Boundary conditions inflow

A convenient concept for introducing the surface boundary condition into the mathematical formulation of migration theory is that of what may be called a diffusional offset length d. Suppose that the external and surface conditions are describable by a set of parameters X, which we do not need to specify in detail we also allow the surface conditions to depend on the internal hydrogen concentration just beneath the surface. If the hydrogen complexes that are continually forming in the crystal are sufficiently immobile, the balance between inflow and outflow across the surface will depend only on X and on the concentration no(0) of H0 just beneath the surface. (If mobile H+ or H are present, the statement just... 

Simulations are then performed for gas bubbles emerging from a single nozzle with 0.4 cm I.D. at an average nozzle velocity of lOcm/s. The experimental measurements of inlet gas injection velocity in the nozzle using an FMA3306 gas flow meter reveals an inlet velocity fluctuation of 3-15% of the mean inlet velocity. A fluctuation of 10% is imposed on the gas velocity for the nozzle to represent the fluctuating nature of the inlet gas velocities. The initial velocity of the liquid is set as zero. An inflow condition and an outflow condition are assumed for the bottom wall and the top walls, respectively, with the free-slip boundary condition for the side walls. 

Subsequently, simulations are performed for the air Paratherm solid fluidized bed system with solid particles of 0.08 cm in diameter and 0.896 g/cm3 in density. The solid particle density is very close to the liquid density (0.868 g/ cm3). The boundary condition for the gas phase is inflow and outflow for the bottom and the top walls, respectively. Particles are initially distributed in the liquid medium in which no flows for the liquid and particles are allowed through the bottom and top walls. Free slip boundary conditions are imposed on the four side walls. Specific simulation conditions for the particles are given as follows Case (b) 2,000 particles randomly placed in a 4 x 4 x 8 cm3 column Case (c) 8,000 particles randomly placed in a 4 x 4 x 8 cm3 column and Case (d) 8,000 particles randomly placed in the lower half of the 4x4x8 cm3 column. The solids volume fractions are 0.42, 1.68, and 3.35%, respectively for Cases (b), (c), and (d). 

During the MC simulation, boundary conditions must be applied at the edges of the flow domain. The four most common types are outflow, inflow, symmetry, and a zero-flux wall. At an outflow boundary, the mean velocity vector will point out of the flow domain. Thus, there will be a net motion of particles in adjacent grid cells across the outflow boundary. In the MC simulation, these particles are simply eliminated. By keeping track of the weights... 

We will assume that and X2 indicate the positive and negative signs, respectively, before the square root in equation (6.19). The final solution for P i) will depend on the inflow concentration, and the determination of Pi and P2 will depend on the boundary conditions. 

Equations (6.16) to (6.18) are still applicable to the pulse input. The boundary conditions and the inflow concentration, however, are different. The inflow concentration at t = 0+ will be zero. Thus, in equation (6.17), fP, = 0. The new boundary conditions are ... 

The boundary conditions for the stream-function-vorticity system requires specifying the stream function on all the boundaries. This is usually straightforward for known inflow and outflow conditions and solid walls. The vorticity boundary conditions comes from evaluating Eq. 3.281 on the boundary. Along the boundary, which usually corresponds with one of the coordinate directions, one of the terms in Eq. 3.281 (i.e., the one in which the derivatives align with the boundary) can be evaluated explicitly since the stream function is already specified. Thus the boundary conditions becomes a relationship between the boundary vorticity and a boundary-normal second derivative of stream function. For example, consider the natural convection in a long horizontal tube. Here, since there is no inflow or outflow, the stream function is simply zero all around the tube wall. Thus the vorticity boundary conditions are... 

Typically, there are two types of boundaries in reacting flows. The first is a solid surface at which a reaction may be occurring, where the flow velocity is usually set to zero (the no-slip condition) and where either a temperature or a heat flux is specified or a balance between heat generated and lost is made. The second type of boundary is an inflow or outflow boundary. Generally, either the species concentration is specified or the Dankwerts boundary condition is used wherein a flux balance is made across the inflow boundary (64). The gas temperature and gas velocity profile are usually specified at an inflow boundary. At outflow boundaries, choices often become more difficult. If the outflow boundary is far away from the reaction zone, the species concentration gradient and temperature gradient in the direction of flow are often assumed to be zero. In addition, the outflow boundary condition on the momentum balance is usually that normal or shear stresses are also zero (64). 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

These forms relate the dependence on the system characteristics. Equation (8.13) describes the concentration c(z, t) of a solute in a tree-like structure that corresponds to the arterial tree of a mammal. Considering also the corresponding venular tree situated next to the arterial tree and appropriate inflow and outflow boundary conditions, we are able to derive an expression for the spatiotemporal distribution of a tracer inside a tree-like transport network. We also make the assumption that the arterial and venular trees are symmetric, that is, have the same volume V then, the total length is L = V/Ag The initial condition is c(z, 0) = 0 and the boundary conditions are ... 

Inflow or outflow can be specified using appropriate boundary conditions, although some characteristic problems are more easily implemented with one than the other. In any case, it is important to insure that the conditions specified are self-consistent. For example, inflow can be specified along the top boundary, by specifying the inflow velocity at V, as in figure (3). This in turn specifies a relative change in vorticity from to o ... 

At the inflow boundary and on top of the computational domain, analytic solution for the disturbance velocity was used in accordance with Eqn. (2.7.1) and (2.7.2). On the flat plate, the no-slip condition simultaneously provides a Dirichlet boundary condition for the stream function and the wall vorticity at every instant of time. 

At the inflow and at the top of the computational domain one calculates the flow variables, as induced by the freestream vortex via Biot-Savart interaction rule. At the outflow, fully developed condition is applied for the wall-normal component of the velocity ( = 0) and using the same in SFE, one can obtain the vorticity boundary condition at the outflow from Equation (3.4.2). At the top frame of Fig. 3.8, one sees incipient unsteady separation on the wall. In subsequent frames, one notices secondary and tertiary events induced by the primary instability. In these computed cases, one does not notice TS waves and the vortices formed on the wall are essentially due to unsteady separation that is initiated by the freestream convecting vortex. These ensemble of events have been noted as the vortex-induced instability or bypass transition in Sengupta et al. (2001, 2003), Sengupta Dey (2004) and in Sengupta Dipankar (2005). 

For Maxwell models, where Tj = 0, we need to distinguish the sub- and the supercritical ceises. In the subcritical case (i.e., U < yT]j pX)) and in two space dimensions, one can prescribe the diagonal components o-p and 7P, whereas in three space dimensions a correct choice of boundary conditions for tP is not simple. A possible choice of four boundary conditions is a nonlocal one (in terms of the Fourier components of rP—see [29]). An alternative approach leading to first order differential boundary conditions at the inflow boundary is described in [30]. 

For Maxwell models in the supercritical case (i.e., U > iJi]l(pX)), the previous choice of boundary conditions leads to an ill-posed problem (as does the Dirichlet boundary condition for a hyperbolic equation), as shown in [17]. In addition to the normal velocities at both boundaries (inflow and outflow) and to the previous inflow conditions on the stresses, one can prescribe the vorticitj and its normal derivative in two space dimensions, or the second and third components of the vorticity and their normal derivatives in three... 

A discussion of the traction boundary conditions—where the totaJ normal stress is prescribed on the inflow and outflow boundaries—for Jeffreys-type fluids is given in [31], and for Maxwell-type fluids in [32]. 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

M. Renaxdy, Inflow boundary conditions for steady flows of viscoelcistic fluids with differential constitutive laws. Rocky Mount. J. Math., 18 (1988) 445-453, and 19 (1989) 561. 

M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990) 419-425. 

R. Talhouk, Unsteady flows of viscoelastic fluids with inflow and outflow boundary conditions, Appl. Math. Letters, (1996) to appear. 

Click on a boundary number (1-4) (Note The corresponding boundary is highlighted in red.). Set the boundary condition for each boundary segment. 1, slip/symmetry 2, inflow/outflow,... 

The model for the geometry description in MOREX contains a complete three dimensional, parametrized description of conveying- and kneading elements. Based on this model a surface mesh can be exported to the BEM-software. For the structure of these meshes the cross section can be seen in Fig. 5.36. Additionally the visualization of the screws in MOREX is based on these meshes. The boundary conditions for the numerical methods as well as the velocity profile at the flow channel inflow and the viscosity can be given in a specified module in MOREX, resp. are overtaken from a previous MOREX calculation. 

The usual way of feeding the microfluidic systems with fluids is to apply either a constant rate of inflow into the chip, or a constant pressirre at the inlet [20]. Formation of droplets or bubbles in systems with such, fixed, boundary conditions for flow is realtively well understood. Two microfluidic geometries are most commonly used a microfluidic T-junction [1] or a microfluidic flow-focusing geometry [6]. 

Inlet swirl inflow conditions are discussed below. The outflow boundary conditions at the combustor outlet involve advection of all flow and species variables with Uc, where the instantaneous mean streamwise outlet boundary velocity Uc is periodically renormalized to ensure that the time-averaged mass flux coincides with that at the inlet these convective boundary conditions are enforced in conjuction with soft relaxation of the outflow pressure to its ambient value. Two types of outlets were considered (Fig. 11.1). Viscous wall regions in the combustor cannot be practically resolved for the moderately-high Reynolds... 

Figure 11.2 (a) Turbulent pipe-flow (open symbols) and GEAE LM-6000 (filled symbols) radial profiles of inlet velocity components (normalized by the peak mean inlet axial velocity) for S = 0.56 1 — axial U 2 — tangential W and 3 — radial V. [b) Sensitivity of LM-6000 normalized centerline velocity to choice of floating inflow boundary condition streamwise variable is scaled with inlet diameter R previous LES [3] 1 — fine grid 2 — coarse grid (fix T and V ) present work 3 — MILES (fix p and V) 4 — MILES (fix Po and float V ) and -5 — OEEV M (fix p and V ). 

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