Rv rrP r pgr559

The three components of this momentum equation, expressed in Cartesian, cylindrical, and spherical coordinates, are given in detail in Appendix E. Note that Eq. (5-59) is simply a microscopic ("local") expression of the conservation of momentum, e.g., Eq. (5-40), and it applies locally at any and all points in any flowing stream.

Note that there are 11 dependent variables, or "unknowns" in these equations (three vi's, six Tj's, P, and p), all of which may depend on space and time. (For an incompressible fluid, p is constant so there are only 10 "unknowns.") There are four conservation equations involving these unknowns (the three momentum equations plus the conservation of mass or continuity equation), which means that we still need six more equations (seven, if the fluid is compressible). These additional equations are the "con stitutive'' equations that relate the local stress components to the flow or deformation of the particular fluid in laminar flow (i.e., these are determined by the constitution or structure of the material) or equations for the local turbulent stress components (the "Reynolds stresses'' see Chapter 6). These equations describe the deformation or flow properties of the specific fluid of interest and relate the six shear stress components (xy) to the deformation rate (i.e., the velocity gradient components). [Note there are only six independent components of the shear stress tensor (xy) because it is symmetrical, i.e., xy = xy, which is a result of the conservation of angular momentum.] For a compressible fluid, the density is related to the pressure through an appropriate equation of state. When the equations for the six xy components are coupled with the four conservation equations, the result is a set of differential equations for the j0 (or jj) unknowns that can be solved (in principle) with appropriate boundary conditions for the velocity components as a function of time and space. In laminar flows, the constitutive equation gives the shear stress components as a unique function ofthe velocity gradient components. For example, the constitutive equation for a Newtonian fluid, generalized from the one dimensional form (i.e., x = py), is s = p[(VV + (YV/] (5-60)

where (Vv/ represents the transpose of the matrix of the Vv components. The component forms of this equation are also given in Appendix E for Cartesian, cylindrical, and spherical coordinate systems. If these equations are used to eliminate the stress components from the momentum equations, the result is called the Navier-Stokes equations, which apply to the laminar flow of any Newtonian fluid in any system and are the starting point for the detailed solution of many fluid flow problems. Similar equations can be developed for non-Newtonian fluids, based upon the appropriate rheologi-cal (constitutive) model for the fluid. For turbulent flows, additional equations are required to describe the momentum transported by the fluctuating ("eddy") components of the flow (see Chapter 6). However, the number of flow problems for which closed analytical solutions are possible is rather limited, so numerical computer techniques are required for many problems of practical interest. These procedures are beyond the scope of this book, but we will illustrate the application of the momentum equations to the solution of an example problem.

Example 5-9: Flow Down an Inclined Plane. Consider the steady laminar flow of a thin layer or film of liquid down a flat plate that is inclined at an angle Q to the vertical, as illustrated in Fig. 540. The width of the plate is W (normal to the plane of the figure). Flow is only in the x direction (parallel to

Figure 5-10 Flow down an inclined plane.

the surface), and the velocity varies only in the y direction (normal to the surface). These prescribed conditions constitute the definition of the problem to be solved. The objective is to determine the film thickness, <5, as a function of the flow rate per unit width of plate (Q/W), the fluid properties (p, p), and other parameters in the problem. Since vy = vz = 0, the microscopic mass balance (continuity equation) reduces to dvr

which tells us that the velocity vx must be independent of x. Hence, the only independent variable is y. Considering the x component of the momentum equation (see Appendix E), and discarding all y and z velocity and stress components and all derivatives except those with respect to the y direction, the result is 3ryx

The pressure gradient term has been discarded, because the system is open to the atmosphere and thus the pressure is constant (or, at most, hydrostatic) everywhere. The above equation can be integrated to give the shear stress distribution in the film:

where the constant of integration is zero, because there is zero (negligible) stress at the free surface of the film (y = 0). Note that this result is valid for any fluid (Newtonian or non-Newtonian) under any flow conditions (laminar or turbulent), because it is simply a statement of the conservation of momentum. If the fluid is Newtonian fluid and the flow is laminar, the shear stress is

Eliminating the stress between the last two equations gives a differential equation for vx(y) that can be integrated to give the velocity distribution:

pgS2cos 0 / y2

vxy1 "

where the boundary condition that vx = 0 at y = S (the wall) has been used to evaluate the constant of integration.

The volumetric flow rate can now be determined from

The film thickness is seen to be proportional to the cube root of the flow rate and the fluid viscosity. The shear stress exerted on the plate is

which is just the component of the weight of the fluid on the plate acting parallel to the plate.

It is also informative to express these results in dimensionless form, i.e., in terms of appropriate dimensionless groups. Because this is a noncircular conduit, the appropriate flow "length" parameter is the hydraulic diameter defined by Eq. (5.48):

The appropriate form for the Reynolds number is thus DhV p 45 V p 4pQ/W

0 0

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