 

As the film properties at the boundary to the substrate are thought to be extrinsic quantities, they are prescribed as Dirichlet-type boundary conditions in... 

In the same way a shear test can be simulated. The geometry and the boundary conditions are shown in Fig. 21.5, while Figs. 21.6 and 21.7 show the distribution of the shear strain, the shear stress, the microstructural flux, and the mi-crostructural parameter, respectively. Again, according to different Dirichlet data for ic, both types of boundary layers are generated. The results are qualitatively the same as in the tension test. 

The first type of boundary conditions (or Dirichlet type) is set by the concentration values at the boundaries ... 

In traditional DSMC simulations of supersonic flows, the Dirichlet type of velocity boundary conditions has generally been used. This approach is often applied in external-flow simulations, which require the downstream boundary to be far away from the base region. However, the flows in microscale systems are often subsonic flows, and the boundary conditions which can be... 

The carbon mass balance (2.1) takes into account the mixing of solids and the combustion reaction and is of reaction diffusion type with Neumann boundary conditions, i,e. the value of the normal derivative dCc/dn of the carbon concentration on the boundary is prescribed. The first two terms in the enthalpy balance (2.3) express the enthalpy flux due to the mixing of solids in the bed, the others the flue gas enthalpy flux, the heat sink due to the heat exchanger tubes and the heat source caused by the combustion. The balance is of convection diffusion type with third type Dirichlet-Neumann boundary conditions, i.e. the temperature values on the boundary depend on the corresponding gradients. Finally, the oxygen balance (2.2) considers the oxygen flux in upward direction and the combustion reaction. This ODE is explicitly solvable in dependence of the carbon concentration Cc and the temperature T ... 

The relevance of interphase gradients distinguishes between two different classes of problems, and this is reflected on the type of boundary condition at the pellet s surface. It is known that specifying the value of the concentration (or temperature) at the surfece (Dirichlet boundary condition) may not be realistic, and thus finite external transfer effects have to be considered (in a Robin-type boundary condition) [72]. Apart from these, a large number of additional effects have also been considered. Some examples include the nonuniformity of the porous pellet structure (distribution of pore sizes [102], bidisperse particles [103], etc.), nonuniformity of catalytic activity [104], deactivation by poisoning [105], presence of multiple reactions [106], and incorporation of additional transport mechanisms such as Soret diffusion [107] or intraparticular convection [108]. 

The velocities at the inlet are specified in Dirichlet-type boundary conditions. At the outlet, an open boundary condition referred to as outflow boundary condition [124] is used. Outflow boundary conditions are usually appropriate to model exit flows where velocity and pressure distributions are not known a priori. They are appropriate when exit flows are fully developed or close to, which impfies, except for pressure, zero gradients for all flow variables normal to the outflow boundary. The liquid holdup at the reactor inlet was evaluated assuming dUg/dz = dut/dz = 0 and combining Equations 5.56 and 5.57 ... 

The velocities at the inlet are specified in Dirichlet-type boundary conditions. At the outlet, an open boimdaiy condition is used [124], which implies, except for pressure, zero gradients... 

The four dependent variables are substrate, 5 oxidized mediator, reduced mediator, and biofilm matrix potential, E. The following are all of the differential equations, initial conditions, and boundary conditions. Each boundary condition is specified as either a Dirichlet (first-type) or a Neumann (second-type) boundary condition. The electrode surface is at x = 0, and the top of the biofilm is at x = L. 

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. 

The partial differential equations defined in fhe previous two sections must be supplied boundary conditions. In general, there are two types of boundary condifions. Neumann conditions specify a flux entering the region and Dirichlet conditions place a constraint on a state variable at the boundary. In the examples in this chapter, we will specify the current density / at t > 0, fix fhe pofenfial af fhe anode side, and fix the water content at both anode and cathode sides. We will specify fhe initial conditions at time f = 0 that would exist if fhe current were zero. There, i/h will have a uniform value of The pofenfial everywhere is zero. We arbitrarily let X vary linearly across the membrane. These conditions are formulized in equation (8.32) ... 

In traditional DSMC simulations of supersonic flows, the Dirichlet type of velocity boundary conditions has generally been used. This approach is often applied in external-flow simulations, which require the downstream boundary to be far away from the base region. However, the flows in microscale systems are often suhsonic flows, and the boundary conditions which can he obtained from the experiment always refer to pressure and temperature, instead of velocity and number density. Wang and Li [5] have proposed a new implicit treatment for a pressure boundary condition, inspired by the characteristic theory of low-speed microscale flows. This new implementation of boundary conditions not only overcomes the instability of particle-based approaches, hut also has a higher efficiency than any other existing methods. The new method is easy to extend to gas flows where the downstream and upstream directions are not opposite, such as in L-shaped and T-shaped channels. 

In the limit of no external mass transfer resistance (Bi, - - ) the Robin type boundary condition (12) becomes a simpler Dirichlet condition of u = 1. This problem was solved also. 

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