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Flow between two stretchable disks—An exact solution of the Navier–Stokes equations
International Communications in Heat and Mass Transfer 35 (2008) 892–895
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i c h m t
Flow between two stretchable disks—An exact solution of the Navier–Stokes equations ☆ Tiegang Fang ⁎, Ji Zhang Department of Mechanical and Aerospace Engineering, North Carolina State University, 3182 Broughton Hall, 2601 Stinson Drive, Campus Box 7910, Raleigh, NC 27695, United States
A R T I C L E
I N F O
Available online 12 June 2008 Keywords: Navier–Stokes equation Exact solution Stretchable disk Similarity solution
A B S T R A C T In this work, an exact solution for the steady state Navier–Stokes equations in cylindrical coordinates is presented by similarity transformation technique. The solution involves the flow between two stretchable infinite disks with accelerated stretching velocity. The similarity equation was solved numerically and the effects of disk stretching parameter and stretching Reynolds number were studied. With the increase of the stretching Reynolds numbers, the fluid begins with a creeping type flow at R = 0 to a typical boundary layer type flow for large Reynolds numbers. The pressure parameter β changes from a positive number to a negative value with the increase of non-zero stretching parameter. The upper wall stretching parameter also greatly affects the velocity distribution between the two disks with a downward net flow for γ ≠ 1. The results are also useful as a benchmark problem for the validation of three-dimensional numerical computation code. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The flow induced by a stretching boundary is important in the extrusion processes in plastic and metal industries [1–3]. The surface stretching problem was first proposed and analyzed by Sakiadis [4,5] based on boundary layer assumption, where the solution was not an exact solution of the Navier–Stokes (NS) equations according to Wang [6]. Crane [7] presented an exact solution of the two-dimensional NS equations for a stretching sheet problem with a closed analytical form, where the surface stretching velocity was proportional to the distance from the slot. This problem was later generalized to a power-law stretching velocity [8,9]. However, for power-law stretching velocity, the solution was not exact any more. The Crane problem with mass suction and injection at the wall was investigated by Gupta et al. [10] and studied by Wang in a rotating system [11]. The stretching boundary problem was extended by Wang [12] to a three-dimensional setting with two lateral stretching velocities. Exact similarity solutions of the NS equations were obtained numerically for a controlling parameter, a, and it was shown that when a changed from zero to one, the problem varied from a two-dimensional stretching problem to an axis-symmetric stretching one. Recently, the axis-symmetric stretching surface problem was extended to a stretchable disk with both disk stretching and rotation [13]. The flow inside a channel or a tube with stretching wall was solved by Brady and Acrivos [14], which turns out to be an exact solution of the NS equations. The flow outside the stretching tube with acceleration was discussed by Wang [15]. The solution was also valid for the whole NS equations. The spatial stability ☆ Communicated by W. J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (T. Fang). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.04.018
[16] and unsteady developments [17–19] for the Brady and Acrivos [14] problem were further investigated. The combined effects of porous and stretching walls for the channel problem were considered by Zaturska and Banks [20]. The flow between two porous disks with mass suction was studied by Rasmusse [21]. In the current work, we shall present the solution of a flow between two coaxial infinite stretching disks, which was not studied in the literature before. If we call the Brady and Acrivos problem a derived problem of the Crane's, the current problem will be a derived one from the stretching disk problem [12,13]. 2. Theoretical derivation Consider the axis-symmetric flow between two stretchable infinite disks with a distance d between them. Both disks are stretched in the radial direction with a velocity proportional to the radii. The bottom disk is located in the z = 0 plane. The stretching velocity ratio of the upper disk to the lower one is γ. For an incompressible fluid without body forces and based on axis-symmetric flow assumption, the steady state NS equations in cylindrical coordinates [22] reads 1A Auz ¼0 ðrur Þ þ r Ar Az
ð1Þ
ur
2 Aur Aur 1 Ap A ur 1 Aur A2 ur ur þm þ uz ¼ þ þ q Ar r Ar Ar Az Ar2 Az2 r2
ð2aÞ
ur
2 Auz Auz 1 Ap A uz 1 Auz A2 uz þm þ uz ¼ þ þ q Az r Ar Ar Az Ar 2 Az2
ð2bÞ
T. Fang, J. Zhang / International Communications in Heat and Mass Transfer 35 (2008) 892–895
Y where the velocity vector is V ¼ ður ; uz Þ, ν is the kinetic viscosity, p is the fluid pressure, and ρ is the density of the fluid. Similarity functions can be sought as follows, 1 ur ¼ rEF ðgÞ; uz ¼ EdH ðgÞ; and p ¼ qEm P ðgÞ þ br 2 =d2 4
H V¼2F FW b ¼ R F 2 þ F VH : P V¼ 2RFH 2F V
and H1 ð0Þ ¼ H1 ð1Þ ¼ 0; H 1Vð0Þ ¼ 0 and H 1Vð1Þ ¼ 0:
H0 ðgÞ ¼ ð2 þ 2gÞg3 þ ð4 þ 2gÞg2 2g
H1 ðgÞ ¼
ð4aÞ–ð4cÞ
with BCs
1 ½g2 12 þ 3g 9g2 þ g3 22 9g þ 12g2 105 þ g5 ð21 þ 21gÞ þ g6 14 21g 7g2 7 2 þ g 3 þ 6g þ 3g :
F ð0Þ ¼ 1; H ð0Þ ¼ H ð1Þ ¼ 0; F ð1Þ ¼ g; and Pð0Þ ¼ 0;
ð5Þ
Ed2 m
where R ¼ is the wall stretching Reynolds number and it is proportional to the disk stretching strength, and γ is the upper disk stretching parameter showing the velocity ratio of the upper disk to the bottom disk. Without loss of generality, it is assumed that 0 ≤ γ ≤ 1. For γ N 1, we can normalize the problem by the upper disk and switch the coordinate direction. Substituting Eq. (4a) into Eq. (4b) yields RH V2 ¼0 2
ð6Þ
ð7Þ
ð8Þ
ð9Þ
3. Results and discussion For very small Reynolds numbers, the solution can be obtained as follows. For R → 0, assume H(η) = H0(η) + RH1(η) + … + RnHn(η) + …. Substituting this relationship into Eq. (8) yields H0iv ¼ 0
ð10Þ
H1iv
ð11Þ
with the associated boundary conditions H0 ð0Þ ¼ H0 ð1Þ ¼ 0; H V0 ð0Þ ¼ 2; and H 0Vð1Þ ¼ 2g;
2 3g2 þ g þ 4 35
ð17Þ
to the first order. pffiffiffi For very large pffiffiffi R, define a new variable as e ¼ Rg and a new function hðeÞ ¼ RHðgÞ. Then plugging h(ε) into Eq. (6) yields hjR þ b RhhW þ
RhV2 ¼0 2
ð18Þ
pffiffiffi pffiffiffi hð0Þ ¼ hð RÞ ¼ 0; hVð0Þ ¼ 2; hVð RÞ ¼ 2g:
ð19Þ
When R is very large, Eq. (6) becomes
The pressure term can be obtained by a simple integration with 2 P ðgÞ ¼ 2 RH2 þ H V. Therefore the pressure at the upper disk will be P (1) = 2 − 2γ. It is obtained that the pressure at the upper disk is higher than the bottom disk when the stretching parameter is less than one. The velocity is governed by Eq. (8) with BCs (9). The similarity equation is similar to the porous disk problem with different boundary conditions [21]. Because β is an unknown parameter, it will be determined through a shooting method. Eq. (8) with BCs (Eq. (9)) was solved by using a shooting method [22] to convert the boundary-value problem to an initial-value problem. A fourth-order Runge–Kutta integration scheme was adopted to solve the corresponding initialvalue problem. The precision in the numerical calculation was controlled less than 10− 7. Because there are two unknowns in the equation, a multiple-shooting method was used.
H0 Hj 0 ¼ 0
ð16Þ
HWð0Þ ¼ ð8 þ 4gÞ þ R
hj hhW þ
with the associated BCs as H ð0Þ ¼ H ð1Þ ¼ 0; H Vð0Þ ¼ 2; H Vð1Þ ¼ 2g; H ð0Þj ¼ b 2R:
2 13 þ 9g 12g2 35
and
A simpler form of the equation can be obtained as follows H iv RHHj ¼ 0
ð15Þ
with the associated BCs as
with the associated BCs as H ð0Þ ¼ H ð1Þ ¼ 0; H Vð0Þ ¼ 2; H Vð1Þ ¼ 2g:
ð14Þ
Therefore, for very small R, one obtains b ¼ ð12 þ 12gÞ þ R
Hj þ b RHHW þ
ð13Þ
The solutions read ð3Þ
where η = z / d is the similarity variable. The quantity E is a parameter corresponding to the disk stretching strength and its unit is 1s . Substituting these functions (3) into the governing equations (1), (2a), (2b) yields a similarity equation group, 8 <
893
ð12Þ
hV2 ¼ 0: 2
ð20Þ
For this equation, when R is approaching to infinity, the flow shows boundary layer behavior near the wall and fluid far from the wall will not be affected by the wall. Therefore the associated boundary conditions become hð0Þ ¼ 0; hVð0Þ ¼ 2; hVðlÞ ¼ 0:
ð21Þ
Eq. (20) and the BCs (Eq. (21)) are exactly the same as the equations for the pure stretching disk problem [12,13], where it is obtained that h″(0) = 2.347442. Hence it is found that when R → ∞, pffiffiffiffiffi HWð0Þ ¼ 2:347442 R:
ð22Þ
For the flow in the central region, itpapproaches an inviscid core ffiffiffi flow. By using a new function as g ðgÞ ¼ RH ðgÞ, it is obtained ggW
g V2 ggj b ¼ pffiffiffi ¼ 0: ðRYlÞ 2 R
ð23Þ
When R → ∞, the right-hand side of Eq. (23) becomes zero. For γ = 1, there is a symmetric stretching on both disks. Therefore, for this case, g(0.5) = 0 and g″(0.5) = 0. There is a special solution for pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Þ . Matching the solution of Eqs. (20) Eq. (23) as g ðgÞ ¼ ð2bÞg ð2b 2 and (23) with h(ε → ∞) = g(η → 0) yields pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2bÞ ¼ hðlÞ ¼ 1:502996: 2
ð24Þ
Therefore, when R → ∞, β → −4.517994. For γ = 0, there is a nontrivial solution for Eq. (23) as g(η) = aη2 − 2aη + a with boundary conditions of g(1) = 0 and g′(1) = 0 after matching with the solution near the stretching disk, where a = h(∞) = −1.502996. For this case,
894
T. Fang, J. Zhang / International Communications in Heat and Mass Transfer 35 (2008) 892–895
Fig. 1. The velocity profiles in the radial and vertical directions for different stretching parameter with R = 2.0.
Fig. 3. The non-dimensional wall shear stress as a function of R under different values of γ. The limiting approximation equations are also illustrated.
Eq. (23) gives β = 0 as R → ∞. These limiting values of β will be compared with their numerical values in a later section. In order to investigate the flow behaviors between the two stretchable disks, results were obtained for different parameters using numerical calculation. The velocity profiles are shown in Fig. 1 for R = 2.0 with different wall stretching parameters. For the radial flow velocity, it is found that the velocity near the wall is stretched by the wall movement. However, away from the wall, there exists an inflow in the negative radial direction to balance the mass stretched out by the wall to be consistent with the force balance. For non-zero stretching parameter, the fluid is stretched along the wall with an outer flow in the radial direction near the upper wall. The vertical velocity is downward near the bottom wall due to the wall stretching. As seen in the previous discussion of pressure, for the situations with γ ≠ 1, the upper disk pressure is higher than the lower disk. So the overall net flow rate is downward to balance the stretching of the bottom disk as seen from the velocity profiles with negative velocity profiles dominant for stretching parameter less than one. For γ = 1 with a symmetric stretching configuration, the net flow in the vertical direction is zero. The maximum negative velocity in the vertical direction decreases with the increase of wall stretching parameter.
Fig. 2. The velocity profiles under different values of R for γ = 0 (a) and γ = 1 (b).
Fig. 4. The pressure parameter β as a function of R for different values of γ.
T. Fang, J. Zhang / International Communications in Heat and Mass Transfer 35 (2008) 892–895
895
4. Summary In this work, the flow between two stretchable disks was investigated. An exact solution of the Navier–Stokes equation was obtained and solved numerically based on similarity transformation. The effects of disk stretching parameter and stretching Reynolds number were studied. With the increase of the stretching Reynolds numbers, the fluid begins with a creeping type flow at R = 0 to a typical boundary layer type flow for large Reynolds numbers. The pressure parameter β changes from a positive number to a negative value with the increase of R for non-zero stretching parameter. The upper wall stretching parameter also greatly affects the velocity distribution between the two disks with a downward net flow for γ ≠ 1. The results are also useful as a benchmark problem for the validation of threedimensional numerical computation code. References
Fig. 5. The pressure parameter β as a function of R for γ = 1 with limiting approximations for small and large stretching Reynolds numbers.
However, in the radial direction, larger maximum negative velocity in the radial direction occurs for symmetric stretching disks. The velocity profiles in the radial direction under different Reynolds numbers are illustrated in Fig. 2(a) for asymmetric stretching with γ = 0. It is obvious that with the increase of R the velocity profiles become closer to the bottom wall. The boundary layer behavior becomes more pronounced for large values of R. The maximum negative velocity decreases with the increase of R. For symmetric stretching disk problem with γ = 1 as shown in Fig. 2(b), the boundary layer type flow becomes more obvious with the increase of R. There is an inviscid core flow in the central region between the two disks. With the increase of the stretching strength or R, the wall shear stress also increases due to higher velocity gradient near the disks. In both Fig. 2 (a) and (b), when R = 0, the fluid shows a creeping flow behavior with parabolic velocity profiles in the radial direction. The non-dimensional shear stress in the radial direction, namely H″(0), is shown in Fig. 3 for different values of γ as a function of stretching Reynolds number, R. It is seen that the shear stress increases with the increase of R. For Rb 10, Eq. (17) provides a good estimation of H″(0). For large values of R, Eq. (22) gives a very close estimation of H″(0) when RN 100. The β value for different values of γ as a function of R is shown in Fig. 4. When γ=0, the value of β decreases with the increase of R and approaches zero for very large R, which is consistent with the previous analysis. But for non-zero values of γ, β first drops to a negative value and reaches a minimum point, then it increases a little bit and finally approaches to a negative value dependent on γ. The values of β for the symmetric stretching case are shown in Fig. 5 for a large value domain of R. As discussed before, when R→∞, β→−4.517994, which is also shown in Fig. 5. The variation trend of β for stretching disks is similar to the solution for stretching channel problem by Brady and Acrivos [14], but with different values. For small R, Eq. (16) provides a close estimation of β for Rb 10.
[1] T. Altan, S. Oh, H. Gegel, Metal Forming Fundamentals and Applications, American Society of Metals, Metals Park, OH, 1979. [2] E.G. Fisher, Extrusion of Plastics, Wiley, New York, 1976. [3] Z. Tadmor, I. Klein, Engineering Principles of Plasticating Extrusion, Polymer Science and Engineering Series, Van Norstrand Reinhold, New York, 1970. [4] B.C. Sakiadis, Boundary-layer behavior on continuous solid surface: I. Boundarylayer equations for two-dimensional and axisymmetric flow, AIChe J. 7 (1961) 26–28. [5] B.C. Sakiadis, Boundary-layer behavior on continuous solid surface: II. Boundarylayer on a continuous flat surface, AIChe J. 7 (1961) 221–225. [6] C.Y. Wang, Exact solutions of the steady state Navier–Stokes equations, Annu. Rev. Fluid Mech. 23 (1991) 159. [7] L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys.(ZAMP) 21 (1970) 645–647. [8] H.K. Kuiken, On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small, IMA J. Appl. Mathem. 27 (1981) 387–405. [9] W.H.H. Banks, Similarity solutions of the boundary-layer equations for a stretching wall, J. Mech. Theor. Appl. 2 (1983) 375–392. [10] 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. [11] C.Y. Wang, Stretching a surface in a rotating fluid, J. Appl. Math. Phys. (ZAMP) 39 (1988) 177–185. [12] C.Y. Wang, The three-dimensional flow due to a stretching flat surface, Phys. Fluids 27 (1984) 1915–1917. [13] T. Fang, Flow over a stretchable disk, Phys. Fluids 19 (2007) 128105. [14] J.F. Brady, A. Acrivos, Steady flow in a channel or tube with an accelerating surface velocity: an exact solution to the Navier–Stokes equations with reverse flow, J. Fluid Mech. 112 (1981) 127–150. [15] C.Y. Wang, Fluid flow due to a stretching cylinder, Phys. Fluids 31 (1988) 466–468. [16] L. Durlofskyl, J.F. Brady, The spatial stability of a class of similarity solutions, Phys. Fluids 27 (5) (1984) 1068–1076. [17] E.B.B. Watson, W.H.H. Banks, M.B. Zaturska, P.G. Drazin, On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls, J. Fluid Mech. 212 (1990) 451. [18] P. Watson, W.H.H. Banks, M.B. Zaturska, P.G. Drazin, ‘‘Laminar channel flow driven by accelerating walls, Eur. J. Appl. Math. 2 (1991) 359. [19] M.B. Zaturska, W.H.H. Banks, Flow in a pipe driven by suction and accelerating wall, Acta Mechanica 110 (1–4) (1995) 111–121. [20] M.B. Zaturska, W.H.H. Banks, New solutions for flow in a channel with porous walls and/or non-rigid walls, Fluid Dynamics Research 33 (1–2) (July 2003) 57–71. [21] H. Rasmusse, Steady viscous flow between two porous disks, J. Appl. Math. Phys. (ZAMP) 21 (2) (1970) 187–195. [22] F.M. White, Viscous Fluid Flow, 2nd ed. McGraw-Hill Company, New York, 1991.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i c h m t
Flow between two stretchable disks—An exact solution of the Navier–Stokes equations ☆ Tiegang Fang ⁎, Ji Zhang Department of Mechanical and Aerospace Engineering, North Carolina State University, 3182 Broughton Hall, 2601 Stinson Drive, Campus Box 7910, Raleigh, NC 27695, United States
A R T I C L E
I N F O
Available online 12 June 2008 Keywords: Navier–Stokes equation Exact solution Stretchable disk Similarity solution
A B S T R A C T In this work, an exact solution for the steady state Navier–Stokes equations in cylindrical coordinates is presented by similarity transformation technique. The solution involves the flow between two stretchable infinite disks with accelerated stretching velocity. The similarity equation was solved numerically and the effects of disk stretching parameter and stretching Reynolds number were studied. With the increase of the stretching Reynolds numbers, the fluid begins with a creeping type flow at R = 0 to a typical boundary layer type flow for large Reynolds numbers. The pressure parameter β changes from a positive number to a negative value with the increase of non-zero stretching parameter. The upper wall stretching parameter also greatly affects the velocity distribution between the two disks with a downward net flow for γ ≠ 1. The results are also useful as a benchmark problem for the validation of three-dimensional numerical computation code. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The flow induced by a stretching boundary is important in the extrusion processes in plastic and metal industries [1–3]. The surface stretching problem was first proposed and analyzed by Sakiadis [4,5] based on boundary layer assumption, where the solution was not an exact solution of the Navier–Stokes (NS) equations according to Wang [6]. Crane [7] presented an exact solution of the two-dimensional NS equations for a stretching sheet problem with a closed analytical form, where the surface stretching velocity was proportional to the distance from the slot. This problem was later generalized to a power-law stretching velocity [8,9]. However, for power-law stretching velocity, the solution was not exact any more. The Crane problem with mass suction and injection at the wall was investigated by Gupta et al. [10] and studied by Wang in a rotating system [11]. The stretching boundary problem was extended by Wang [12] to a three-dimensional setting with two lateral stretching velocities. Exact similarity solutions of the NS equations were obtained numerically for a controlling parameter, a, and it was shown that when a changed from zero to one, the problem varied from a two-dimensional stretching problem to an axis-symmetric stretching one. Recently, the axis-symmetric stretching surface problem was extended to a stretchable disk with both disk stretching and rotation [13]. The flow inside a channel or a tube with stretching wall was solved by Brady and Acrivos [14], which turns out to be an exact solution of the NS equations. The flow outside the stretching tube with acceleration was discussed by Wang [15]. The solution was also valid for the whole NS equations. The spatial stability ☆ Communicated by W. J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (T. Fang). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.04.018
[16] and unsteady developments [17–19] for the Brady and Acrivos [14] problem were further investigated. The combined effects of porous and stretching walls for the channel problem were considered by Zaturska and Banks [20]. The flow between two porous disks with mass suction was studied by Rasmusse [21]. In the current work, we shall present the solution of a flow between two coaxial infinite stretching disks, which was not studied in the literature before. If we call the Brady and Acrivos problem a derived problem of the Crane's, the current problem will be a derived one from the stretching disk problem [12,13]. 2. Theoretical derivation Consider the axis-symmetric flow between two stretchable infinite disks with a distance d between them. Both disks are stretched in the radial direction with a velocity proportional to the radii. The bottom disk is located in the z = 0 plane. The stretching velocity ratio of the upper disk to the lower one is γ. For an incompressible fluid without body forces and based on axis-symmetric flow assumption, the steady state NS equations in cylindrical coordinates [22] reads 1A Auz ¼0 ðrur Þ þ r Ar Az
ð1Þ
ur
2 Aur Aur 1 Ap A ur 1 Aur A2 ur ur þm þ uz ¼ þ þ q Ar r Ar Ar Az Ar2 Az2 r2
ð2aÞ
ur
2 Auz Auz 1 Ap A uz 1 Auz A2 uz þm þ uz ¼ þ þ q Az r Ar Ar Az Ar 2 Az2
ð2bÞ
T. Fang, J. Zhang / International Communications in Heat and Mass Transfer 35 (2008) 892–895
Y where the velocity vector is V ¼ ður ; uz Þ, ν is the kinetic viscosity, p is the fluid pressure, and ρ is the density of the fluid. Similarity functions can be sought as follows, 1 ur ¼ rEF ðgÞ; uz ¼ EdH ðgÞ; and p ¼ qEm P ðgÞ þ br 2 =d2 4
H V¼2F FW b ¼ R F 2 þ F VH : P V¼ 2RFH 2F V
and H1 ð0Þ ¼ H1 ð1Þ ¼ 0; H 1Vð0Þ ¼ 0 and H 1Vð1Þ ¼ 0:
H0 ðgÞ ¼ ð2 þ 2gÞg3 þ ð4 þ 2gÞg2 2g
H1 ðgÞ ¼
ð4aÞ–ð4cÞ
with BCs
1 ½g2 12 þ 3g 9g2 þ g3 22 9g þ 12g2 105 þ g5 ð21 þ 21gÞ þ g6 14 21g 7g2 7 2 þ g 3 þ 6g þ 3g :
F ð0Þ ¼ 1; H ð0Þ ¼ H ð1Þ ¼ 0; F ð1Þ ¼ g; and Pð0Þ ¼ 0;
ð5Þ
Ed2 m
where R ¼ is the wall stretching Reynolds number and it is proportional to the disk stretching strength, and γ is the upper disk stretching parameter showing the velocity ratio of the upper disk to the bottom disk. Without loss of generality, it is assumed that 0 ≤ γ ≤ 1. For γ N 1, we can normalize the problem by the upper disk and switch the coordinate direction. Substituting Eq. (4a) into Eq. (4b) yields RH V2 ¼0 2
ð6Þ
ð7Þ
ð8Þ
ð9Þ
3. Results and discussion For very small Reynolds numbers, the solution can be obtained as follows. For R → 0, assume H(η) = H0(η) + RH1(η) + … + RnHn(η) + …. Substituting this relationship into Eq. (8) yields H0iv ¼ 0
ð10Þ
H1iv
ð11Þ
with the associated boundary conditions H0 ð0Þ ¼ H0 ð1Þ ¼ 0; H V0 ð0Þ ¼ 2; and H 0Vð1Þ ¼ 2g;
2 3g2 þ g þ 4 35
ð17Þ
to the first order. pffiffiffi For very large pffiffiffi R, define a new variable as e ¼ Rg and a new function hðeÞ ¼ RHðgÞ. Then plugging h(ε) into Eq. (6) yields hjR þ b RhhW þ
RhV2 ¼0 2
ð18Þ
pffiffiffi pffiffiffi hð0Þ ¼ hð RÞ ¼ 0; hVð0Þ ¼ 2; hVð RÞ ¼ 2g:
ð19Þ
When R is very large, Eq. (6) becomes
The pressure term can be obtained by a simple integration with 2 P ðgÞ ¼ 2 RH2 þ H V. Therefore the pressure at the upper disk will be P (1) = 2 − 2γ. It is obtained that the pressure at the upper disk is higher than the bottom disk when the stretching parameter is less than one. The velocity is governed by Eq. (8) with BCs (9). The similarity equation is similar to the porous disk problem with different boundary conditions [21]. Because β is an unknown parameter, it will be determined through a shooting method. Eq. (8) with BCs (Eq. (9)) was solved by using a shooting method [22] to convert the boundary-value problem to an initial-value problem. A fourth-order Runge–Kutta integration scheme was adopted to solve the corresponding initialvalue problem. The precision in the numerical calculation was controlled less than 10− 7. Because there are two unknowns in the equation, a multiple-shooting method was used.
H0 Hj 0 ¼ 0
ð16Þ
HWð0Þ ¼ ð8 þ 4gÞ þ R
hj hhW þ
with the associated BCs as H ð0Þ ¼ H ð1Þ ¼ 0; H Vð0Þ ¼ 2; H Vð1Þ ¼ 2g; H ð0Þj ¼ b 2R:
2 13 þ 9g 12g2 35
and
A simpler form of the equation can be obtained as follows H iv RHHj ¼ 0
ð15Þ
with the associated BCs as
with the associated BCs as H ð0Þ ¼ H ð1Þ ¼ 0; H Vð0Þ ¼ 2; H Vð1Þ ¼ 2g:
ð14Þ
Therefore, for very small R, one obtains b ¼ ð12 þ 12gÞ þ R
Hj þ b RHHW þ
ð13Þ
The solutions read ð3Þ
where η = z / d is the similarity variable. The quantity E is a parameter corresponding to the disk stretching strength and its unit is 1s . Substituting these functions (3) into the governing equations (1), (2a), (2b) yields a similarity equation group, 8 <
893
ð12Þ
hV2 ¼ 0: 2
ð20Þ
For this equation, when R is approaching to infinity, the flow shows boundary layer behavior near the wall and fluid far from the wall will not be affected by the wall. Therefore the associated boundary conditions become hð0Þ ¼ 0; hVð0Þ ¼ 2; hVðlÞ ¼ 0:
ð21Þ
Eq. (20) and the BCs (Eq. (21)) are exactly the same as the equations for the pure stretching disk problem [12,13], where it is obtained that h″(0) = 2.347442. Hence it is found that when R → ∞, pffiffiffiffiffi HWð0Þ ¼ 2:347442 R:
ð22Þ
For the flow in the central region, itpapproaches an inviscid core ffiffiffi flow. By using a new function as g ðgÞ ¼ RH ðgÞ, it is obtained ggW
g V2 ggj b ¼ pffiffiffi ¼ 0: ðRYlÞ 2 R
ð23Þ
When R → ∞, the right-hand side of Eq. (23) becomes zero. For γ = 1, there is a symmetric stretching on both disks. Therefore, for this case, g(0.5) = 0 and g″(0.5) = 0. There is a special solution for pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Þ . Matching the solution of Eqs. (20) Eq. (23) as g ðgÞ ¼ ð2bÞg ð2b 2 and (23) with h(ε → ∞) = g(η → 0) yields pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2bÞ ¼ hðlÞ ¼ 1:502996: 2
ð24Þ
Therefore, when R → ∞, β → −4.517994. For γ = 0, there is a nontrivial solution for Eq. (23) as g(η) = aη2 − 2aη + a with boundary conditions of g(1) = 0 and g′(1) = 0 after matching with the solution near the stretching disk, where a = h(∞) = −1.502996. For this case,
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T. Fang, J. Zhang / International Communications in Heat and Mass Transfer 35 (2008) 892–895
Fig. 1. The velocity profiles in the radial and vertical directions for different stretching parameter with R = 2.0.
Fig. 3. The non-dimensional wall shear stress as a function of R under different values of γ. The limiting approximation equations are also illustrated.
Eq. (23) gives β = 0 as R → ∞. These limiting values of β will be compared with their numerical values in a later section. In order to investigate the flow behaviors between the two stretchable disks, results were obtained for different parameters using numerical calculation. The velocity profiles are shown in Fig. 1 for R = 2.0 with different wall stretching parameters. For the radial flow velocity, it is found that the velocity near the wall is stretched by the wall movement. However, away from the wall, there exists an inflow in the negative radial direction to balance the mass stretched out by the wall to be consistent with the force balance. For non-zero stretching parameter, the fluid is stretched along the wall with an outer flow in the radial direction near the upper wall. The vertical velocity is downward near the bottom wall due to the wall stretching. As seen in the previous discussion of pressure, for the situations with γ ≠ 1, the upper disk pressure is higher than the lower disk. So the overall net flow rate is downward to balance the stretching of the bottom disk as seen from the velocity profiles with negative velocity profiles dominant for stretching parameter less than one. For γ = 1 with a symmetric stretching configuration, the net flow in the vertical direction is zero. The maximum negative velocity in the vertical direction decreases with the increase of wall stretching parameter.
Fig. 2. The velocity profiles under different values of R for γ = 0 (a) and γ = 1 (b).
Fig. 4. The pressure parameter β as a function of R for different values of γ.
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4. Summary In this work, the flow between two stretchable disks was investigated. An exact solution of the Navier–Stokes equation was obtained and solved numerically based on similarity transformation. The effects of disk stretching parameter and stretching Reynolds number were studied. With the increase of the stretching Reynolds numbers, the fluid begins with a creeping type flow at R = 0 to a typical boundary layer type flow for large Reynolds numbers. The pressure parameter β changes from a positive number to a negative value with the increase of R for non-zero stretching parameter. The upper wall stretching parameter also greatly affects the velocity distribution between the two disks with a downward net flow for γ ≠ 1. The results are also useful as a benchmark problem for the validation of threedimensional numerical computation code. References
Fig. 5. The pressure parameter β as a function of R for γ = 1 with limiting approximations for small and large stretching Reynolds numbers.
However, in the radial direction, larger maximum negative velocity in the radial direction occurs for symmetric stretching disks. The velocity profiles in the radial direction under different Reynolds numbers are illustrated in Fig. 2(a) for asymmetric stretching with γ = 0. It is obvious that with the increase of R the velocity profiles become closer to the bottom wall. The boundary layer behavior becomes more pronounced for large values of R. The maximum negative velocity decreases with the increase of R. For symmetric stretching disk problem with γ = 1 as shown in Fig. 2(b), the boundary layer type flow becomes more obvious with the increase of R. There is an inviscid core flow in the central region between the two disks. With the increase of the stretching strength or R, the wall shear stress also increases due to higher velocity gradient near the disks. In both Fig. 2 (a) and (b), when R = 0, the fluid shows a creeping flow behavior with parabolic velocity profiles in the radial direction. The non-dimensional shear stress in the radial direction, namely H″(0), is shown in Fig. 3 for different values of γ as a function of stretching Reynolds number, R. It is seen that the shear stress increases with the increase of R. For Rb 10, Eq. (17) provides a good estimation of H″(0). For large values of R, Eq. (22) gives a very close estimation of H″(0) when RN 100. The β value for different values of γ as a function of R is shown in Fig. 4. When γ=0, the value of β decreases with the increase of R and approaches zero for very large R, which is consistent with the previous analysis. But for non-zero values of γ, β first drops to a negative value and reaches a minimum point, then it increases a little bit and finally approaches to a negative value dependent on γ. The values of β for the symmetric stretching case are shown in Fig. 5 for a large value domain of R. As discussed before, when R→∞, β→−4.517994, which is also shown in Fig. 5. The variation trend of β for stretching disks is similar to the solution for stretching channel problem by Brady and Acrivos [14], but with different values. For small R, Eq. (16) provides a close estimation of β for Rb 10.
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