The present paper deals with laminar flow of a viscous incompressible
fluid through a pipe bounded by a hyperbola and a line. The expressions,
for the discharge of flux, maximum velocity and the coefficients
k, k' as introduced by Boussinosqu (Bateman et al 1932) have been
obtained. Graphs representing variations of velocity within the pipe
give the final result

Singh, S N

Tripathi, R C

English

IACS Institutional Repository

I n d ia n  J .  P h y a ., 47. 326-333, (1973).On the laminar flow o f viscous incompressible fluid through a pipe bounded by a hyperbola and a line.S. N. Singh and R. C. TripathiDepartment of Mathematics, Banaras Hindu University, Varanasi-5{Received 15 May 1972, revised 30 June 1972 and 3 October 1972)The present paper deals with lami?iar flow of a viscous incompressible fluid through a pipe bounded ))y a hyperbola and a lino. The expres­sions, for the discharge of flux, maximum velocity and the coeflfleients k, k' as introduced by Roussinosqu (Bateman et al 1932) have been obtained. Graphs ropresonting variations of velocity within the pipe give the final results.IntroductionFlow through various curved pipes have been discussed by several authors Dube(1967) has discussed a problem on the hyperbolic tube and Lai (1965) has consi­dered the flow through a tube whose cross-section is formed by the intersection of two circles.In the present paper, flow through a pipe bounded by a hyperbola and a lino, has been discussed. It is seen that if the cross sectional area is reduced, the ratio ' of maximum to average velocity increases sliglitly.Basic E quations and SolutionsLot the axis of the pipe be along the z-axis. Assuming =  0 — u and w ~J{x, y), th(i basic equations (Ijamb 1916) aredx dy ’d^ w , d^ w _  _  p dx^  dy^  fiSolution of equation (2) is(1)(2)(9), ^  =  0 da:“ ^with a boundary condition that w =  6, on the boundary of the pipe and... (4)Let us take ^  =  A{x^-^xy^)~{-B{x^—y -^\-x), ... (^ )where A and B are constants to be determined from the boundary contitions.326Laminar flow of viscous incompressible fluid etc, 3 2 7The equation (3) with the help of (6) beoomeB. . .  (6)If one of tlio boundaries of the cross-sections of the pipe be a; — a, wo have w =  6, (jii X a and (2a+l)a;2+4aa:—(6a+3)«/® =  0.Hence solving for the constants and substituting in (6), we getThe asymptotes are given by jy iL-l/V3(:K+(2a)/(2«+l)), hence tracing i,hc curve wo get for section as shown in figure 1. The cross sectional area of tlie ])ijx! is :A^.2fydx= y  j^ (2 a + l/4 )» » (2 a + 5/2)l  ^ (2 g + l/4 )i(2ffl+5/4)i-^3 ' 2a-l-l - 1  sinh-x8 2 J (8)The co-ordinates of the centre of gravity of the area are1®2a-[-l (2a+l/4)i(2a+5/4)t{^?^+-^+ -^ s in h -xy - 0 .(9)328 S. N . Singh and R . C. TripathiNow by evaluating 2Hpwdxdy over the whole cross section, we have the dis­charge of flux per second as(2a+5/4)* (2a+l)4 f (2a+l)» , (2o+l/4)* 4 1 4 68(2a+l)* 1 pp 2 a + l /  4a \®2 J p a(2a+3)33/3 \ 2 a + l /24 4 ^16J 128r (2u+l)l (2a+5p f (2fl+l/4)« I (2 a + lW  7 (2a+l)aL 4“  4^  ' 4  I '  1 0 “  80 480 47 2 u + l _  7 1 , 7  i 1 i (2u+ l )M  I 4a 1 /  4a y256 6 256 J ' 256 2 J (2a+3) 32 \ 2 a + l /f ( 2 a + l ) i  (2a+5)i f (2a+l/4)'^ 2a-}-l/4 _  1 \ 1 (2a+l)* ]L 4“ ‘ 4 1 6  ■' "24“  1 6 / 1 6  2 J+4a \ H / 2 a + l  \ * /  2 a ~h 5  \ U ( 2 a H - l / 4 ) «  l l / 2 a + l  >,3 [i ’a('2aH-3)33/'3' \ 2 «+ i / [1 4 / \ 4 M  " 10 '^80\ 4 “  )2a^-H4. 1 (2 a + ] )^  , 3 1 2 a + l"^16 4 ■^  256 / 256 '2 J p128 (2a+3)33/34a \^f(2aH-l)l (2a+5)l f (2«+l/4)3 9 (2a+l)3 , 2a-|-l/4^ l 2 a + l /  L 4 " 4 1  " ' 8  ■■^48 4" 64 '- A l + A  sinh-i128 1 128 2 1From equation (7), we have for the maximum or minimum dw _  —p - [(6a+3)rH (6a-4a2)r-4a2] =  0Or,(1 0 )(11)(12)(13)dr 4/oi(2a“}-3)r =  (4a2-6a)±{(6a-4a3)2+16a2(6a+3)}V2(6a+3).Thus the maximum velocity is"" =  4 7 ^ ^  [-4aV+(3a-2a>H(2a+i).»], where r is given by (12).The velocity at centre of gravity, w is obtained from the equations (7) and (9) by eliminating x with the help ofX =  X =  <(say).It is clear that the elimination is much more complicated.Hence for a =  1, 2, we get® =  [(0.0105), (0.0333), (0.0949)]. ... (14)From tlio above equation, we see that the velocity is increased with the increment of n and the ratio among W is 1:3:9 if the ratio in a is 1:2:4. Hence the ratio of velocity at centre of gravity is greater than the ratio of the span of the pipe.ExampleFor the purpose of comparison with previous results wo consider a numerical example. We put a =  \, and calculate the coefficients k, k'.Using the notations of Boussinosq, we haveLaminar flow of viscous incompressible fl,uid etc, 32 9Q =  UA =  k 2 . A\I*G =  K 'A  ^AvlioroQ -= total How per second in the pit>o, c =  maximum or axial velocity in the pipe,U =  the mean velocity in the j)ipe,A ^  cross seclioual ai-ea of the pipe and k and k' arc the coefficients.It. will be easy to draw conclusions from the following table ;(15)... (16)Table 1Cross sectional Max. vel FluxareaCircular pq)e Bateman et al (1932)3.1416c2i/i  ^ ^ /IElliptical pipe Lamb (1916)7Tai%l—e^ )^ P. 77a,»6,a p2/1Tube bounded by two circles 1.91225“^ Lai (1965)^  (0.1047)62 0.09466V ^P'Channel bounded by twohyperbolasTripathi (1973)1.744rf2 ^  (0.0125)rf2 0.0853dV^PBipo bounded by a hyper­bola and a line (present work)1.05250 0.06998 0.034/?^PThus, for the pipe under studyh =  0.0306, ¥  =  0.0665,wliioh give the ratio of maximum and average velocity as jj- ■ 2.1732.330 S. K . Singh and R . C. T ripath i,From the table, it can bo easily seen that A ^ ~  A  when c® =  0.3352,A.^  =  A provided that =  0.3352(1—(17)(18)A  ^=  A when d'^  =  0.6034,whore A, A^  ^ ylg, A  ^ and A  ^ are the cross-sectional areas of the present pipe, circular pipe, elliptical pipe, tube bounded by two circles and the channel bounded by two hyperbolas, respectively.Disgttssion and Physical Interpretation The expression for velocity distribution in the pipe under study, when a =: 1,i s ;w{x, y) =  — l)(3a;2+4a:—9t/3). (19)Let and denote velocities along x-axis and along y — 0.88x, which is the intersection of the line x — 1 and the hyperbola and is denoted as ON in figure 1. Obviously ic =  0 at x =  1 and at Sx^-fix—9?/- =  0, which gives y =  0.88x.Fig. 2The functions Wy =  Wi{x, 0) and =  Wa(a;, 0.88aj) are plotted in figure 2. Wy luis a maximutn at 0.56 and at 0.36. Thus there are two points along x at which fluid particles will have some velocity, since any line parallel to a;-axis will cut the curve at two pointsLaminar flow o f viscous incompressible fluid etc. 331Difference, 6.6996p , 3fl ^— ^0/t—  IS maximum at a: =  0,666.Again, lot and denote velocities along a: =  0 56 and along »  — 0.178. The functions =  i/;3(0.66, y) and =  1^ 4^(0.178, y) are plotted in figure 3. Since PQ and RS are parallel lines, is never equal to w^ . Both the velocities attainm.'iximum at 0.Difference, s= 3.4299?/“*+0.7332, is maximum at y =  —0.469.T1h‘ velocity at P is w^ a^) ~  0.06998(p//t)and the velocity at Q is iWgfb) == 0.0224 ^ . Purther, the velocities at R and S are :w^ ia) — 0.0333 ^ , w^ ib) =  0.0238^./t [IThus, we have. . .  (20). . .  (21)w^ (b) : w^ ata) =  1:3.12 ■ ^4(») == 1:1.40 M'a(6) : w^ 4(6) =  1:1.04^4(0).: M?a(«) =  1:2.1. . .  ( 2 2 )332 S. N . Singh and R . C . TripathiHence the velocity at P is three times greater than the velocity Q and the velocity at S js slightly loss than the velocity at R. ]3y this we mean that the particles near P and II are moving faster than particles near Q and S. That is, the fluid partich^s along x- axis are moving more speedily than i)articles farther from the ic-axis. It is also concluded that the velocity at S is slightly greater than velocity at Q and the velocity at P is about two times larger than velocity at R.Tims the fluid particles near P are moving with larger speed than particles near Q, R and S, There will bo points given by y =  iO .6  and ibO.3, where and are zero.In figure 2 the variations of velocity along a;-axis and along tho liney =  0.88a;  ^ with difference values of x have been shown. Rinoe CLW^jdiX 0at X — 0.56 and x ~  —0.79 it can ho easily seen that as x moves from 0 to 0.56, Wy increases and attains maximum at a; =  0 56 and then it decreases as x increases. Also as X moves in tho negative direction oi' the .r-axis, Wy decreases and attains minimum at x — —0.79 and then it increases as x further decreases. Since anj^  line parallel to x axis must cut the curve at two points, tho velocities at those points are the same.From tho curve in figure 2 we see that the velocity is maximum ata; — 0 345 ajid minimum at x — 1.041. Thus it is concluded that tho particles between a; =  0 and x =  0.345 are moving faster than particles onwards to x =  0.345.From the figure 2 we see that(23)Figure 3 gives the velocity distribution along tho line through the centre of gravity through maximum point w {^fij'p). From figure 3, it is clear that the symmetrical points have the same velocity and (^ 4)0-05 ■ (^ 3^)0-05 ~  1:2.14(^4)01 : («^3)oi =  1:2.33 I  ... (24)(■*^ 4)0-15 ■ (^d)o-ib — 1:2.62From equations (23) and (24), we conclude that the ratio of the velocity at centre o f gravity and the maximum velocity is nearly two for all values of y. l i  y =  0 tho same ratio is obtained for tho points along the lines parallel to 4/-axis passing through the centre of gravity and the point whore the maximum velocity lies.(^a)o-i ' 1:1.18 1(^2)0 - 4  ■ (^ i)o--i ~  1 :l-40(’^ 2 ) 0  0 ■(^i)o'c ~  1:3 26(^2 )0 *3 : (^ i)o-n “  1:5.29 JCONOLTTSIONWe see iliat the maximum velocity in the present pipe is less than the circular and elliptic pipes and the cliamxol bounded by two hyperbolas, whereas it is more in case of a tube bounded by two circles, when the oross-sootional areas of all the pipes are equal. The maximum velocity in the present pipe is twice the velocity at centre of gravity.Also, it is found that the discharge from the present pipe is less than the dis­charge from the circular and elliptic pipes, and it is more than the discharge through the sections which are formed by the intersections of (i) I wo circles and (ii) two liyperbolas.It is seen that if the cross sectional area is reduced the ratio of maximum to average velocity inoreasos. Thus it is concluded that the present pipe acts more efficiently than a tube bounded by two circles and a channel bounded by two hyperbolas.ReferekohsBatomau H . Drydon H L. & Miiruaghau F P 19.‘12 H yd rod y tia rm ca , Dover Piibliration, p.l 78. nul)o S. N. 1967 Ph.D. Thefiis, Bamxras Hindu University.Ijamb H. 1916 H yd rod ifiia yn ics , 4th Edition, pp. C79-680 Liil K. 1965 Q a n ita , Vol. 16 No. 1-r^npathi H C. 1979 ./. o f  B n c n t i fw  P csc a r ch  Baiiaras Jlindu University, {to  a p p e a r )Laminar flow of viscous incompressible fluid etc. 333

On the Laminar Flow of Viscous Incompressible Fluid Through a Pipe Bounded by a Hyperbola and a Line

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On the Laminar Flow of Viscous Incompressible Fluid Through a Pipe Bounded by a Hyperbola and a Line

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