- Email: [email protected]
stretching sheet
International Journal of Heat and Mass Transfer 93 (2016) 17–22
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass transfer induced slip effect on viscous gas flows above a shrinking/stretching sheet Lin Wu School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China
a r t i c l e
i n f o
Article history: Received 14 January 2015 Received in revised form 20 September 2015 Accepted 29 September 2015
Keywords: Viscous gas flows Moving boundary Mass transfer Slip flows
a b s t r a c t Viscous gas flows above linearly shrinking and stretching sheets with mass transfer are studied via theoretical analysis. Effect of mass-transfer induced slip at a moving surface on gas flows, which has not been considered in all previous studies, is systematically investigated. Net mass-transfer on a moving surface is demonstrated by a newly developed slip flow model (Wu, 2014) to introduce a gas slip velocity component in addition to the slip velocity component due to velocity shearing considered in the original 1st or 2nd order slip flow models. Our results show that mass suction induces a slip velocity against the sheet motion. The competition between flow driven effects of mass-suction induced slip and sheet motion significantly expands the available solution space, and adds to the solution space one more solution sub-region, which is absent when mass-suction induced slip is not considered. Mass-suction induced slip may even achieve a dominant flow driven role by totally reversing the flow direction of adjacent gases to flow against the sheet motion. Mass-injection induced slip enhances the flow driven effect of the moving sheet. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Viscous gas flows driven by a shrinking or stretching sheet with mass suction or injection exist in many industrial applications such as plastic sheet extrusion, paper production, fiber spinning, etc., to mention just a few. The related viscous gas flow problems have been extensively studied in the past four decades due to their industrial relevance [1–9]. Miklavcic and Wang [1] were the first to obtain an exact similarity solution for gas flows over a linearly shrinking sheet with mass suction, and their results suggest that dual solutions exist in a certain range of mass suction rate. The shrinking sheet driven flow problem was extended afterwards to include power-law shrinking velocity [2], and MHD flows [3,4], etc. Gas flows over a stretching sheet have been studied for both cases without [5] and with mass transfer [6]. One aspect distinguishing micro/nanoscale gas flows from their macroscale counterparts is that wall-slip effect usually becomes so vital at micro/nanoscales that its negligence may lead to predictions unacceptably deviated from physical reality. The Navier– Stokes equation, together with adequate slip velocity boundary conditions that properly incorporate gas molecule and wall interaction kinetics, has been demonstrated to provide accurate results for micro/nanoscale gas flows [10].
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.080 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
Maxwell’s 1st order slip flow model [11] was employed by Wang [7] in the study of stretching sheet problem. Fang et al. [8,9] considered wall-slip effect on shrinking/stretching sheet driven flows by adopting a 2nd order slip flow model developed by Wu [10], and found that the flow solutions highly depend on the 1st and 2nd order slip coefficients. We very recently derived a slip velocity boundary condition for rarefied gas flows above a moving surface with net mass transfer from kinetic theory [12]. The wall-slip velocity can be obtained by equating the total tangential momentum transfer rate of gas molecules at the wall to the viscous wall shear stress [10–13]. When mass is sucked from or injected into the flow domain through a moving wall, it is usually unavoidable for the gas molecules leaving/entering the gas flow domain to have a non-zero average tangential velocity. Consequently, mass transfer contributes to the total tangential momentum transfer rate of gas molecules at the moving wall, and in turn introduces an additional gas slip velocity component linearly proportional to the mass transfer rate and the average tangential velocity of gas molecules leaving/entering the gas flow domain through the moving wall [12]. The mass transfer induced slip velocity component is very different from the previously studied gas slip velocity component due to velocity shearing [10,11,13], and may have a magnitude comparable to or even larger than the latter.
18
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
Mass-transfer induced slip effect has not been considered in all previous studies of viscous flows over shrinking/stretching sheets [1–9]. In this paper we theoretically study viscous gas flows over both shrinking and stretching sheets with mass transfer by fully considering the effect of aforementioned mass transfer induced slip. Exact similarity solutions are obtained for both shrinking and stretching sheet driven flows. Our results show that masstransfer induced slip has a non-negligible effect and thus has to be considered in gas flows driven by a moving sheet with mass transfer.
The viscous gas flows under consideration are steady state 2-D flows driven by an underneath sheet moving horizontally at a speed uw (Fig. 1). Similar gases are sucked from or injected into the gas flow domain through the moving surface at a wallnormal speed mw . The continuity and Navier–Stokes equations governing the flows are
@u @ v þ ¼ 0; @x @y
u
ð1Þ ! @2u @2u ; þ @x2 @y2
! @v @v 1 @p @2v @2v ; þm þ þv ¼ @x @y q @y @x2 @y2
2
ð2Þ
ð3Þ
where u is the velocity in x direction, v is the velocity in y direction,
ity, respectively. The corresponding boundary conditions are
uðx; 0Þ ¼ uw þ uslip ;
ð4Þ
v ðx; 0Þ ¼ v w ;
ð5Þ
uðx; þ1Þ ¼ 0;
ð6Þ
where uw ¼ ax is the shrinking (negative sign) or stretching (positive sign) speed of the sheet with a being the magnitude of the shrinking or stretching rate. The gas slip velocity at the moving surface is [12]
_ @u @ 2 u 4M ¼ bs k u ; cs k2 2 þ @y @y aqv w
ð7Þ
ð9Þ
are the first and second order slip coefficients due to velocity shearing, k is the mean free path of gas molecules, K n is the Knudsen number, f ¼ min½1=K n ; 1 is a dimensionless function, and v is the mean molecular speed, respectively. The accommodation coefficient a has a value in the range from 0 to 1 depending on the surface _ ¼ qv w , consequently the slip property. The mass flux at wall is M velocity at the moving surface can be written as
@u @2u cs k2 2 m1 ax; @y @y
ð10Þ
where
m1 ¼ 4v w =ðav Þ;
ð11Þ
is a dimensionless mass-transfer induced slip parameter. The third term on the right-hand-side (RHS) of Eq. (10) has a negative sign for a shrinking sheet and a positive sign for a stretching sheet. When mass transfer is due to evaporation or condensation, m1 can be further simplified to
m1 ¼
q is the density, p is the pressure, and m is the gas kinematic viscos-
uslip
4
cs ¼ f =4 þ ð1 f Þ=ð2K 2n Þ;
uslip ¼ bs k
2. Theoretical models
@u @u 1 @p þv ¼ þm u @x @y q @x
and
pev p ; ap
ð12Þ
where pev is the equilibrium vapor pressure of the evaporating/condensing gases [12]. The mass transfer induced slip parameter m1 can be large in a wide range of practical applications. For example when the mass flux is large, or when the surface accommodation coefficient a is small (ranging from 0 to 1). One practical example for m1 to be large is for the case when the mass transfer is due to evaporation/condensation and the magnitude of the difference between the equilibrium vapor pressure pev and the vapor pressure p is large compared with the vapor pressure as shown by Eq. (12). In a slightly different geometry setup, i.e., a gas bearing system, we already demonstrated that evaporation/condensation induced slip may have a significant impact on gas flows [12]. We now introduce the stream function and similarity variable
wðx; yÞ ¼ f ðgÞx
pffiffiffiffiffiffi m a;
ð13Þ
rffiffiffi a g¼y ;
ð14Þ
m
so that 0
uðx; yÞ ¼ axf ðgÞ;
ð15Þ
pffiffiffiffiffiffi
v ðx; yÞ ¼ m af ðgÞ:
where 3
2
bs ¼ 2ð3 a f Þ=ð3aÞ ð1 f Þ=K n ;
ð8Þ
ð16Þ
With the introduction of the stream function the continuity equation (1) is automatically satisfied, and the momentum equations (2) and (3) transform into 000
00
f þ ff f
02
¼ 0;
ð17Þ
subject to the following boundary conditions
vw
f ð0Þ ¼ pffiffiffiffiffiffi ¼ m2 ; ma 0
ð18Þ 00
000
f ð0Þ ¼ ð1 þ m1 Þ þ bf ð0Þ þ c f ð0Þ; 0
f ðþ1Þ ¼ 0;
ð19Þ ð20Þ
where m2 is the mass transfer parameter, and
rffiffiffi a ;
b ¼ bs k Fig. 1. System setup for viscous gas flow above a shrinking/stretching sheet with mass suction/injection.
m
a
c ¼ cs k2 ; m
ð21Þ ð22Þ
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
are the first and second order slip parameters, respectively. Positive sign of the first term on the RHS of Eq. (19) corresponds to a stretching sheet, while negative sign corresponds to a shrinking sheet. The two parameters, m1 and m2 , always have signs opposite to each other. Mass suction/injection yields a positive/negative m2 . The first order slip parameter b is always positive and the second order slip parameter c is always negative according to Eqs. (21) and (22). The solution to Eq. (17) subject to boundary conditions (18)– (20) is
f ðgÞ ¼ d þ ðm2 dÞ ExpðdgÞ;
ð23Þ
which allows for a stable steady state solution when d P 0, while d is the solution of a fourth order algebra equation
d4 þ bd3 þ cd2 þ dd þ e ¼ 0;
d¼
b m2 1
c m2
c
;
ð25Þ
;
ð26Þ
;
ð27Þ
1 þ m1
c
;
ð28Þ
are four dimensionless coefficients. The RHS of (28) has a positive sign for a stretching sheet, and has a negative sign for a shrinking sheet. The solutions d to the fourth order Eq. (24) are equivalent to the solutions of the following two second order algebra equations
d2 þ
b þ c m2
c
e¼
ð24Þ
where
b¼
c¼
19
ðb
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 8h þ b 4cÞ bh d d þ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; 2 2 8h þ b 4c
ð29Þ
where h is the real solution of the following third order algebra equation
h3 þ c1 h2 þ c2 h þ c3 ¼ 0;
ð30Þ
with
c c1 ¼ ; 2
ð31Þ
c2 ¼
bd e; 4
c3 ¼
eð4c b Þ d : 8 8
ð32Þ 2
2
ð33Þ
From Eq. (23) we can readily calculate the dimensionless horizontal velocity function 0
f ðgÞ ¼ ðd2 dm2 ÞExpðdgÞ;
Fig. 2. Gas flows above a shrinking sheet with mass suction: (a) solution d as a function of mass transfer parameter m2 , (b) d as a function of mass transfer induced 00 slip parameter m1 , and (c) wall shear stress f ð0Þ as a function of m1 . Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions. The first and second order slip parameters are fixed at b ¼ 0:1 and c ¼ 0:1.
ð34Þ
Fig. 3. Velocity distribution for flows above a shrinking sheet with mass suction: (a) 0 vertical velocity f ðgÞ, and (b) horizontal velocity f ðgÞ. Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions. Other parameters are fixed at b ¼ 0:1, c ¼ 0:1, and m2 ¼ 2.
20
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
and the dimensionless shear stress function 00
f ðgÞ ¼ ðd3 d2 m2 ÞExpðdgÞ:
ð35Þ
3. Results and discussion Our formulations are validated by reproducing the results of Fang et al. [8,9] for both the shrinking [8] and stretching [9] sheet driven flows when the mass-transfer induced slip effect is turned off by setting m1 ¼ 0 in our formulations. To illustrate how mass-transfer induced slip affects moving sheet driven flows, we first consider the gas flows above a shrinking sheet with mass suction. The first and second order slip parameters are assumed to be fixed at b ¼ 0:1 and c ¼ 0:1 in plotting Fig. 2. The mass-suction induced slip parameter m1 is negative due to the negative wall-normal velocity of gases v w (see Eq. (11)), and the decreasing of m1 represents a strengthening of mass-suction induced slip effect. Fig. 2(a) plots the solution d as a function of the mass transfer parameter m2 at different values
Fig. 4. Gas flows above a stretching sheet with mass suction: (a) the solution d as a function of the mass transfer parameter m2 , (b) d as a function of the mass transfer 00 induced slip parameter m1 , and (c) the wall shear stress f ð0Þ as a function of m1 . The first and second order slip parameters are fixed at b ¼ 0:1, and c ¼ 0:1. Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions.
of m1 . The whole solution space is divided into two sub-regions by one particular solution d ¼ m2 to Eq. (24) when m1 ¼ 1. The first sub-region is bounded by two straight lines d ¼ m2 and d ¼ 0. The line d ¼ 0 is another solution to Eq. (24) when m1 ¼ 1. Within the first sub-region, d has two solution branches. The existence of dual solutions for shrinking sheet driven flows with mass-suction has been reported in previous studies without considering the mass-suction induced slip effect (m1 ¼ 0) [1,8]. The upper branch of d increases while the lower branch decreases with the increase of m2 and the decrease of m1 . At each particular value of m1 , m2 has a lower limit, below which there exists no stable solution, and above which the continuous solution curve bends into an upper branch and a lower branch. Fig. 2(a) shows that the lower limit value of m2 decreases towards zero as m1 approaches 1. In this sense the available dual solution space has been significantly expanded by the mass-suction induced slip. Fig. 2(a) shows that a second sub-region with a unique solution, which is above the line d ¼ m2 , appears when m1 < 1. The unique solution d increases with the decrease of m1 and the increase of m2 . The unique solution sub-region has not been identified in all previous studies on shrinking sheet driven flows with mass-suction [1– 4,8]. Fig. 2(b) plots the solution d as a function of m1 at different values of m2 . Fig. 2(b) shows that dual solutions exist only when m1 is in the range from 1 to an upper limit for each particular value of m2 . No solution exists when m1 is increased beyond the upper limit. The upper limit value of m1 increases with the increase of m2 until the upper limit value of m1 reaches zero. Further increase of m2 leads to the break-up of the upper and lower branch 00 solution curves at m1 ¼ 0. Fig. 2(c) plots the wall shear stress f ð0Þ as a function of m1 at different values of m2 . Within the dual solu00 tion region with m1 > 1, f ð0Þ is positive and the two branches of 00 f ð0Þ gradually decrease to zero when m1 reduces towards 1. 00 Fig. 2(c) shows that f ð0Þ as a function of m1 may form a closed curve in the dual solution region when m2 is small enough. The closed curve opens up at m1 ¼ 0 when m2 becomes large enough.
Fig. 5. Velocity distribution for flows above a stretching sheet with mass suction: 0 (a) vertical velocity f ðgÞ, and (b) horizontal velocity f ðgÞ. Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions. Other parameters are fixed at b ¼ 0:1, c ¼ 0:1, and m2 ¼ 2.
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
In the unique solution region with m1 < 1, a single solution tail of 00 00 f ð0Þ of negative value is formed, and the magnitude of f ð0Þ increases with the decrease of m1 . Fig. 2(c) indicates that the wall shear stress has a direction opposite to the sheet motion when m1 > 1, and has a direction similar to the sheet motion when m1 < 1. The reason is that the mass-suction induced slip tends to drive flows in a direction opposite to the sheet motion. When m1 > 1, mass-suction induced slip is not large enough to overcome the flow driven effect of the sheet motion. The masssuction induced slip only retards the flow by reducing the flow speed and the magnitude of wall shear stress. As a result, the directions of flow and wall shear stress are determined by the sheet motion driven flow. When m1 < 1, mass-suction induced slip dominates the flow driven effect of sheet motion, and consequently both the flow and wall shear stress reverse their directions. 0 Fig. 3 plots the velocity distribution functions f ðgÞ and f ðgÞ at different values of m1 when m2 is fixed at 2. At m1 ¼ 1, the mass-suction induced slip velocity exactly cancels the sheet mov-
Fig. 6. Gas flows above a stretching sheet with mass injection: (a) the solution d as a function of the mass transfer parameter m2 , (b) d as a function of the mass transfer 00 induced slip parameter m1 , and (c) the wall shear stress function f ð0Þ as a function of m1 . The first and second order slip parameters are fixed at b ¼ 0:1 and c ¼ 0:1.
21
ing speed. As a result, the flow becomes vertical and has a uniform velocity equal to the wall normal velocity v w . When m1 > 1, the boundary layer thickness of the upper branch solution is thinner than the lower branch. The boundary layer thickness of the upper branch decreases with the decrease of m1 , while the lower branch has exactly the opposite evolving trend. When m1 < 1, the boundary layer thickness of the unique solution decreases with the decrease of m1 . Our results indicate that there is no physically stable solution when the flow is driven by a shrinking sheet with mass injection. In fact the mass-injection induced slip further destabilizes the flow. For gas flows driven by a stretching sheet with mass suction, previous studies without considering mass-suction induced slip only identified a unique solution region [5,6,9]. Our results plotted in Fig. 4(a) and (b) show that mass-suction induced slip adds one more dual solution region to the solution space. The solution space is divided into a unique solution region and a dual solution region by one particular solution d ¼ m2 to Eq. (24) when m1 ¼ 1. The unique solution region is above the line d ¼ m2 . The dual solution region particularly due to mass-suction induced slip is bounded by the lines of d ¼ m2 and d ¼ 0. The line d ¼ 0 is another solution to Eq. (24) when m1 ¼ 1. The first and second order slip parameters are assumed to be fixed at b ¼ 0:1 and c ¼ 0:1 in plotting Fig. 4. In the unique solution region with m1 > 1, d and the magnitude of wall shear stress increase with the increase of m1 as shown by Fig. 4(a)–(c). In the dual solution region with m1 < 1, the upper branch of d increases with the increase of m1 and m2 , while the lower branch has exactly the opposite evolving trend. The magnitude of wall shear stress increases with the decrease of m1 (see Fig. 4(c)). Fig. 4(a) shows that dual solutions exist only when m2 is larger than a lower limit value, which decreases towards zero as m1 approaches 1. At each particular value of m2 , there is also a lower limit value for m1 , which allows for the existence of dual solutions as shown by Fig. 4(b), and the lower
Fig. 7. Velocity distribution for flows above a stretching sheet with mass injection: 0 (a) vertical velocity f ðgÞ, and (b) horizontal velocity f ðgÞ. Other parameters are fixed at b ¼ 0:1, c ¼ 0:1, and m2 ¼ 2.
22
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
limit value of m1 decreases with the increase of m2 . The boundary layer thickness increases with the decrease of m1 in the unique solution region as shown by Fig. 5. In the dual solution region, the upper branch solution has a thinner boundary thickness than the lower branch. The boundary layer thickness of the upper branch solution increases with the decrease of m1 , while the lower branch has exactly the opposite evolving trend. Only unique solution exists for stretching sheet driven flows with mass injection as shown by Fig. 6(a) and (b). The solution d increases with the increase of m1 and m2 . The magnitude of wall shear stress increases with the increase of m1 and m2 (see Fig. 6 (c)). The boundary layer thickness decreases with the increase of m1 as shown by Fig. 7. Fig. 7(a) shows an interesting phenomenon, i.e., the gases have a positive vertical velocity in regions adjacent to the sheet due to the mass blowing, but the vertical velocity gradually becomes negative far away enough from the sheet.
tion for flows driven by a shrinking sheet. Mass-injection induced slip forces flows driven by a stretching sheet to flow faster. In summary, our results demonstrate that mass-transfer induced slip has a non-negligible impact on flows driven by a moving sheet. Mass-transfer induced slip may even be able to dominate the flow driven effect of the moving sheet by changing the flow direction to flow against the sheet motion. Conflict of interest None declared. Acknowledgments This work is partially supported by the program for professor of special appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.
4. Conclusions References In this paper we systematically studied mass-transfer induced slip effect on viscous gas flows driven by linearly shrinking/ stretching sheets with mass suction/injection. Mass-transfer induced slip has not been considered in all previous studies on moving sheet driven flows. Mass-suction induced slip at a moving surface tends to drive gas flows in a direction opposite to the surface motion, while mass-injection induced slip drives flows in a direction similar to the surface motion. Our results show that mass-transfer induced slip effect significantly expands the available solution space for the shrinking/stretching sheet driven flows. Additional solution sub-regions for flows with mass-suction, which are absent in all previous studies, are discovered by the incorporation of mass-suction induced slip effect. For flows with mass-suction, the competition between flow driven effects of mass-suction induced slip and sheet motion leads to much enriched nonlinear dynamics. Depending on the relative strength of mass-suction induced slip, one more unique solution region may appear in addition to the previously studied dual solution region of flows driven by a shrinking sheet, and a dual solution region may appear in addition to the previously studied unique solution region for flows driven by a stretching sheet. Mass-injection induced slip simply enhances the flow driven effect of sheet motion, and has an effect equivalent to increasing the sheet moving speed. As a result, mass-injection induced slip makes flows more unstable, and thus yields no steady state solu-
[1] M. Miklavcic, C.Y. Wang, Viscous flow due to a shrinking sheet, Q. Appl. Math. 64 (2006) 283–290. [2] T. Fang, Boundary layer flow over a shrinking sheet with power-law velocity, Int. J. Heat Mass Transfer 51 (2008) 5838–5843. [3] T. Hayat, Z. Abbas, M. Sajid, On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet, J. Appl. Mech. Trans. ASME 74 (2007) 1165–1171. [4] M. Sajid, T. Hayat, T. Javed, MHD rotating flow of a viscous fluid over a shrinking surface, Nonlinear Dyn. 51 (2008) 259–265. [5] L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645– 647. [6] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744–746. [7] C.Y. Wang, Analysis of viscous flow due to a stretching sheet with surface slip and suction, Nonlinear Anal. Real World Appl. 10 (2009) 375–380. [8] T. Fang, S. Yao, J. Zhang, A. Aziz, Viscous flow over a shrinking sheet with a second order slip flow model, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1831–1842. [9] T. Fang, A. Aziz, Viscous flow with second-order slip velocity over a stretching sheet, Z. Naturforsch. 65 (2010) 1087–1092. [10] L. Wu, A slip model for rarefied gas flows at arbitrary Knudsen number, Appl. Phys. Lett. 93 (2008) 253103. [11] A. Burgdorfer, The influence of the molecular mean free path on the performance of hydrodynamic gas lubricated bearings, J. Basic Eng. ASME 81 (1959) 94–99. [12] L. Wu, A slip model for rarefied gas flows above a moving surface with mass transfer, J. Appl. Phys. 116 (2014) 054503. [13] Y.T. Hsia, G.A. Domoto, An experimental investigation of molecular rarefaction effects in gas lubricated bearings at ultra-low clearances, J. Lubr. Technol. ASME 105 (1983) 120–130.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass transfer induced slip effect on viscous gas flows above a shrinking/stretching sheet Lin Wu School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China
a r t i c l e
i n f o
Article history: Received 14 January 2015 Received in revised form 20 September 2015 Accepted 29 September 2015
Keywords: Viscous gas flows Moving boundary Mass transfer Slip flows
a b s t r a c t Viscous gas flows above linearly shrinking and stretching sheets with mass transfer are studied via theoretical analysis. Effect of mass-transfer induced slip at a moving surface on gas flows, which has not been considered in all previous studies, is systematically investigated. Net mass-transfer on a moving surface is demonstrated by a newly developed slip flow model (Wu, 2014) to introduce a gas slip velocity component in addition to the slip velocity component due to velocity shearing considered in the original 1st or 2nd order slip flow models. Our results show that mass suction induces a slip velocity against the sheet motion. The competition between flow driven effects of mass-suction induced slip and sheet motion significantly expands the available solution space, and adds to the solution space one more solution sub-region, which is absent when mass-suction induced slip is not considered. Mass-suction induced slip may even achieve a dominant flow driven role by totally reversing the flow direction of adjacent gases to flow against the sheet motion. Mass-injection induced slip enhances the flow driven effect of the moving sheet. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Viscous gas flows driven by a shrinking or stretching sheet with mass suction or injection exist in many industrial applications such as plastic sheet extrusion, paper production, fiber spinning, etc., to mention just a few. The related viscous gas flow problems have been extensively studied in the past four decades due to their industrial relevance [1–9]. Miklavcic and Wang [1] were the first to obtain an exact similarity solution for gas flows over a linearly shrinking sheet with mass suction, and their results suggest that dual solutions exist in a certain range of mass suction rate. The shrinking sheet driven flow problem was extended afterwards to include power-law shrinking velocity [2], and MHD flows [3,4], etc. Gas flows over a stretching sheet have been studied for both cases without [5] and with mass transfer [6]. One aspect distinguishing micro/nanoscale gas flows from their macroscale counterparts is that wall-slip effect usually becomes so vital at micro/nanoscales that its negligence may lead to predictions unacceptably deviated from physical reality. The Navier– Stokes equation, together with adequate slip velocity boundary conditions that properly incorporate gas molecule and wall interaction kinetics, has been demonstrated to provide accurate results for micro/nanoscale gas flows [10].
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.080 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
Maxwell’s 1st order slip flow model [11] was employed by Wang [7] in the study of stretching sheet problem. Fang et al. [8,9] considered wall-slip effect on shrinking/stretching sheet driven flows by adopting a 2nd order slip flow model developed by Wu [10], and found that the flow solutions highly depend on the 1st and 2nd order slip coefficients. We very recently derived a slip velocity boundary condition for rarefied gas flows above a moving surface with net mass transfer from kinetic theory [12]. The wall-slip velocity can be obtained by equating the total tangential momentum transfer rate of gas molecules at the wall to the viscous wall shear stress [10–13]. When mass is sucked from or injected into the flow domain through a moving wall, it is usually unavoidable for the gas molecules leaving/entering the gas flow domain to have a non-zero average tangential velocity. Consequently, mass transfer contributes to the total tangential momentum transfer rate of gas molecules at the moving wall, and in turn introduces an additional gas slip velocity component linearly proportional to the mass transfer rate and the average tangential velocity of gas molecules leaving/entering the gas flow domain through the moving wall [12]. The mass transfer induced slip velocity component is very different from the previously studied gas slip velocity component due to velocity shearing [10,11,13], and may have a magnitude comparable to or even larger than the latter.
18
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
Mass-transfer induced slip effect has not been considered in all previous studies of viscous flows over shrinking/stretching sheets [1–9]. In this paper we theoretically study viscous gas flows over both shrinking and stretching sheets with mass transfer by fully considering the effect of aforementioned mass transfer induced slip. Exact similarity solutions are obtained for both shrinking and stretching sheet driven flows. Our results show that masstransfer induced slip has a non-negligible effect and thus has to be considered in gas flows driven by a moving sheet with mass transfer.
The viscous gas flows under consideration are steady state 2-D flows driven by an underneath sheet moving horizontally at a speed uw (Fig. 1). Similar gases are sucked from or injected into the gas flow domain through the moving surface at a wallnormal speed mw . The continuity and Navier–Stokes equations governing the flows are
@u @ v þ ¼ 0; @x @y
u
ð1Þ ! @2u @2u ; þ @x2 @y2
! @v @v 1 @p @2v @2v ; þm þ þv ¼ @x @y q @y @x2 @y2
2
ð2Þ
ð3Þ
where u is the velocity in x direction, v is the velocity in y direction,
ity, respectively. The corresponding boundary conditions are
uðx; 0Þ ¼ uw þ uslip ;
ð4Þ
v ðx; 0Þ ¼ v w ;
ð5Þ
uðx; þ1Þ ¼ 0;
ð6Þ
where uw ¼ ax is the shrinking (negative sign) or stretching (positive sign) speed of the sheet with a being the magnitude of the shrinking or stretching rate. The gas slip velocity at the moving surface is [12]
_ @u @ 2 u 4M ¼ bs k u ; cs k2 2 þ @y @y aqv w
ð7Þ
ð9Þ
are the first and second order slip coefficients due to velocity shearing, k is the mean free path of gas molecules, K n is the Knudsen number, f ¼ min½1=K n ; 1 is a dimensionless function, and v is the mean molecular speed, respectively. The accommodation coefficient a has a value in the range from 0 to 1 depending on the surface _ ¼ qv w , consequently the slip property. The mass flux at wall is M velocity at the moving surface can be written as
@u @2u cs k2 2 m1 ax; @y @y
ð10Þ
where
m1 ¼ 4v w =ðav Þ;
ð11Þ
is a dimensionless mass-transfer induced slip parameter. The third term on the right-hand-side (RHS) of Eq. (10) has a negative sign for a shrinking sheet and a positive sign for a stretching sheet. When mass transfer is due to evaporation or condensation, m1 can be further simplified to
m1 ¼
q is the density, p is the pressure, and m is the gas kinematic viscos-
uslip
4
cs ¼ f =4 þ ð1 f Þ=ð2K 2n Þ;
uslip ¼ bs k
2. Theoretical models
@u @u 1 @p þv ¼ þm u @x @y q @x
and
pev p ; ap
ð12Þ
where pev is the equilibrium vapor pressure of the evaporating/condensing gases [12]. The mass transfer induced slip parameter m1 can be large in a wide range of practical applications. For example when the mass flux is large, or when the surface accommodation coefficient a is small (ranging from 0 to 1). One practical example for m1 to be large is for the case when the mass transfer is due to evaporation/condensation and the magnitude of the difference between the equilibrium vapor pressure pev and the vapor pressure p is large compared with the vapor pressure as shown by Eq. (12). In a slightly different geometry setup, i.e., a gas bearing system, we already demonstrated that evaporation/condensation induced slip may have a significant impact on gas flows [12]. We now introduce the stream function and similarity variable
wðx; yÞ ¼ f ðgÞx
pffiffiffiffiffiffi m a;
ð13Þ
rffiffiffi a g¼y ;
ð14Þ
m
so that 0
uðx; yÞ ¼ axf ðgÞ;
ð15Þ
pffiffiffiffiffiffi
v ðx; yÞ ¼ m af ðgÞ:
where 3
2
bs ¼ 2ð3 a f Þ=ð3aÞ ð1 f Þ=K n ;
ð8Þ
ð16Þ
With the introduction of the stream function the continuity equation (1) is automatically satisfied, and the momentum equations (2) and (3) transform into 000
00
f þ ff f
02
¼ 0;
ð17Þ
subject to the following boundary conditions
vw
f ð0Þ ¼ pffiffiffiffiffiffi ¼ m2 ; ma 0
ð18Þ 00
000
f ð0Þ ¼ ð1 þ m1 Þ þ bf ð0Þ þ c f ð0Þ; 0
f ðþ1Þ ¼ 0;
ð19Þ ð20Þ
where m2 is the mass transfer parameter, and
rffiffiffi a ;
b ¼ bs k Fig. 1. System setup for viscous gas flow above a shrinking/stretching sheet with mass suction/injection.
m
a
c ¼ cs k2 ; m
ð21Þ ð22Þ
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
are the first and second order slip parameters, respectively. Positive sign of the first term on the RHS of Eq. (19) corresponds to a stretching sheet, while negative sign corresponds to a shrinking sheet. The two parameters, m1 and m2 , always have signs opposite to each other. Mass suction/injection yields a positive/negative m2 . The first order slip parameter b is always positive and the second order slip parameter c is always negative according to Eqs. (21) and (22). The solution to Eq. (17) subject to boundary conditions (18)– (20) is
f ðgÞ ¼ d þ ðm2 dÞ ExpðdgÞ;
ð23Þ
which allows for a stable steady state solution when d P 0, while d is the solution of a fourth order algebra equation
d4 þ bd3 þ cd2 þ dd þ e ¼ 0;
d¼
b m2 1
c m2
c
;
ð25Þ
;
ð26Þ
;
ð27Þ
1 þ m1
c
;
ð28Þ
are four dimensionless coefficients. The RHS of (28) has a positive sign for a stretching sheet, and has a negative sign for a shrinking sheet. The solutions d to the fourth order Eq. (24) are equivalent to the solutions of the following two second order algebra equations
d2 þ
b þ c m2
c
e¼
ð24Þ
where
b¼
c¼
19
ðb
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 8h þ b 4cÞ bh d d þ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0; 2 2 8h þ b 4c
ð29Þ
where h is the real solution of the following third order algebra equation
h3 þ c1 h2 þ c2 h þ c3 ¼ 0;
ð30Þ
with
c c1 ¼ ; 2
ð31Þ
c2 ¼
bd e; 4
c3 ¼
eð4c b Þ d : 8 8
ð32Þ 2
2
ð33Þ
From Eq. (23) we can readily calculate the dimensionless horizontal velocity function 0
f ðgÞ ¼ ðd2 dm2 ÞExpðdgÞ;
Fig. 2. Gas flows above a shrinking sheet with mass suction: (a) solution d as a function of mass transfer parameter m2 , (b) d as a function of mass transfer induced 00 slip parameter m1 , and (c) wall shear stress f ð0Þ as a function of m1 . Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions. The first and second order slip parameters are fixed at b ¼ 0:1 and c ¼ 0:1.
ð34Þ
Fig. 3. Velocity distribution for flows above a shrinking sheet with mass suction: (a) 0 vertical velocity f ðgÞ, and (b) horizontal velocity f ðgÞ. Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions. Other parameters are fixed at b ¼ 0:1, c ¼ 0:1, and m2 ¼ 2.
20
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
and the dimensionless shear stress function 00
f ðgÞ ¼ ðd3 d2 m2 ÞExpðdgÞ:
ð35Þ
3. Results and discussion Our formulations are validated by reproducing the results of Fang et al. [8,9] for both the shrinking [8] and stretching [9] sheet driven flows when the mass-transfer induced slip effect is turned off by setting m1 ¼ 0 in our formulations. To illustrate how mass-transfer induced slip affects moving sheet driven flows, we first consider the gas flows above a shrinking sheet with mass suction. The first and second order slip parameters are assumed to be fixed at b ¼ 0:1 and c ¼ 0:1 in plotting Fig. 2. The mass-suction induced slip parameter m1 is negative due to the negative wall-normal velocity of gases v w (see Eq. (11)), and the decreasing of m1 represents a strengthening of mass-suction induced slip effect. Fig. 2(a) plots the solution d as a function of the mass transfer parameter m2 at different values
Fig. 4. Gas flows above a stretching sheet with mass suction: (a) the solution d as a function of the mass transfer parameter m2 , (b) d as a function of the mass transfer 00 induced slip parameter m1 , and (c) the wall shear stress f ð0Þ as a function of m1 . The first and second order slip parameters are fixed at b ¼ 0:1, and c ¼ 0:1. Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions.
of m1 . The whole solution space is divided into two sub-regions by one particular solution d ¼ m2 to Eq. (24) when m1 ¼ 1. The first sub-region is bounded by two straight lines d ¼ m2 and d ¼ 0. The line d ¼ 0 is another solution to Eq. (24) when m1 ¼ 1. Within the first sub-region, d has two solution branches. The existence of dual solutions for shrinking sheet driven flows with mass-suction has been reported in previous studies without considering the mass-suction induced slip effect (m1 ¼ 0) [1,8]. The upper branch of d increases while the lower branch decreases with the increase of m2 and the decrease of m1 . At each particular value of m1 , m2 has a lower limit, below which there exists no stable solution, and above which the continuous solution curve bends into an upper branch and a lower branch. Fig. 2(a) shows that the lower limit value of m2 decreases towards zero as m1 approaches 1. In this sense the available dual solution space has been significantly expanded by the mass-suction induced slip. Fig. 2(a) shows that a second sub-region with a unique solution, which is above the line d ¼ m2 , appears when m1 < 1. The unique solution d increases with the decrease of m1 and the increase of m2 . The unique solution sub-region has not been identified in all previous studies on shrinking sheet driven flows with mass-suction [1– 4,8]. Fig. 2(b) plots the solution d as a function of m1 at different values of m2 . Fig. 2(b) shows that dual solutions exist only when m1 is in the range from 1 to an upper limit for each particular value of m2 . No solution exists when m1 is increased beyond the upper limit. The upper limit value of m1 increases with the increase of m2 until the upper limit value of m1 reaches zero. Further increase of m2 leads to the break-up of the upper and lower branch 00 solution curves at m1 ¼ 0. Fig. 2(c) plots the wall shear stress f ð0Þ as a function of m1 at different values of m2 . Within the dual solu00 tion region with m1 > 1, f ð0Þ is positive and the two branches of 00 f ð0Þ gradually decrease to zero when m1 reduces towards 1. 00 Fig. 2(c) shows that f ð0Þ as a function of m1 may form a closed curve in the dual solution region when m2 is small enough. The closed curve opens up at m1 ¼ 0 when m2 becomes large enough.
Fig. 5. Velocity distribution for flows above a stretching sheet with mass suction: 0 (a) vertical velocity f ðgÞ, and (b) horizontal velocity f ðgÞ. Solid curves in the dual solution region correspond to the upper branch solutions, while the dashed curves correspond to the lower branch solutions. Other parameters are fixed at b ¼ 0:1, c ¼ 0:1, and m2 ¼ 2.
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
In the unique solution region with m1 < 1, a single solution tail of 00 00 f ð0Þ of negative value is formed, and the magnitude of f ð0Þ increases with the decrease of m1 . Fig. 2(c) indicates that the wall shear stress has a direction opposite to the sheet motion when m1 > 1, and has a direction similar to the sheet motion when m1 < 1. The reason is that the mass-suction induced slip tends to drive flows in a direction opposite to the sheet motion. When m1 > 1, mass-suction induced slip is not large enough to overcome the flow driven effect of the sheet motion. The masssuction induced slip only retards the flow by reducing the flow speed and the magnitude of wall shear stress. As a result, the directions of flow and wall shear stress are determined by the sheet motion driven flow. When m1 < 1, mass-suction induced slip dominates the flow driven effect of sheet motion, and consequently both the flow and wall shear stress reverse their directions. 0 Fig. 3 plots the velocity distribution functions f ðgÞ and f ðgÞ at different values of m1 when m2 is fixed at 2. At m1 ¼ 1, the mass-suction induced slip velocity exactly cancels the sheet mov-
Fig. 6. Gas flows above a stretching sheet with mass injection: (a) the solution d as a function of the mass transfer parameter m2 , (b) d as a function of the mass transfer 00 induced slip parameter m1 , and (c) the wall shear stress function f ð0Þ as a function of m1 . The first and second order slip parameters are fixed at b ¼ 0:1 and c ¼ 0:1.
21
ing speed. As a result, the flow becomes vertical and has a uniform velocity equal to the wall normal velocity v w . When m1 > 1, the boundary layer thickness of the upper branch solution is thinner than the lower branch. The boundary layer thickness of the upper branch decreases with the decrease of m1 , while the lower branch has exactly the opposite evolving trend. When m1 < 1, the boundary layer thickness of the unique solution decreases with the decrease of m1 . Our results indicate that there is no physically stable solution when the flow is driven by a shrinking sheet with mass injection. In fact the mass-injection induced slip further destabilizes the flow. For gas flows driven by a stretching sheet with mass suction, previous studies without considering mass-suction induced slip only identified a unique solution region [5,6,9]. Our results plotted in Fig. 4(a) and (b) show that mass-suction induced slip adds one more dual solution region to the solution space. The solution space is divided into a unique solution region and a dual solution region by one particular solution d ¼ m2 to Eq. (24) when m1 ¼ 1. The unique solution region is above the line d ¼ m2 . The dual solution region particularly due to mass-suction induced slip is bounded by the lines of d ¼ m2 and d ¼ 0. The line d ¼ 0 is another solution to Eq. (24) when m1 ¼ 1. The first and second order slip parameters are assumed to be fixed at b ¼ 0:1 and c ¼ 0:1 in plotting Fig. 4. In the unique solution region with m1 > 1, d and the magnitude of wall shear stress increase with the increase of m1 as shown by Fig. 4(a)–(c). In the dual solution region with m1 < 1, the upper branch of d increases with the increase of m1 and m2 , while the lower branch has exactly the opposite evolving trend. The magnitude of wall shear stress increases with the decrease of m1 (see Fig. 4(c)). Fig. 4(a) shows that dual solutions exist only when m2 is larger than a lower limit value, which decreases towards zero as m1 approaches 1. At each particular value of m2 , there is also a lower limit value for m1 , which allows for the existence of dual solutions as shown by Fig. 4(b), and the lower
Fig. 7. Velocity distribution for flows above a stretching sheet with mass injection: 0 (a) vertical velocity f ðgÞ, and (b) horizontal velocity f ðgÞ. Other parameters are fixed at b ¼ 0:1, c ¼ 0:1, and m2 ¼ 2.
22
L. Wu / International Journal of Heat and Mass Transfer 93 (2016) 17–22
limit value of m1 decreases with the increase of m2 . The boundary layer thickness increases with the decrease of m1 in the unique solution region as shown by Fig. 5. In the dual solution region, the upper branch solution has a thinner boundary thickness than the lower branch. The boundary layer thickness of the upper branch solution increases with the decrease of m1 , while the lower branch has exactly the opposite evolving trend. Only unique solution exists for stretching sheet driven flows with mass injection as shown by Fig. 6(a) and (b). The solution d increases with the increase of m1 and m2 . The magnitude of wall shear stress increases with the increase of m1 and m2 (see Fig. 6 (c)). The boundary layer thickness decreases with the increase of m1 as shown by Fig. 7. Fig. 7(a) shows an interesting phenomenon, i.e., the gases have a positive vertical velocity in regions adjacent to the sheet due to the mass blowing, but the vertical velocity gradually becomes negative far away enough from the sheet.
tion for flows driven by a shrinking sheet. Mass-injection induced slip forces flows driven by a stretching sheet to flow faster. In summary, our results demonstrate that mass-transfer induced slip has a non-negligible impact on flows driven by a moving sheet. Mass-transfer induced slip may even be able to dominate the flow driven effect of the moving sheet by changing the flow direction to flow against the sheet motion. Conflict of interest None declared. Acknowledgments This work is partially supported by the program for professor of special appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.
4. Conclusions References In this paper we systematically studied mass-transfer induced slip effect on viscous gas flows driven by linearly shrinking/ stretching sheets with mass suction/injection. Mass-transfer induced slip has not been considered in all previous studies on moving sheet driven flows. Mass-suction induced slip at a moving surface tends to drive gas flows in a direction opposite to the surface motion, while mass-injection induced slip drives flows in a direction similar to the surface motion. Our results show that mass-transfer induced slip effect significantly expands the available solution space for the shrinking/stretching sheet driven flows. Additional solution sub-regions for flows with mass-suction, which are absent in all previous studies, are discovered by the incorporation of mass-suction induced slip effect. For flows with mass-suction, the competition between flow driven effects of mass-suction induced slip and sheet motion leads to much enriched nonlinear dynamics. Depending on the relative strength of mass-suction induced slip, one more unique solution region may appear in addition to the previously studied dual solution region of flows driven by a shrinking sheet, and a dual solution region may appear in addition to the previously studied unique solution region for flows driven by a stretching sheet. Mass-injection induced slip simply enhances the flow driven effect of sheet motion, and has an effect equivalent to increasing the sheet moving speed. As a result, mass-injection induced slip makes flows more unstable, and thus yields no steady state solu-
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