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A note on hydrodynamic entrance lengths of non-Newtonian laminar falling liquid films
Shorter Commumcations
566
for physlcaUy possible values of N, Q and l The absolute value of A, IS large1 by at least a factor of three than IA21when
the r&-hand side of (A4) vamshes and c, IS then sven by c, = B exp (AZ@,
,so
NHe 1+&+, wluch 1s. u-respective
19,
t(A, - A3el>
(JW
3
A plot of ln c, vs tune now IS a stra&t lme of slope AZ The value of N can be obtamed from the slope, smce
>
of the value of (NHe/.&
relative to the second term,
A,=
fulfilled If
-
(l:S)S03s
-+F)+ J[(l+&J+yy-*] 1‘f(l_Ns) 2
(A61
For values of, say, I(A, - AZ)01> 3 the value of the first term on
A note on hydrodynamic entrance lengths of non-Newtmian
lamiuar falliug liquid lilms
(Received 15 September 1976) The present note mveswates the hydrodynamic entrance lengths of non-Newtoman lammar fallmg hqmd films It IS observed that entrance lengths are dependant on modtied forms of (FrlRe) and the mdex “n” of the power law relation between the shear stress and shear deformatton rate Very recently Stucheh and &islk[l] obtamed a closed form of solution to evaluate the entrance lengths for the no drag condlhon at the vapor-bqmd mterface of a fallmg Newtoman hqmd film down an mclmed plane The analysis was basically developed on the assumption that the parabohc velocity profile at the outlet of the slit gets rapidly rearranged to a slrmlar velocity profile of a semr-parabola The present article 1s more general m approach to mclude both ddatant and pseudoplastic flulds in addition to the Newtonian flulds which are already presented by [l] ANALYSS The physlcai model 1s Identical to the one presented m Ref [l] The equation of motion m mtegral form for zero shear con&tion at the vapor-lrquld mterface can be wntten as pu2dy = - 7w + gp8 sm where the shear stress expression
at the wall IS aven
e by the power
where
the boundary
condttlon bemg at A = 0, fl= 1 Equation (4) degenerates to eqn (7) of Ref [l] when ?I” is made equal to umty Thus eqn (4) IS very general m nature wluch can be solved for different values of “n” Expbcit form of the solution IS possible only when “R” IS an integer Further, the hnutmg value of p can be obtamed by takmg d@dA = 0 Thus, (5) The dtierentml equation (4) 1s converted mto a dtierent follows whde the numerical integration 1s performed
where law
B = 818.. r) = (2&.‘/n)A Thus, eqn (6) LSsolved numencally q = 0, B = l/s.. on IBM 1130
The mdex “n” IS the charactenshc value whtch differs from umty for both ddatant and pseudoplastic fluids The velocity profile at any location 1s aven by the expression
20
16 6
where
i3*
u,,
=
= (r”/2”-‘)@‘_‘”
12
?7G/2Pho
Thus, eqn (1) can be mampulated with the help of eqns (2) and (3) to obtam the dlfferenhal equation 2
subject to the comhtion, VIZ at
,,2
(3)
s 1 -c-r h, A,
form as
-2(#T’/a)
0 0
Fg
001
002
1 Van&on n=O4, ---,
003
004
of bquld n=O6,----,
005
006
007
006
009
7 iilm thickness P = 0 01 -, n=lO.-x-x-x-, n=14
01
567
Shorter Commumcattons -Ts
Three typical graphs are shown plotted mdtcatmg the vanatton of non-dunenstonal tihn thuzkness, B agamst 1 (Fgs l-3) For low values of u (a = 0 001 and 0 01) as ?I” mcreases, E decreases for a fixed value of 7 For low values of (1. I e for acceleratmg con&tions of the film. the van&on m film tickness along the length follows a systematic pattern as obvious from Ftgs 1 and 2 However, for (I around unity, 1 e when vtscous and gravttattonal forces are of the same order of magnitude, the Itqtud films become decelerating and the thckness mcreases along the flow duection and the mfluence of “n” on B at a particular value of 7 ts rather complex at distances far away from the extt of the sltt (Ftg 3) In all the cases, I e for ail values of a, tt IS observed that as 7 + 01, B + 1 For Newtonian fluids, 1 e for n = 1 the values are tn perfect agreement with Ref [l] and further comparison for nonNewtoman flmds could not be undertaken due to the lack of data m the developing reeon of the hquld films Thus, m concluston, the present note can be consrdered as more general tn approach and Ref [l] reduces to a particular case of the present analysis
7 Ftg 2 Vacation of hqutd tilm tbckness (I = 0 001 n=O4 ,_____, ~=06,----,~=08,-x-x-x-,n=10,-,n=12.- - -,n=16
Department of MechanIcal Engrneenng Andhra Umverslty, Waltair 530013, In&a -,
E
0
87
1
I
u
I
A
I
ratlo of film thtckness (= @fi_ m eqn 6) acceleration due to gravity mass flow rate per umt width mtttal film thickness, Le at x = 0 flow behavtour tndex tn eqn (2) bqmd veloctty component along x-duection maximum veloctty tn the sltt, I e at x = 0 Cartesian co-ordinates Froude number (u&JghO SUI @) modified Reynolds number @uzh,“/p)
cg n
0 88
NARAYANA MURTHY P K SARMA
NOTATlON
_ _
ho I
V
kll,
x9 Y Fr Re
.
0 B
00 8261 0
001
002
1 003
004
005
006
007
006
009
-I 01
7
Fig 3 Vmatton of ltqutd film tluckness a = 1 1 -, -_-_ , n=O6, ----, n=lO, --x--x--x--, n=14, n=16
n = 0 4, - - - -
symbols dunenslonless number (FrlRe) dtmenslonless film thickness (S/h*) ltmtttng value of @(S,/hO) dlmenslonless distance (pu:zxlphO”+‘) function of x in eqn (3) local film thickness density of ltqutd tndex of conststency (and for n = 1, p becomes Newtoman vtscoslty) wall shear angle of mclmatton to the honon REFERENCE
[l] Stiicheh A and 6z1s& M N
, Chem Engng Scl 1976 31 369
566
for physlcaUy possible values of N, Q and l The absolute value of A, IS large1 by at least a factor of three than IA21when
the r&-hand side of (A4) vamshes and c, IS then sven by c, = B exp (AZ@,
,so
NHe 1+&+, wluch 1s. u-respective
19,
t(A, - A3el>
(JW
3
A plot of ln c, vs tune now IS a stra&t lme of slope AZ The value of N can be obtamed from the slope, smce
>
of the value of (NHe/.&
relative to the second term,
A,=
fulfilled If
-
(l:S)S03s
-+F)+ J[(l+&J+yy-*] 1‘f(l_Ns) 2
(A61
For values of, say, I(A, - AZ)01> 3 the value of the first term on
A note on hydrodynamic entrance lengths of non-Newtmian
lamiuar falliug liquid lilms
(Received 15 September 1976) The present note mveswates the hydrodynamic entrance lengths of non-Newtoman lammar fallmg hqmd films It IS observed that entrance lengths are dependant on modtied forms of (FrlRe) and the mdex “n” of the power law relation between the shear stress and shear deformatton rate Very recently Stucheh and &islk[l] obtamed a closed form of solution to evaluate the entrance lengths for the no drag condlhon at the vapor-bqmd mterface of a fallmg Newtoman hqmd film down an mclmed plane The analysis was basically developed on the assumption that the parabohc velocity profile at the outlet of the slit gets rapidly rearranged to a slrmlar velocity profile of a semr-parabola The present article 1s more general m approach to mclude both ddatant and pseudoplastic flulds in addition to the Newtonian flulds which are already presented by [l] ANALYSS The physlcai model 1s Identical to the one presented m Ref [l] The equation of motion m mtegral form for zero shear con&tion at the vapor-lrquld mterface can be wntten as pu2dy = - 7w + gp8 sm where the shear stress expression
at the wall IS aven
e by the power
where
the boundary
condttlon bemg at A = 0, fl= 1 Equation (4) degenerates to eqn (7) of Ref [l] when ?I” is made equal to umty Thus eqn (4) IS very general m nature wluch can be solved for different values of “n” Expbcit form of the solution IS possible only when “R” IS an integer Further, the hnutmg value of p can be obtamed by takmg d@dA = 0 Thus, (5) The dtierentml equation (4) 1s converted mto a dtierent follows whde the numerical integration 1s performed
where law
B = 818.. r) = (2&.‘/n)A Thus, eqn (6) LSsolved numencally q = 0, B = l/s.. on IBM 1130
The mdex “n” IS the charactenshc value whtch differs from umty for both ddatant and pseudoplastic fluids The velocity profile at any location 1s aven by the expression
20
16 6
where
i3*
u,,
=
= (r”/2”-‘)@‘_‘”
12
?7G/2Pho
Thus, eqn (1) can be mampulated with the help of eqns (2) and (3) to obtam the dlfferenhal equation 2
subject to the comhtion, VIZ at
,,2
(3)
s 1 -c-r h, A,
form as
-2(#T’/a)
0 0
Fg
001
002
1 Van&on n=O4, ---,
003
004
of bquld n=O6,----,
005
006
007
006
009
7 iilm thickness P = 0 01 -, n=lO.-x-x-x-, n=14
01
567
Shorter Commumcattons -Ts
Three typical graphs are shown plotted mdtcatmg the vanatton of non-dunenstonal tihn thuzkness, B agamst 1 (Fgs l-3) For low values of u (a = 0 001 and 0 01) as ?I” mcreases, E decreases for a fixed value of 7 For low values of (1. I e for acceleratmg con&tions of the film. the van&on m film tickness along the length follows a systematic pattern as obvious from Ftgs 1 and 2 However, for (I around unity, 1 e when vtscous and gravttattonal forces are of the same order of magnitude, the Itqtud films become decelerating and the thckness mcreases along the flow duection and the mfluence of “n” on B at a particular value of 7 ts rather complex at distances far away from the extt of the sltt (Ftg 3) In all the cases, I e for ail values of a, tt IS observed that as 7 + 01, B + 1 For Newtonian fluids, 1 e for n = 1 the values are tn perfect agreement with Ref [l] and further comparison for nonNewtoman flmds could not be undertaken due to the lack of data m the developing reeon of the hquld films Thus, m concluston, the present note can be consrdered as more general tn approach and Ref [l] reduces to a particular case of the present analysis
7 Ftg 2 Vacation of hqutd tilm tbckness (I = 0 001 n=O4 ,_____, ~=06,----,~=08,-x-x-x-,n=10,-,n=12.- - -,n=16
Department of MechanIcal Engrneenng Andhra Umverslty, Waltair 530013, In&a -,
E
0
87
1
I
u
I
A
I
ratlo of film thtckness (= @fi_ m eqn 6) acceleration due to gravity mass flow rate per umt width mtttal film thickness, Le at x = 0 flow behavtour tndex tn eqn (2) bqmd veloctty component along x-duection maximum veloctty tn the sltt, I e at x = 0 Cartesian co-ordinates Froude number (u&JghO SUI @) modified Reynolds number @uzh,“/p)
cg n
0 88
NARAYANA MURTHY P K SARMA
NOTATlON
_ _
ho I
V
kll,
x9 Y Fr Re
.
0 B
00 8261 0
001
002
1 003
004
005
006
007
006
009
-I 01
7
Fig 3 Vmatton of ltqutd film tluckness a = 1 1 -, -_-_ , n=O6, ----, n=lO, --x--x--x--, n=14, n=16
n = 0 4, - - - -
symbols dunenslonless number (FrlRe) dtmenslonless film thickness (S/h*) ltmtttng value of @(S,/hO) dlmenslonless distance (pu:zxlphO”+‘) function of x in eqn (3) local film thickness density of ltqutd tndex of conststency (and for n = 1, p becomes Newtoman vtscoslty) wall shear angle of mclmatton to the honon REFERENCE
[l] Stiicheh A and 6z1s& M N
, Chem Engng Scl 1976 31 369