Description of a Fluid in Motion

T

he development of an analytical description offluidflow is based upon the expression of the physical laws related tofluidflow in a suitable mathematical form. Accordingly, we shall present the pertinent physical laws and discuss the methods used to describe afluid in motion.

▶ 3.1

FUNDAMENTAL PHYSICAL LAWS

There are three fundamental physical laws that, with the exception of relativistic and nuclear phenomena, apply to each and every flow independently of the nature of thefluid under consideration. These laws are listed below with the designations of their mathematical formulations.

Law Equation

1. The law of conservation of mass Continuity equation 2. Newton’s second law of motion Momentum theorem 3. Thefirst law of thermodynamics Energy equation

The next three chapters will be devoted exclusively to the development of a suitable working form of these laws.¹

In addition to the above laws, certain auxiliary or subsidiary relations are employed in describing a fluid. These relations depend upon the nature of thefluid under consideration.

Unfortunately, most of these auxiliary relations have also been termed“laws.”Already in our previous studies, Hooke’s law, the perfect gas law, and others have been encountered. However accurate these“laws”may be over a restricted range, their validity is entirely dependent upon the nature of the material under consideration. Thus, while some of the auxiliary relations that will be used will be called laws, the student will be responsible for noting the difference in scope between the fundamental physical laws and the auxiliary relations.

▶ 3.2

FLUID-FLOW FIELDS: LAGRANGIAN AND EULERIAN REPRESENTATIONS

The termfieldrefers to a quantity defined as a function of position and time throughout a given region. There are two different forms of representation forfields influid mechanics,

1The second law of thermodynamics is also fundamental tofluid-flow analysis. An analytic consideration of the second law is beyond the scope of the present treatment.

29

Lagrange’s form and Euler’s form. The difference between these approaches lies in the manner in which the position in thefield is identified.

In the Lagrangian approach, the physical variables are described for a particular element of fluid as it traverses the flow. This is the familiar approach of particle and rigid-body dynamics. The coordinates (x, y, z) are the coordinates of the element offluid and, as such, are functions of time. The coordinates (x, y, z) are therefore dependent variables in the Lagrangian form. Thefluid element is identified by its position in thefield at some arbitrary time, usually tˆ0. The velocityfield in this case is written in functional form as

vˆv…a;b;c;t† (3-1)

where the coordinates (a, b, c) refer to theinitialposition of thefluid element. The other fluid-flow variables, being functions of the same coordinates, may be represented in a similar manner. The Lagrangian approach is seldom used influid mechanics, as the type of information desired is usually the value of a particularfluid variable at afixed point in the flow rather than the value of afluid variable experienced by an element offluid along its trajectory. For example, the determination of the force on a stationary body in aflowfield requires that we know the pressure and shear stress at every point on the body. The Eulerian form provides us with this type of information.

The Eulerian approach gives the value of afluid variable at a given point and at a given time. In functional form the velocityfield is written as

vˆv…x;y;z;t† (3-2)

wherex,y,z, andtareallindependent variables. For a particular point (x₁,y₁,z₂) andt₁, equation (3-2) gives the velocity of the fluid at that location at timet1. In this text the Eulerian approach will be used exclusively.

▶ 3.3

STEADY AND UNSTEADY FLOWS

In adopting the Eulerian approach, we note that thefluidflow will, in general, be a function of the four independent variables (x, y, z, t). If the flow at every point in the fluid is independent of time, theflow is termedsteady. If theflow at a point varies with time, the flow is termedunsteady. It is possible in certain cases to reduce an unsteadyflow to a steady flow by changing the frame of reference. Consider an airplane flying through the air at constant speed v0, as shown in Figure 3.1. When observed from the stationary x, y, z coordinate system, the flow pattern is unsteady. The flow at the point P illustrated, for example, will vary as the vehicle approaches it.

Now consider the same situation when observed from thex⁰,y⁰,z⁰coordinate system, which is moving at constant velocityv0 as illustrated in Figure 3.2.

Theflow conditions are now independent of time at every point in theflowfield, and thus theflow is steady when viewed from the moving coordinate system. Whenever a body moves through afluid with a constant velocity, theflowfield may be transformed from an unsteadyflow to a steadyflow by selecting a coordinate system that isfixed with respect to the moving body.

In the wind-tunnel testing of models, use is made of this concept. Data obtained for a static model in a moving stream will be the same as the data obtained for a model moving through a static stream. The physical as well as the analytical simplifications afforded by this transforma- tion are considerable. We shall make use of this transformation whenever applicable.

▶ 3.4

STREAMLINES

A useful concept in the description of aflowingfluid is that of astreamline. A streamline is defined as the line-drawn tangent to the velocity vector at each point in aflowfield. Figure 3.3 shows the streamline pattern for idealflow past a football-like object. In steadyflow, as all velocity vectors are invariant with time, the path of afluid particle follows a streamline, hence a streamline is the trajectory of an element of fluid in such a situation. In unsteadyflow, streamline patterns change from instant to instant. Thus, the trajectory of afluid element will be different from a streamline at any particular time. The actual trajectory of afluid element as it y

x

z

P

Figure 3.1 Unsteadyflow with respect to afixed coordinate system.

y

x

z

P y'

x'

z'

Figure 3.2 Steadyflow with respect to a moving coordinate system.

Figure 3.3 Illustration of streamlines.

3.4 Streamlines ◀ 31

traverses theflow is designated asa path line. It is obvious that path lines and streamlines are coincident only in steadyflow.

The streamline is useful in relating thatfluid velocity components to the geometry of the flow field. For two-dimensionalflow this relation is

vy

vxˆdy

dx (3-3)

as the streamline is tangent to the velocity vector havingxandycomponentsvxandvy. In three dimensions this becomes

dx vx ˆdy

vy ˆdz

vz (3-4)

The utility of the above relations is in obtaining an analytical relation between velocity components and the streamline pattern.

Some additional discussion is provided in Chapter 10 regarding the mathematical description of streamlines around solid objects.

▶ 3.5

SYSTEMS AND CONTROL VOLUMES

The three basic physical laws listed in Section 3.1 are all stated in terms of asystem. A system is defined as a collection of matter of fixed identity. The basic laws give the interaction of a system with its surroundings. The selection of the system for the application of these laws is quite flexible and is, in many cases, a complex problem. Any analysis utilizing a fundamental law must follow the designation of a specific system, and the difficulty of solution varies greatly depending on the choice made.

As an illustration, consider Newton’s second law,Fˆma. The terms represented are as follows:

F ˆthe resultant force exerted by the surroundings on the system: mˆthe mass of the system:

a ˆthe acceleration of the center of mass of the system:

(3-4) In the piston-and-cylinder arrange- ment shown in Figure 3.4, a conve- nient system to analyze, readily identified by virtue of its confine- ment, is the mass of material enclosed within the cylinder by the piston.

In the case of the nozzle shown in Figure 3.5, the fluid occupying the nozzle changes from instant to instant. Thus, different systems occupy the nozzle at different times.

A more convenient method of analysis of the nozzle would be to consider the region bounded by

System

Figure 3.4 An easily identifiable system.

Figure 3.5 Control volume for analysis offlow through a nozzle.

the dotted line. Such a region is acontrol volume. A control volume is a region in space through whichfluidflows.²

The extreme mobility of a fluid makes the identification of a particular system a tedious task. By developing the fundamental physical laws in a form that applies to a control volume (where the system changes from instant to instant), the analysis of fluid flow is greatly simplified. The control-volume approach circumvents the difficulty in system identification. Succeeding chapters will convert the fundamental physical laws from the system approach to a control-volume approach. The control volume selected may be either finite or infinitesimal.

2A control volume may befixed or moving uniformly (inertial), or it may be accelerating (noninertial). Primary consideration here will be given to inertial control volumes.

3.5 Systems and Control Volumes ◀ 33

▶

C H A P T E R

4

Conservation of Mass:

Control-Volume Approach

T

he initial application of the fundamental laws offluid mechanics involves the law of conservation of mass. In this chapter, we shall develop an integral relationship that expresses the law of conservation of mass for a general control volume. The integral relation thus developed will be applied to some often-encounteredfluid-flow situations.

▶ 4.1

INTEGRAL RELATION

The law of conservation of mass states that mass may be neither created nor destroyed. With respect to a control volume, the law of conservation of mass may be simply stated as

rate of mass efflux from

control volume 8>

><

>>

:

9>

>=

>>

;

rate of mass flow into

control volume 8>

><

>>

:

9>

>=

>>

;‡

rate of accumulation of mass within control volume 8>

><

>>

:

9>

>=

>>

;ˆ0

Consider now the general control volume located in a fluid-flow field, as shown in Figure 4.1.

For the small element of area dA on the control surface, the rate of mass efflux ˆ …rv†…dAcosq†, wheredAcosqis the projection of the areadAin a plane normal to the velocity vector,v, andqis the angle between the velocity vector,v, and theoutwarddirected unit normal vector,n, todA.From vector algebra, we recognize the product

rvdA cosqˆrdAjvjnjcosq v

n dA

Streamlines at time t

Figure 4.1 Fluidflow through a control volume.

34

as the“scalar”or“dot”product

r…v_?n†dA

which is the form we shall use to designate the rate of mass efflux throughdA.The productrv is the massflux, often called the mass velocity,G.Physically, this product represents the amount of massflowing through a unit cross-sectional area per unit time.

If we now integrate this quantity over the entire control surface, we have Z Z

c:s:r…v_?n†dA

which is the net outwardflow of mass across the control surface, or thenet mass effluxfrom the control volume.

Note that if mass is entering the control volume—that is,flowing inward across the control surface—the product v

^{nˆ|v|n| cosq} is negative, since q>90°, and cosq is therefore negative. Thus, if the integral is

positive, there is a net efflux of mass negative, there is a net influx of mass

zero, the mass within the control volume is constant.

The rate of accumulation of mass within the control volume may be expressed as

@

@t Z Z Z

c:v:rdV

and the integral expression for the mass balance over a general control volume becomes Z Z

c:s:r…v_?n†dA‡@

@t Z Z Z

c:v:rdVˆ0 (4-1)

▶ 4.2

SPECIFIC FORMS OF THE INTEGRAL EXPRESSION

Equation (4-1) gives the mass balance in its most general form. We now consider some frequently encountered situations where equation (4-1) may be applied.

Ifflow is steady relative to coordinatesfixed to the control volume, the accumulation term, @/@tRRR

c.v.rdV, will be zero. This is readily seen when one recalls that, by the definition of steadyflow, the properties of aflowfield are invariant with time, hence the partial derivative with respect to time is zero. Thus, for this situation the applicable form of the continuity expression is

Z Z

c:s:r…v_?n†dAˆ0 (4-2)

Another important case is that of an incompressibleflow withfluidfilling the control volume. For incompressibleflow the density,r, is constant, and so the accumulation term involving the partial derivative with respect to time is again zero. Additionally, the density term in the surface integral may be canceled. The conservation-of-mass expression for incompressibleflow of this nature thus becomes

Z Z

c:s:…v_?n†dAˆ0 (4-3)

4.2 Specific Forms of the Integral Expression ◀ 35

The following examples illustrate the application of equation (4-1) to some cases that recur frequently in momentum transfer.

Example 1

As ourfirst example, let us consider the common situation of a control volume for which mass efflux and influx are steady and one-dimensional. Specifically, consider the control volume indicated by dashed lines in Figure 4.2.

Equation (4-2) applies. As mass crosses the control surface at positions (1) and (2) only, our expression is Z Z

c:s:r…v?n†dAˆ Z Z

A₁

r…v?n†dA‡ Z Z

A₂

r…v?n†dAˆ0

The absolute value of the scalar product (v?n) is equal to the magnitude of the velocity in each integral, as the velocity vectors and outwardly directed normal vectors are collinear both at (1) and (2). At (2) these vectors have the same sense, so this product is positive, as it should be for an efflux of mass. At (1), where massflows into the control volume, the two vectors are opposite in sense—hence the sign is negative. We may now express the continuity equation in scalar form

Z Z

c:s:r…v?n†dAˆ Z Z

A1

rvdA‡ Z Z

A2

rvdAˆ0 Integration gives the familiar result

r1v1A1ˆr2v2A2 (4-4)

In obtaining equation (4-4), it is noted that theflow situation inside the control volume was unspecified. In fact, this is the beauty of the control-volume approach; theflow inside the control volume can be analyzed from information (measurements) obtained on the surface of the control volume. The box-shaped control volume illustrated in Figure 4.2 is defined for analytical purposes; the actual system contained in this box could be as simple as a pipe or as complex as a propulsion system or a distillation tower.

In solving Example 1, we assumed a constant velocity at sections (1) and (2). This situation may be approached physically, but a more general case is one in which the velocity varies over the cross-sectional area.

Example 2

Let us consider the case of an incompressibleflow, for which theflow area is circular and the velocity profile is parabolic (see Figure 4.3), varying according to the expression

vˆvmax 1 r R

2

wherevmaxis the maximum velocity, which exists at the center of the circular passage (i.e., atrˆ0), andRis the radial distance to the inside surface of the circular area considered.

A₁

1 A2

1

2

2 Figure 4.2 Steady one-dimensionalflow into and out of a control volume.

The above velocity-profile expression may be obtained experimentally. It will also be derived theoretically in Chapter 8 for the case of laminarflow in a circular conduit. This expression represents the velocity at a radial distance,r,from the center of the flow section. As the average velocity is of particular interest in engineering problems, we will now consider the means of obtaining the average velocity from this expression.

At the station where this velocity profile exists, the mass rate offlow is …rv†avgAˆ

Z Z

A

rvdA

For the present case of incompressibleflow, the density is constant. Solving for the average velocity, we have vavg ˆ 1

A Z Z

A

vdA

ˆ 1 pR²

Z 2p 0

Z R 0

vmax 1 r R

2

r dr dq

ˆ vmax

2

In the previous examples, we were not concerned with the composition of the fluid streams. Equation (4-1) applies tofluid streams containing more than one constituent as well as to the individual constituents alone. This type application is common to chemical processes in particular. Ourfinal example will use the law of conservation of mass for both the total mass and for a particular species, in this case, salt.

Example 3

Let us now examine the situation illustrated in Figure 4.4. A tank initially contains 1,000 kg of brine containing 10% salt by mass.

An inlet stream of brine containing 20% salt by massflows into the tank at a rate of 20 kg/min. The mixture in the tank is kept uniform by stirring. Brine is removed from the tank via an outlet pipe at a rate of 10 kg/min. Find the amount of salt in the tank at any timet, and the elapsed time when the amount of salt in the tank is 200 kg.

R

r cL

cL Figure 4.3 A parabolic velocity profile

in a circularflow passage.

20 kg/min

10 kg/min Salt content

20% by mass

Tank, initial content 1,000 kg

Control volume

Figure 4.4 A mixing process.

4.2 Specific Forms of the Integral Expression ◀ 37

Wefirst apply equation (4-1) to express the total amount of brine in the tank as a function of time. For the control volume shown

Z Z

c:s:r…v?n†dAˆ10 20ˆ 10 kg=min

@

@t Z Z Z

c:v:rdVˆ d dt

Z M 1000

dMˆ d

dt…M 1000†

whereMis the total mass of brine in the tank at any time. Writing the complete expression, we have Z Z

c:s:r…v?n†dA‡ @

@t Z Z Z

c:v:rdV ˆ 10‡d

dt…M 1000† ˆ0 Separating variables and solving forMgives

Mˆ1000‡10t …kg†

We now letSbe the amount of salt in the tank at any time. The concentration by weight of salt may be expressed as S

Mˆ S

1000‡10t

kg salt kg brine Using this definition, we may now apply equation (4-1) to the salt, obtaining

Z Z

c:s:r…v?n†dAˆ 10S

1000‡10t …0:2†…20† kg salt min and

@

@t Z Z Z

c:v:rdVˆ d dt

Z S S0

dSˆdS dt

kg salt min The complete expression is now

Z Z

c:s:r…v?n†dA‡@

@t Z Z Z

c:v:rdVˆ S

100‡t 4‡dS dtˆ0 This equation may be written in the form

dS dt‡ S

100‡tˆ4

which we observe to be afirst-order linear differential equation. The general solution is Sˆ2t…200‡t†

100‡t ‡ C 100‡t

The constant of integration may be evaluated, using the initial condition thatSˆ100 attˆ0 to giveCˆ10,000. Thus, the first part of the answer, expressing the amount of salt present as a function of time, is

Sˆ10;000‡400t‡2t² 100‡t

The elapsed time necessary forSto equal 200 kg may be evaluated to givetˆ36.6 min.

Example 4

A large well-mixed tank of unknown volume, open to the atmosphere initially, contains pure water. The initial height of the solution in the tank is unknown. At the start of the experiment a potassium chloride solution in water startsflowing into the tank from two separate inlets. Thefirst inlet has a diameter of 1 cm and delivers a solution with a specific gravity of 1.07 and a velocity of 0.2 m/s. The second inlet with a diameter of 2 cm delivers a solution with a velocity of 0.01 m/s and a density of 1053 kg/m³. The single outflow from this tank has a diameter of 3 cm. A colleague helps you by taking samples of the tank and outflow in your absence. He samples the tank and determines that the tank contains 19.7 kg of potassium chloride. At the same moment he measures theflow rate at the outlet to be 0.5 L/s, and the concentration of potassium chloride to be 13 g/L. Please calculate the exact time when your colleague took the samples. The tank and all inlet solutions are maintained at a constant temperature of 80°C.

Solution

First, let’s make a roughfigure with all the information we are given so we can visualize the problem.

Inlet #1

Diameter = 1 cm Specific gravity = 1.07 Velocity = 0.2 m/s

Inlet #2

Diameter = 2 cm Density = 1053 kg/m³ Velocity = 0.01 m/s

Outlet

Diameter = 3 cm Flow Rate = 0.5 L/s Concentration = 13 g/L Tank

Initially contains pure water Analyzed to have 19.7 kg KCl at some time t.

Beginning with equation (4.1),

Z Z

r…u?n†dA‡@

@t Z Z Z

rdV ˆ0

_

mout m_in‡dM dt ˆ0 Since we have two inputs to the tank,

_

mout ‰m_in1‡m_in2Š ‡dM dt ˆ0 _

moutˆ …0:5 L=s†…13 g=L† ˆ6:5 g=s

_

minˆm_in1‡m_in2ˆ

…1:07† 971:8kg m³

0:2 m=s

… † p…0:01 m†² 4

! 1000g

kg

‡ …1053 kg=m³†…0:01 m=s† p…0:02 m†²

4

! 1000 g

kg

ˆ19:64 g=s 6:5 g=s 19:64 g=s‡dS

dt ˆ0

4.2 Specific Forms of the Integral Expression ◀ 39