Wednesday, November 12, 2014

The Differential Equations : The Boundary-Value Problem and The Differential Continuity Equation

The Differential

Equations

This chapter may be omitted in an introductory course. The derivations in subsequent chapters will either not require these differential equations or there will be two methods to derive the equations: one using differential elements and one utilizing the differential equations.

In Chap. 4 problems were solved using integrals for which the integrands were known or could be approximated. Differential equations are needed in order to solve for those quantities in the integrands that are not known, such as the velocity distribution in a pipe or the velocity and pressure distributions on and around an airfoil. The differential equations may also contain information of interest, such as a point of separation of a fluid from a surface.

5.1 The Boundary-Value Problem

To solve a partial differential equation for the dependent variable, certain conditions are required, i.e., the dependent variable must be specified at certain values of the independent variables. If the independent variables are space coordinates (such as the velocity at the wall of a pipe), the conditions are called boundary conditions. If the independent variable is time, the conditions are called initial conditions. The general problem is usually referred to as a boundary- value problem.

Boundary conditions typically result from one or more of the following:

• The no-slip condition in a viscous flow. Viscosity causes any fluid, be it a gas or a liquid, to stick to the boundary. Most often the boundary is not moving.

• The normal component of the velocity in an inviscid flow. In an inviscid flow where the viscosity is neglected, the velocity vector is tangent to the boundary at the boundary, providing the boundary is not porous.

• The pressure at a free surface. For problems involving a free surface, a pressure condition is known at the free surface.

For an unsteady flow, initial conditions are required, e.g., the initial velocity and pressure must be specified at some time, usually at t = 0.

The differential equations in this chapter will be derived using rectangular coordinates. It may be easier to solve problems using cylindrical or spherical coordinates, so the differential equations using those two coordinate systems will be presented in Table 5.1.

The differential energy equation will not be derived in this book. It would be needed if there are temperature differences on the boundaries or if viscous effects are so large that temperature gradients are developed in the flow. A course in heat transfer would include such effects.

5.2 The Differential Continuity Equation

To derive the differential continuity equation, the infinitesimal element of Fig. 5.1 is needed. It is a small control volume into and from which the fluid flows. It is shown in the xy-plane with depth dz. Let us assume that the flow is only in the xy- plane so that no fluid flows in the z-direction. Since mass could be changing inside the element, the mass that flows into the element minus that which flows out must equal the change in mass inside the element. This is expressed as

Table 5.1 The Differential Continuity, Momentum Equations, and Stresses for Incompressible Flows in Cylindrical and Spherical Coordinates

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Table 5.1 The Differential Continuity, Momentum Equations, and Stresses for Incompressible Flows in Cylindrical and Spherical Coordinates (Continued)

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Figure 5.1 Infinitesimal control volume.

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where the products Pu and Pv are allowed to change across the element.1 Simplifying the above, recognizing that the elemental control volume is fixed, results in

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Differentiate the products and include the variation in the z-direction. Then the differential continuity equation can be put in the form

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The first four terms form the material derivative [see Eq. (3.11)], so Eq. (5.3) becomes

imageproviding the most general form of the differential continuity equation expressed using rectangular coordinates.

The differential continuity equation is often written using the vector operator

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where the velocity vector is V = ui + vj + wk. The scalar Æ·V is called the diver- gence of the velocity vector.

For an incompressible flow, the density of a fluid particle remains constant as it travels through a flow field, that is,

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so it is not necessary that the density be constant. If the density is constant, as it often is, then each term in Eq. (5.7) is zero. For an incompressible flow, Eqs. (5.4) and (5.6) also demand that

imageThe differential continuity equation for an incompressible flow is presented in cylindrical and spherical coordinates in Table 5.1.

EXAMPLE 5.1

Air flows with a uniform velocity in a pipe with the velocities measured along the centerline at 40-cm increments as shown. If the density at point 2 is 1.2 kg/m3, estimate the density gradient at point 2.

imageSolution

The continuity equation (5.3) is used since the density is changing. It is simplified as follows:

imageCentral differences2 are used to approximate the velocity gradient ∂u/∂ x at point 2 since information at three points is given:

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