Flux boundary conditions

NUh2 is the Nusselt number for uniform heat flux boundary condition along the flow direction and periphery. 

The concentration at the wall, a(7), is found by applying the zero flux boundary condition. Equation (8.14). A simple way is to set a(I) = a(I — 1) since this gives a zero first derivative. However, this approximation to a first derivative converges only 0(Ar) while all the other approximations converge O(Ar ). A better way is to use... 

For a continuous steady-state release the concentration flux at any point r from the origin must equal the release rate Qm (with units of mass/time). This is represented mathematically by the following flux boundary condition ... 

As in the full-field formulation, we assigned a zero flux boundary condition, i.e. j = 0 at the outer boundary of the domain as well as on the axis of symmetry ahead of the crack tip (Fig. 5b). Also, along the crack surface, we assumed the NILS hydrogen concentration CL to be in equilibrium... 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

The flux boundary condition accounting for vaporization and condensation kinetics at membrane-vapor interfaces is... 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

For a given olivine crystal with radius a, treat it as a plane sheet along the c-axis with half-thickness a. Assume that the initial zonation is symmetric with respect to the center. Approximate the initial profile as C = Cq + CiCOs(7ix/a), where Q is the amplitude of the variation. Assume no flux boundary condition. The solution to the diffusion problem can be found by separation of variables as... 

For radial concentration profiles, a quadratic representation may not be adequate since application of the zero flux boundary conditions at r, = cp0 and r, = 1.0 leads to d2 = d3 = 0. Thus a quadratic representation for the concentration profiles reduces to the assumption of uniform radial concentrations, which for a highly exothermic system may be significantly inaccurate. 

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

Fig. 17. Temporal dissipative structure after various time intervals during the period of oscillation. The reaction medium is a circle with zero flux boundary conditions. The lines correspond to isoconentrations. A =2, B = 5.4, Dt = 8 10 3, D2 = 4- 1G"3. Curves of equal concentration for Y are represented by full or broken lines when the concentration is, respectively, larger or smaller than the unstable steady state. The radius of the circle r0 = 0.5861. 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

Fig. 10.2. A typical non-uniform stationary-state profile. Note the vanishing spatial derivative at the end walls (r = 0 and r = a0) appropriate to zero-flux boundary conditions.

Murray (1982) has confirmed this pattern of behaviour empirically for a variety of two-variable models with zero-flux boundary conditions such as those considered here. In general, the dominant mode increases in wave number n as the size of the reaction zone y increases, but decreases as the ratio of diffusivities increases—as shown in Fig. 10.6. 

Section 16.6.1.1 discusses a mass-flux boundary condition for a burner stabilized flame. Based on a surface mass balance, reformulate this boundary condition, assuming that elementary heterogeneous chamistry may occur at the burner face. 

Model the growth of Si02 in this reactor for the baseline conditions, testing the following two alternate methods for specifying inlet gas composition, (a) Specify that the gas composition (mole fractions) at the inlet manifold are those given above (i.e., a composition boundary condition) (b) Specify that the ratio of the molar fluxes at the manifold are equal to the ratio above (i.e., a flux boundary condition). 

Fig. 6. Outline of the successive eigenfunction model. Chemical patterns grow in domains when no flux boundary conditions are satisfied and > 0. For more detail, see Appendix.

For no-flux boundary conditions, the spatial gradient at the boundary must have zero component normal to the boundary.49 In a circle of radius ro, this means that dx(r, o, t)ldr = dy(r, o, t)/dr = 0 at r = r0- The zeros in the derivatives of J (z) occur at particular values of the argument z = z. 50 Therefore, the spatial mode J k r) cos m >, which we abbreviate by J j, is obtained when the jth zero in the derivative of J occurs at the boundary that is, when k ro = z. This fixes the value of knJ associated with the mode J for any given radius ro. As the radius changes, the value of k j changes in inverse proportion. 

Boundary conditions for electron density, ion density, and electron energy are all flux boundary conditions, wherein an expression is given for the flux of the quantity over which the balance is made. The net flux of electrons to the surface (assumed to be conducting) is the difference between the rate of recombination and the rate of creation through secondary electron... 

Yet another boundary condition encountered in polymer processing is prescribed heat flux. Surface-heat generation via solid-solid friction, as in frictional welding and conveying of solids in screw extmders, is an example. Moreover, certain types of intensive radiation or convective heating that are weak functions of surface temperature can also be treated as a prescribed surface heat-flux boundary condition. Finally, we occasionally encounter the highly nonlinear boundary condition of prescribed surface radiation. The exposure of the surface of an opaque substance to a radiation source at temperature 7 ,-leads to the following heat flux ... 

For a uniform heat flux boundary condition, we define r , u and z as before, and the dimensionless temperature as... 

The contaminant transport model, Eq. (28), was solved using the backwards in time alternating direction implicit (ADI) finite difference scheme subject to a zero dispersive flux boundary condition applied to all outer boundaries of the numerical domain with the exception of the NAPL-water interface where concentrations were kept constant at the 1,1,2-TCA solubility limit Cs. The ground-water model, Eq. (31), was solved using an implicit finite difference scheme subject to constant head boundaries on the left and right of the numerical domain, and no-flux boundary conditions for the top and bottom boundaries, corresponding to the confining layer and impermeable bedrock, respectively, as... 

The numerical solution for the solute-humic cotransport model was obtained by an unconditionally stable, fully implicit finite difference discretization method. The three governing transport Eqs. (38), (48), and (54) in conjunction with the initial and boundary conditions given by Eqs. (39)-(41), (51)—(53), (58) and (59) were solved simultaneously [57]. All flux boundary conditions were estimated using a second-order accurate one sided approximation [53]. 

The above averaging procedure described for Neumann boundary conditions may be extended to general flux boundary conditions of the form... 

The critical Reynolds number Reor is typically taken as 5 x 105, lie, < Re, < 3 x 107, and 0.7 < Pr < 400. The fluid properties are evaluated at the film temperature (7, + I )/2 where 7, is the free-stream temperature and 7 is the surface temperature. Equation (5-60) also apphes to the uniform heat flux boundary condition provided h is based on the average temperature difference between 7 and 7, ... 

Laplace transformation to the constant-flux boundary condition (4.50). Laplace transformation on the left-hand side of the boundary condition leads to (dc /dx), and the same operation performed on the right-hand side, to - l/Dp (Appendix 4.2). Thus, from the boundary condition (4.50) one gets... 

The effects of dry deposition are included as a flux boundary condition in the vertical diffusion equation. Dry deposition velocities are calculated from a big leaf multiple resistance model (Wesely 1989 Zhang et al. 2002) with aerodynamic, quasi-laminar layer, and surface resistances acting in series. The process assumes 15 land-use types and takes snow cover into account. 

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