The Governing Equations

The governing equations are given in the next section. The mean flow, whose stability will be studied, is given in the section 6.3. The stability equations and related numerical methods for CMM is given in section 6.4. The results and discussion follow in section 6.5. The chapter closes with some comments and outlook in section 6.6. 

We consider the laminar two-dimensional motion of fluid past a hot semi-infinite plate, with the free stream velocity and temperature denoted by, Uoo and Too- We will focus our attention on the top of the plate, for which the temperature is T - that is greater than Too, while assuming the leading edge of the plate as the stagnation point. Governing equations are written in dimensional form (indicated by the quantities with asterisk), along with the Boussinesq approximation to represent the buoyancy effect, 

If we introduce a length scale (L), velocity scale U ), temperature scale (AT = T — Too) and pressure scale pU ), then the above equations can be represented in non-dimensional form by, 

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Equations (1.6) and (1.7) are used to formulate explicit relationships between the extra stress components and the velocity gradients. Using these relationships the extra stress, t, can be eliminated from the governing equations. This is the basis for the derivation of the well-known Navier-Stokes equations which represent the Newtonian flow (Aris, 1989). 

Components of the governing equations of the process can be decoupled to develop a solution scheme for a three-dimensional problem by combining one- and two-dimensional analyses. 

The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

In the continuous penalty technique prior to the discretization of the governing equations, the pressure in the equation of motion is substituted from Fquation (3.6) to obtain... 

For simplicity, we define T - and T (A iooTe/At). As explained by Luo and Tanner (1989), the decoupled method requires a suitable variable transfonna-tion in the governing equations (3.20) and (3.21). This is to ensure that the discrete momentum equations always contain the real viscous term required to recover the Newtonian velocity-pressure formulation when Ws approaches zero. This is achieved by decomposing the extra stress T as... 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

To develop the scheme we start with the normalization of the governing equations by letting... 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... 

In order to account for the heat loss through the metallic body of the cone, a heat conduction equation, obtained by the elimination of the convection and source terms in Equation (5.25), should also be incorporated in the governing equations. 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymraetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the shear-dependent viscosity in Equations (4.11) with a constant viscosity p, as... 

The functional relation ia equation 53 or 54 cannot be determined by dimensional analysis alone it must be suppHed by experiments. The significance is that the mean-free-path problem is reduced from an original relation involving seven variables to an equation involving only three dimensionless products, a considerable saving ia terms of the number of experiments required ia determining the governing equation. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

A kinetic model originally derived by Nyholm is distinguished from Monod s model by the fate of a hmiting substrate. Instead of immediate metabolism, the substrate in Nyholm s model is sequestered. The governing equations are ... 

There are few analytic solutions to the governing equations for interesting problems. The conservation equations are typically solved approximately on digital computers. It is assumed that the sound speeds are real and the system... 

The governing equations, (9.1)-(9.4), are approximated with discrete equations on the computational mesh. The discrete equations can be derived... 

Boundary conditions are special treatments used for internal and external boundaries. For example, the center line in cylindrical geometry is an internal boundary that is modeled as a plane of symmetry. External boundaries model the world outside the mesh. The outermost row of elements is often used to implement the boundary condition as shown in Fig. 9.13. The mass, stress, velocity, etc., of the boundary elements are defined by the boundary conditions rather than the governing equations. External boundary conditions are typically prescribed through user input. 

The governing equations for the combined effect of concentration and temperature gradient are ... 

To eompare the different meehanisms of filtration, the governing equation of filtration must be rearranged. The starting expression is ... 

The governing equations (Equations 42, 67, 74 and 81) describing the filtration mechanisms are expressed as linear relationships with parameters conveniently grouped into constants that are functions of the specific operating conditions. The exact form of the linear functional relationships depends on the filtration mechanism. Table 1 lists the coordinate systems that will provide linear plots of filtration data depending on the controlling mechanism. 

Integrating the governing equations of fluid flow over all the finite control volumes of the solution domain. 

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