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Unformatted text preview: " c./ 6.1 The velocity in a certain twodimen
sional ﬂow ﬁeld is given by the equation ‘V = 2xti — 2ytj ‘ Where the velocity is in ft/s when x, y, and t are
in feet and seconds, respectively. Determine
expressions for the local and convective compo}:
nents of acceleration in the x and y directions;
What is the magnitude and direction of the veA;
locity and the acceleration at the point x 4—r y =
2 ft at the time t = 0? ‘ Fm”? “Was/m fur velacl+5/ LL: zxA 4nd 7;. 4512
Since xx =' '3’? + a. 9.5 1— v3.3“
*ﬁﬂn
4x {/007/)" %' 7" X
and 1
a; (ea/w) = u ’3}: Mg; = (Zx£)(22£)+ (432%)
= 43*,"
S/hi/ar/g/ :
au 9w Mr
and f7
= lfﬂt’z'
’415 2C=°J=2sz and zf=~0
u= 2(2)/o)=c> 1r: Z(z)[o)—=0
59 7714171: 7:0 WM 4 = 2x+ amt"? == 200+ 9(2Mo) = 4752/51
~2j+6ﬁ7iL~‘ “2(2) 1" ‘I‘fzJ/D) :”""Ci'/52 ’ 2;... L/fsaj‘ pie/sf M75, IZI=W”ﬁiﬁ/§ 6.2 Repeat Problem 6.1 if the ﬂow ﬁeld is described by the
equation = (3x2 + 1)? — 6x313 where the velocity is in ft/s when x and y are in feet. 4 IL 3 2
” 36X (5’)! +/) = [9(1) [30) 2+1} #211! ~3E/5 Z: n9»: 9 m. =Ixmm~im= 129% Thus
: 25‘? 2+ 32:; #151 = VQV)1+(IZ)1 : 2L8 9575/51 Gz V6.3 ‘ The velocity in a certain flow ﬁeld is given by the equa
tion
V=xl~lfx223+yzl2 Determine the expressions for the three rectangular components
of acceleration. 1 Fram apex/kn 4r tuba/{17) a = X 0: {22
Since
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a2! 22H“ “ ”+7753 “”22
50 ﬂat l 2
4%: 0+ (X‘ We»); +(X?)(2)+f:rz)/s)
Z
= 52w 5%
6:"3 6.7 For a certain incompressible, two—dimensional ﬂow
ﬁeld the velocity component in the y direction is given by the
equation ' U=3xy—x2y% Determine the velocity component in theix direction so that the
continuity equation is satisﬁed. ‘ 72> Sal/37% 771? cvnéliﬂiaiﬂy ezueélémj __ ——“— 0 (I; 31, 1 $31?
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a = :Xzf 5 + {(31) where 75(9) is an undekrmined ﬁmeézéw 0:5 g. 6.10 For a certain incompressible ﬂow ﬁeld it
is suggested that the velocity components are given
by the equations ‘ u=2xy v=—x1y :WZO
Is this a physically possible ﬂow ﬁeld? Explain. Any Physical/g PaSS/Z/é: I‘ncom/oress/Z/e, F/uu [fa/A
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Fir 7712 1/5 ADC/1‘7 c/ui 7‘r/Jmi1/fm 7I'um __ :0
ﬁg; 2‘ >
g)? = 23 M :'  X 2 aw
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twodimensional ﬂow ﬁeld (122 = O) is u, = 2r + 3r2 gin 0 Determine the corresponding tangential velocity component,
1),}, required to satisfy conservation of mass. ’90V/e’ / 9% 2V” f5 43:
l’ 3‘!» +FZH3+_é2;0 7' J
Smce ’V;—=d}
aw? : _ 20m)
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r‘?}‘ 2P2+3r35m 6
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along a horizontal streamline that coincides with
the x axis in a plane, two dimensional incom
pressible ﬂow ﬁeld was experimentally found to
be described by the equation u = chz. Along this
streamline determine an expression for: (a) the
rate of change of the v—component of velocity with
respect to y; (b) the acceleration of the particle; ‘ and (c) the pressure gradient 1n the x direction.
The ﬂuid 15 Newtonian. ' (a) From The Conimurfg 6514.119!“ an. US: 1
31¢”; 23 0 so 7714!: ail/771 u=xz
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(A) “x: “3—32 +7; 2%: (ix‘)(2x)+ (~2263)(0) = 2x3
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50 ﬂat 2X3: “F§£+ i&{2+o: I and (I) 6. '13 Two horizontal, inﬁnite, parallel plates are spaced w
a distance (3 apart. A viscous liquid is contained between the T ’ ’ ‘ '
plates. The bottom plate is fixed and the upper plate moves parallel to the bottom plate with a velocity U.EE‘»eczause of "'V M.
the noslip boundary condition (see Video V6.5“), the liquid b motion is caused by the liquid being dragged along; by the i 3 moving boundary. There is no pressure gradient in the di—
rection of ﬂow Note that this 15 a so called simple Couette *
ﬂow discussed in Section 6. 9 2 (a) Start with the Navier— Fixed Pl‘ie
Stokes equations and determine the velocity distribution be
tween the plates (b) Determine an expression for the flowrate passing between the plates (for a unit lwidth) Ex
press your answer in terms of b and (I (a) Far sie'aa’y f/ou M7711 'Vztha ff {cl/tau: 7747f We
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’ U
«LL (724: AL: U and C/ = E Merefme U
u=7§ﬂ
b i A "
I U U W  Ii]?
(1,) g: “(1)497; 5‘13 = 1: a: a” z
o o m where % /'5 i’i’he. ﬁ/owrai'e per Lin/'1' WIdfh. 6.75 Two fixed, hormonal, parallel plates are spaced 0.4 m.
apart. A viscous liquid (5:. = 3 X 10—3 lb  s/ftz. $0 = 0.9)
flows between the plates with a them) velocity of 0.5 ft/s. The
flow is laminar. Determine the pressure drop per unit length in
the direction of flow. What is tic maximum velocity in the
channel? ’ iéqu Direction of flow 6.77 7 A viscous, incompressible ﬂuid ﬂows betf
7' tween the two inﬁnite, vertical, parallelplates of
Fig. P637. Determine, by use of the Navijeir
Stokes equations, an expression for the pressure
gradient in the direction of ﬂow. Express Your
answer in terms of the mean velocity. Assume
that the ﬂow is laminar, steady, and uniform; l \\\\\\\\‘\\\\\\\\\\\
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s _ l_..L £12412 V: 2): " 3 /’ 1+ Mm 77m: . 6.80 An incompressible, viscous ﬂuid 1 is
placed between horizontal, inﬁnite, parallel
plates as is shown in Fig“ P680. The two plates
move in opposite directions with constant veloc—
ities, U, and U2, as shown. The pressure gradient
in the x direction is zero and the only body forice
is due to the ﬂuid weight. Use the Navier~Stoli1es
equations to derive an expression for the velocity 1 distribution between the plates. Assume lamiriar
ﬂow. 1 FIGURE P630 l
x
i
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Thus) t) 14»e7 Jé,/ 6.81 Two immiscible, incompressible, viscous
ﬂuids having the same densities but different \fis
cosities are contained between two inﬁnite, hbr
izontal, parallel plates (Fig. P6.8l . The bottom
plate is ﬁxed and the upper plate moves with a
constant velocity U Determine the velocity at the
interface. Express your answer in terms of U, yl,‘
and #2 The motion of the ﬂuid' is caused entirely  ‘_
by the movement of the upper plate; that IS there 7 i. ‘ ' ’ o is no pressure gradient in the x direction. The . FIGURE 1'6 3 l‘
ﬂuid velocity and shearing stress is continuous 7 across the interface between the two ﬂuids. As sume laminar ﬂow Far We spew/ﬂed Canal/from: 21—0 arr: ~0 53? :0} and ai" 0’ so ﬂmt 7712 X— wmponenf 0/; 7‘"): IVaV/erJ/v/(e: (iuaézéus {55,é,lz7a)
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pressure gradient in a direction parallel to the
plates 18 suddenly applied and the? ﬂuid starts to
move. Determine the appropriate differential
equation(s), initial condition and [Eoundary con
ditions that govern this type of ﬂow. You need not solve the equation(s) DI fwzenwémY eguaz’wﬁm aErc 77/2 Mme as Ezs. 4127/ 5/30, 4/14 é/B/ E’xae/aé 7712:! j“ 7':E 0 (5/9“, ﬁre f/au Ii «hymn/7)
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* 1?: " la? 49115251011) 6.3 2‘ (a) Show that for Poiseuille ﬂow in a
tube of radius R the magnitude of the wall shear ing stress, 1'”, can be obtained from the relation
ship i(Trz)walli = 4"uRQ E E for a Newtonian ﬂuid of viscosity ,1; The volume
rate of ﬂow is Q. ([1) Determine the magnitude
of the wall shearing stress for a ﬂuid having a
viscosity of 0.004 Ns/m2 ﬂowing with an average
velocity of 130 mm/s in a 2—mmdiameter tube. (a) T =,u(~b—1:’i’§: (5;. 6.1257”) . . E
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centric cylinders as shown in Fig P6315 The
outer cylinder is ﬁxed, but the inner cylinder
moves with a longitudinal velocity V0 as shown.
For what value of V0 will the drag on the inner
cylinder be zero? Assume that the ﬂow IS laminar,
axisymmetric, and fully developed.‘ ©71/l/l/l/l/I/I/Il/I/II/Ill III/II/H/WIIIIIII Ill. Ezua‘ £10»: 4 /’f 7 which was gJamie/calcxd rér 1C/ow m Cl/‘Cu/ay fakes,
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der of radius R is located in an inﬁnite mass‘ of
an incompressible ﬂuid. Start with the Navier~
Stokes equation in the 6 direction and derive an
expression for the velocity distribution for the steady ﬂow case in which the cylinder 15 rotating
about a ﬁxed axis with a constant angular velodity
a). You need not consider body forces Assume
that the ﬂow 15 axisymmetric and the ﬂuid is? at
rest at inﬁnity.‘ (Ea. 4.35) 11‘ 7/0/0445 771;:
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(214% 2 AWN? 6. ‘75 A viscous ﬂuid is contained between two
inﬁnitely long vertical concentric cylinders. The
outer cylinder has a radius r and rotates with an
angular velocity co. The inner cylinder 13 ﬁxed Eand
has a radius r,. Make use of the Navier—Stokes
equations to obtain an exact solution for theE ve
locity distribution in the gap. Assume that the
ﬂow in the gap is axisyrnmetric (neither velocity
nor pressure are functions of angular position 6
within gap) and that there are no velocity com—
ponents other than the tangential component
The only body force is the weight. E The Velociig disirlbubo'n In ﬁve annular Space is 7112614 by We 9161141:on =C‘"
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