Flows with curved streamlines

This chapter discusses flows that form many different patterns, and the same analysis techniques apply to mostly straight flows as to those that are strongly curving. Flows that are strongly curving, however, present a particular challenge to our intuition because rotational motion is more complex and can lead to counterintuitive results. For this reason, flows with curved streamlines typically are covered only in advanced courses in fluid mechanics. An introduction to flows with curved streamlines is in Section10.5.

Many important flows have curved streamlines. A tornado is an extreme exam-ple of such a flow, and understanding their velocity and pressure distributions can be of great humanitarian importance. Other curved flows include fluids stirred in a vessel, water flowing in curving rivers or through pipe bends, blood flowing throughout the human body, vessels being drained of their contents, plastic flow-ing into a mold, smoke rflow-ings, and vortices formed at the tips of airplane wflow-ings or in the wake following a propeller (Figure2.43).

An unusual phenomenon associated with curved flow is the development of secondary flows near boundaries. We experience this type of flow when stirring

Figure 2.43 Various flows with curved streamlines are observed: (a) vortices shed by a stationary object in a flow; (b) tornadoes;

and (c) whirlpools. Images courtesy of the National Science Foundation (nsf.gov), the National Oceanic and Atmospheric Administration’s National Severe Storm Laboratory, and tippecanoe.in.gov.

loose tea leaves in a cup (Figure 2.44).⁷ After stirring, the circular flow dies slowly, and the brewed tea comes to rest. It is interesting to observe that the tea leaves collect in the center of the cup. Because of the inertia⁸of the spinning tea leaves, it seems intuitive that the leaves would be thrown to the outer perimeter of the cup rather than collect in the center. They collect in the center rather than at the periphery because of a weak radial flow in the boundary layer near the bottom of the cup. We use this example to frame a brief discussion of secondary flows.

The strong circular flow in the teacup experiment is called the primary flow, and the weaker radial flow that takes place at the bottom is called the secondary flow [154]. In the teacup, the secondary flow occurs because the fluid near the bottom is slowed down by its proximity to the motionless bottom surface. Away from the bottom, in the strong primary flow, a pressure distribution builds up, resulting in a larger pressure near the outer edges of the cup compared to the center. Near the bottom wall of the teacup, the slowed fluid is unable to maintain this pressure gradient and becomes subject to it instead, and fluid is pushed toward

7In some cultures, the teabag has replaced the practice of brewing loose tea in a cup, so this phenomenon may not be familiar; a little fieldwork therefore may be required to observe the secondary flow discussed.

8Recall that inertia is the tendency of a body once in motion to remain in motion unless an outside force acts on it. Thus, inertial forces in a circular flow refer to the tendency of fluid particles to experience an outward force pushing them toward larger radial positions.

Figure 2.44 After azimuthal stirring, tea leaves tend to gravitate to the center of a cup. This effect is due to the secondary flow near the bottom of the cup.

the center of the cup. The tea leaves, which are heavier than water and therefore settle to the bottom of the cup, are dragged along in this inward flow and collect in the center of the flow (see Figure2.44).

A second example of secondary flow induced by curved streamlines occurs near the bottom of a riverbed. This flow is partially responsible for the tendency of rivers and streams to develop exaggerated bends and turns. If a mild bend develops in a river or stream, the induced secondary flow drags silt and other sediments from the outer bank of the river and deposits them on the inner bank, accentuating the bend and strengthening the secondary flow [112].

Secondary flows can be beneficial in applications that require good mixing, such as in a heart–lung machine (HLM). The HLM, or pump oxygenator [54], is an instrument used in surgery when the heart must be stopped to allow a surgeon to perform repairs. A body cannot survive without a heart; thus, the duties of the heart are taken over by the HLM. An important function of the heart is to pump blood to the lungs, where carbon dioxide is removed from the blood and oxygen is replenished. The heart also pumps the newly restored blood to the rest of the body, where it is needed. When the HLM takes over for the heart, it pumps blood to an external device in which oxygen is added to the blood and carbon dioxide is removed (Figure2.45). The transfer of gases to and from the blood in a membrane oxygenator is effected through gas-permeable circular tubing arranged in coils (Figure2.46). The primary flow is down the length of the tube, but the tube is curved intentionally to induce a secondary flow. The streamlines for this

O2 heat

exchanger

pump

membrane oxygenator – oxygen moves into blood;

carbon dioxide comes out

O2

MO

spent blood

fresh blood

Figure 2.45 Schematic of the surgical use of a membrane oxygenator, a type of heart–lung machine. In a HLM, blood returning to the heart is pumped outside of the body and through the membrane oxygenator. In the membrane oxygenator, oxygen diffuses through membranes and dissolves into the blood, and carbon dioxide diffuses back through the membranes and exits the oxygenator. The oxygen-laden blood exiting the membrane oxygenator returns to the body.

secondary flow are shown in Figure2.46. The vortices in a helical tube were first described by W. R. Dean [38,39] and are called Dean vortices. This secondary flow in the HLM moves blood from the walls to the center of the tube and back again as the fluid progresses downstream [21,53,110,140]. Thus, the secondary flow stirs up the blood and results in an improvement of a factor between two and four in the blood oxygenation that occurs [110].

The subtle nature of curved flow makes these flows a challenge to study. Flows with curvature are usually analyzed with the help of the concept of vorticity, which is a vector quantity related to the amount of rotational character in a flow

Figure 2.46 The oxygen that diffuses into the blood near the tube surface mixes efficiently with the rest of the blood because of a secondary flow that takes place in the curved tubes.

field at a particular point. In terms of vector calculus, vorticityω is defined as ω ≡ ∇ × v, where v is the vector that describes the local direction and magnitude of fluid velocity, and∇ is the spatial differentiation operator (see Section1.3).

Like velocity, vorticity forms a field, and we speak of a pattern of vortex lines in a flow that map out the local vorticity vector by tracing lines that are everywhere tangent to the vorticity (see Section8.3). Vortex lines drawn through every point on a closed curve form a vortex tube. Various mathematical theorems based on momentum, mass, and energy conservation apply to vorticity and vortex tubes and can be helpful in understanding fluid motions involving strong amounts of curvature. For example, the product of the magnitude of the vorticity and the cross-sectional area of a vortex tube must be constant for a vortex tube [72].

Vorticity is introduced in Section8.3[114], and flows with curved streamlines are discussed in Section10.5[123]. Many resources in the literature [9,79,154,168]

can guide further study of highly rotational flows once the basics in this text have been mastered.

2.9 Magnetohydrodynamics

The fluid behaviors described in the preceding sections are exhibited by normal fluids including air, water, oils, and foods. In addition to these behaviors, there are specialized types of fluid behaviors characteristic of more esoteric fluids, such as the molten core of the Earth. Research fields have arisen around unusual fluids, and basic fluid mechanics is the entry point to the study of these advanced topics, one of which is the field of magnetohydrodynamics (MHD), which helps us to understand flows in the core of the Earth or on the surface of the sun.

As discussed in this chapter, flow and deformation of fluids is caused by the imposition of forces such as a knife spreading peanut butter or gravity pulling water over Niagara Falls. Three types of forces cause most flows: pressure dif-ferences, imposed forces that act on the boundaries of a fluid, and gravity (see Chapter6). A more unusual source of flow driving force is a magnetic field. When a fluid is electrically conductive, forces are induced in the fluid by an external electric field. These forces cause fluid motion; in turn, the fluid motion alters the magnetic field. To understand the effect of magnetic field on the motion of a conductive fluid, the electromagnetic and the fluid-mechanics equations must be considered simultaneously. The electromagnetic equations are the Maxwell differential equations [167] and are taught in physics and chemistry courses. The fluid-mechanics equations are those that are discussed in this book (see Chap-ter6). Both types of equations are vector-field differential equations and are best described with vector calculus.

The phenomenon of MHD is due to the mutual interaction of a magnetic field B and a fluid velocity fieldv [35]. For convenience, we divide the process into three parts. In the first part, relative movement of a conducting fluid and a magnetic field causes an electromotive force (e.m.f.) to develop. This is a consequence of Faraday’s law of induction [167], and when a conducting fluid moves in a magnetic field, a current begins to flow in the conducting fluid. The induced current in the fluid must itself create a magnetic field, in accordance with Amp`ere’s law. In the second part, the induced magnetic field adds to the

V=0

Magnetic south

Magnetic north

V

Magnetic south

Magnetic north

Figure 2.47 When a conductive rod is drawn through a vertical magnetic field, the induced current in the rod creates an induced magnetic field [35]. The effect on the magnetic field lines is that they bend. The visual effect is as if the rod is dragging the magnetic field lines in the direction of its motion. The effect is similar for a conductive fluid, although more complicated because the moving conductor is deformable in that case. Reprinted with the permission of Cambridge University Press.

original magnetic field, altering the field lines. The change is usually such that the fluid appears to drag the magnetic field lines in the direction of the flow (Figure2.47). The third step in this simplified explanation of MHD is when the modified magnetic field interacts with the induced current density to give rise to a Lorentz force. This is a force exerted on moving charged particles, and it acts perpendicular to both the direction of the motion of the charged particles and the magnetic field lines [167]. In MHD, the Lorentz force is directed so that it inhibits the relative movement of the magnetic field and the fluid [35].

MHD figures prominently in astronomy, geology, and metallurgy (Figure2.48).

The Earth’s magnetic field is a result of fluid motion in its core, and the solar magnetic field generates sunspots and solar flares due to MHD. Because liquid metals are conductive, MHD is used in the metallurgical industry to heat, pump, stir, and levitate liquid metals. MHD also is used to damp surface motion in metallurgical processing [35]. MHD flows are highly rotational, and vorticity is an important tool in their study.

This discussion is only a summary of an advanced application of fluid mechanics, but it demonstrates that a mathematical understanding of basic fluid flow is essential before attempting to master complex fluid motions, such as those induced in conducting fluids by magnetic fields. Investing in the study of basic fluid mechanics opens up a wide variety of avenues for advanced fluid applications.