Momentum equation boundary conditions

For solid walls, no-penetration and no-slip are typically applied to the momentum equation. Boundary conditions such as velocity, pressure and temperature at the inlet are usually known and specified, whereas their counterparts at the outlet are derived from assumptions of no-stress or fully developed and simulated flow. The thermal wall boundary conditions influence the flow significantly. Simple assumptions of constant wall temperature, insulated side walls and constant wall heat transfer flux have been used extensively for simple applications. More specifically, the following assumptions are normally made for a retort as shown in Figure 6.29 (the directions of the velocity in the following description refer to a cylindrical coordinate system in this figure). 

To solve the conservation equations, boundary conditions are needed. For the momentum equation, the flow velocity at wall is fixed. It is generally assumed that the fluid molecules near the wall are in equilibrium with those of the wall and the fluid velocity is written as ... 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

From the continuity and momentum equations for the fluid and solid phases along with the boundary conditions, the following groups of independent dimensionless parameters are found to control the hydrodynamics, noting our assumption that the particle-particle forces are only dependent on hydrodynamic parameters,... 

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

A more sophisticated approach is to avoid the postulate of a shock and instead to state the differential equations of conservation of mass, momentum, and energy to include more properties of a real fluid. Including the effects of viscosity, heat conditions, and diffusion along with chem reaction gives eqs with a unique solution for given boundary conditions and so solves the determinacy problem. The boundary conditions are restricted by the assumption that the reaction begins and is completed with the region considered. 

Since the equations of continuity, momentum, energy, and state do not suffice to determine the five unknowns, it. is necessary to inquire into the conditions under which solutions exist and whether solns are unique. The information which has thus far been omitted is a specification of the flow field of die reaction products, that is to say, since this section is restricted to one--dimensional flow, of the rear boundary condition. Before discussing the question of determinancy it is necessary to deduce from the equations of Section II of Ref 66, the general properties of flow ahead and behind reaction waves. To do this the Hugoniot curve for the products... 

It is important to determine the partial-differential-equation order. One of the most important reasons to understand order relates to consistent boundary-condition assignment. All the equations are first order in time. The spatial behavior can be a bit trickier. The continuity equation is first order in the velocity and density. The momentum equations are second order on the velocity and first order in the pressure. The species continuity equations are essentially second order in the composition (mass fraction Yy), since (see Eq. 3.128)... 

Understanding the order of the hydrodynamics equations, continuity and momentum, can be somewhat confusing and possibly not the same from problem to problem. The continuity and momentum equations must be viewed as a closely coupled system. Again, it is clear that the momentum equations are second order in velocity and first order in pressure. The continuity equation is first order in density. However, an equation of state requires that density be a function of pressure, and vice versa. Density and pressure must be dependent on each other through an algebraic equation. Therefore a substitution could be done to eliminate either pressure or density. As a result the coupled system is third order, which can present some practical issues for boundary-condition assignment. The first-order behavior must carry information from some portions of the boundary into the domain, but it does not communicate information back. Therefore, over some portions of a problem... 

Assuming axisymmetric flow (i.e., r and z as independent variables, neglecting any 6 variations), state (not derive) the full mass-continuity and momentum equations that describe the flow in the annulus. Identify the dependent variables. Considering the characteristics and order of the system, state and discuss a set of boundary conditions that could be used to solve the system. Clearly, some approximation is required around the steam-feed entrance to retain axisymmetry. 

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