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Unformatted text preview: Fr{_5.;‘1y6 (0‘7/174'0‘.’ (f EX’ *Cw—uxﬁe/ 4 lﬁéJpn/‘g 7&4// “~77 Mij/x3,4/C Ihrﬁ¢¢ 7‘!~;.‘04—, W4,V( /r.q,(_; 4 Mcy— W Aptf/Z; ' IC/Lu'tx/ /€4m¢; 77'! [V‘r7(a(/ “40% ’57]?
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ivior after eopectic
/ _Common
ﬂuids \ 
lOlli’OPIC Time tin rate; yield may also be nonlinear. An example of a yielding ﬂuid is toothpaste, which
will not ﬂow out of the tube until a ﬁnite stress is applied by squeezing. A further complication of nonnewtonian behavior is the transient effect shown
in Fig. 1.6b. Some ﬂuids require a gradually increasing shear stress to maintain a
constant strain rate and are called rheopectic. The opposite case of a ﬂuid which
thins out with time and requires decreasing stress is termed thixotropic. We shall
neglect nonnewtonian effects in this book; see Ref. 5 for further study. Surface Tension A liquid, being unable to expand freely, will form an interface with a second liquid
or gas. The physical chemistry of such interfacial surfaces is quite complex, and
whole textbooks are devoted to this specialty [l3]. Molecules deep within the
liquid repel each other because of their close packing. Molecules at the surface are
less dense and attract each other. Since half of their neighbors are missing, the
mechanical effect is that the surface is in tension. We can account adequately for
surface effects in ﬂuid mechanics with the concept of surface tension. If a cut of length dL is made in an interfacial surface, equal and opposite forces
of magnitude T dL are exposed normal to the cut and parallel to the surface,
where T is called the coefﬁcient of surface tension. The dimensions of T are {F /L},
with SI units of newtons per meter and BG units of pounds force per foot. An alternate concept is to open up the cut to an area dA; this requires work to be
done of amount T (M. Thus the coefﬁcient T can also be regarded as the surface energy per unit area of the interface, in newtonmeters per square meter or foot
pounds force per square foot. The trvo most common interfaces are waterair and mercuryair. For a clean
surface at 20°C = 68°F, the measured surface tension is Y _ 0.0050 lbf/ft = 0.073 N/m airwater (I 48)
— 0.033 lbf/ft = 0.48 N/m airmercury ' These are design values and can change considerably if the surface contains con
taminants like detergents or slicks. Generally Y decreases with liquid temperature
and is zero at the critical point. If the interface is curved, a mechanical balance shows that there is a pressure
difference across the interface, the pressure being higher on the concave side. This is illustrated in F ig. 1.7. In Fig. 1.7a, the pressure increase in the interior of a liquid
cylinder is balanced by two surfacetension forces 2RL Ap = ZTL
T
or Ap = R (1.49) We are not considering the weight of the liquid in this calculation. In Fig. 1.7b, the
pressure increase in the interior of a spherical droplet balances a ring of surface VISCOSITY AND OTHER SECONDARY PROPERTIES/ 1.7 .: m Mr...” .nl 2RL Ap 1R2 Ap (a) (b) Ap (M .m. l e
A (C) ‘n . Fig. 1.7 Pressure change across a curved interface due to surfatx tension: (a) interior of
a liquid cylinder; (b) interior of a spherical droplet; (c) general curved interface. tension force 11R2 Ap = 21rRY
' 2r
or Ap = i— (1.50) We can use this result to predict the pressure increase inside a soap bubble, which
has two interfaces with air, an inner and outer surface of nearly the same radius R 4T
Apbubble z 2 Apdrnplet = 17 32 INTRODUCTION 1R2 4p ZIRT (b) :e tension: (a) interior of
curved interface. (1.50) a soap bubble, which
rly the same radius R (1.51) Figure 1.7:: shows the general case of an arbitrarily curved interface whose princi pal radii of curvature are R , and R,. A force balance normal to the surface will
show that the pressure increase on the concave side is Ap=T(R;" +R;‘) (1.52) Equations (1.49) to (1.51) can all be derived from this general relation; e.g.. in Eq.
(1.49) R, = R and R2 a co. A second important surface effect is the contact angle 9 which appears when a
liquid interface intersects with a solid surface, as in Fig. 1.8. The force balance
would then involve both T and 0. If the contact angle is less than 90°. the liquid is
said to wet the solid; if 0 > 90°, the liquid is termed nonwetting. For example.
water wets soap but does not wet wax. Water is extremely wetting to a clean glass
surface, with 9 z 0°. Like T, the contact angle 0 is sensitive to the actual physi
cochemical conditions of the solidliquid interface. For a clean mercuryairglass
interface, 0 = 130°. Example 1.13 illustrates how surface tension causes a ﬂuid interface to rise or
fall in a capillary tube. EXAMPLE 1.13 Derive an expression for the change in height h in a circular tube of a
liquid with surface tension T and contact angle 0, as in Fig. £1.13. solution The vertical component of the ring surfacetension force at the interface in the tube
must balance the weight of the column of ﬂuid of height h 21tRT cos 0 = pgnR 2h Solving for h, we have the desired result = 2]” cos 0 h
09R Ans. Thus the capillary height decreases inversely with tube radius R and is positive if 0 < 90°
(wetting liquid) and negative (capillary depression) if 9 > 90". Gas Nonwetting ﬂ
Solid Fig. 1.8 Contactangle effects at a liquidgassolid interface. If 0 < 90°, the liquid “wets”
the solid; if 9 > 90°, the liquid is nonwetting. vrscosnv AND OTHER SECONDARY PROPERTIES/1.7 33 Fig. 121.13 Suppose that R = 1 mm. Then the capillary rise for a waterairglass interface, 0 2 0°,
T = 0.073 N/m, and p = 1000 kg/m3 is _ 2(0.073 N/m)(cos 0°) . 2 —. 
h _ 1000 kg/m (9.81 m/s )(0'001 m) 0.015 (N 5 )/kg — 0.015 m — 1.5 cm For a mercuryairglass interface, with 9 = 130°, T = 0.48 N/m, and p = 13,600 kg/m’, the
capillary rise is h _ 2(0.48)(cos 130°) ‘ 131300931 )(0.001) = ‘0'46 cm When a smalldiameter tube is used to make pressure measurements (Chap. '2), these
capillary effects must be corrected for. Vapor Pressure Vapor pressure is the pressure at which a liquid boils and is in equilibrium with its
own vapor. For example, the vapor pressure of water at 68°F is 49 lbf/ft’, while
that of mercury is only 0.0035 lbf/ftz. If the liquid pressure is greater than the
vapor pressure, the only exchange between liquid and vapor is evaporation at the
interface. If, however, the liquid pressure falls below the vapor pressure, vapor
bubbles begin to appear in the liquid. If water is heated to 212°F, its vapor
pressure rises to 2116 lbf/ft2 and thus water at normal atmospheric pressure will
boil. If the liquid pressure is dropped below the vapor pressure due to a ﬂow
phenomenon, we call the process cavitation. As we shall see in Chap. 2, if water is
accelerated from rest to about 50 ft/s, its pressure drops by about 15 lbf/inz, or
1 atm. This can cause cavitation. Figure 1.9 shows cavitation occurring in the low
pressure region associated with the tip vortices shed by a marine propeller. The dimensionless parameter describing ﬂowinduced boiling is the cavitation
number Ca = Fiﬁ—{E (1.53) 34 INTRODUCTION ...
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This note was uploaded on 04/07/2008 for the course AE 2020 taught by Professor Ruffin during the Summer '07 term at Georgia Tech.
 Summer '07
 Ruffin

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