Appl. Math. Mech. -Engl. Ed., 32(10), 1221–1230 (2011) DOI 10.1007/s10483-011-1495-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011
Applied Mathematics and Mechanics (English Edition)
Analytical solution of magnetohydrodynamic sink flow∗ Ji ZHANG (
)1 , Tie-gang FANG ()1 , Yong-fang ZHONG ()2
(1. Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, NC 27695, USA; 2. School of Engineering, Penn State Erie, the Behrend College, Erie, PA 16563-1701, USA)
Abstract An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of –1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of this equation. Key words similarity solution, Falkner-Skan equation, stretching surface, magnetohydrodynamic (MHD), analytical solution, sink flow Chinese Library Classification O327 2010 Mathematics Subject Classification
1
74D05
Introduction
The famous Falkner-Skan equation obtained for the boundary layer flow with the streamwise pressure gradient[1–2] is f + f f + β(1 − f 2 ) = 0
(1)
with the boundary conditions (BCs) f (0) = λ,
(2a)
f (0) = γ,
(2b)
f (∞) = 1,
(2c)
where λ is the wall mass transfer parameter showing the strength of the mass transfer at the wall, γ is the wall movement parameter indicating the strength of the wall stretching, and β 2m is a parameter of the stream-wise pressure gradient with β = m+1 , in which m is the power ∗ Received Jun. 22, 2010 / Revised Jun. 17, 2011 Corresponding author Tie-gang FANG, Ph. D., E-mail:
[email protected]
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Ji ZHANG, Tie-gang FANG, and Yong-fang ZHONG
index of the free stream velocity. This celebrated equation plays an important role in the development of the boundary layer theory in fluid mechanics. There have been many related works published in the literature thereafter for different conditions of the equations[3–8] . The solution to the Falkner-Skan equation with mass transfer and wall stretching, namely, f (0) = 0 and/or f (0) = 0 were also investigated for different values of β [9–23] . Recently, a special case for the Falkner-Skan equation with m = −1, namely, the sink flow or flow into a convergent channel corresponding to β → −∞, was analytically solved considering wall stretching, and the solution was given in a closed form[24] . In engineering applications, there has been increasing interest in magnetohydrodynamic (MHD) flows. These MHD flows can be used in devices such as power generators, the cooling of reactors, electrostatic filters, and MHD accelerators[25]. In general, the magnetic fields could have stabilizing effects on the boundary layer flow. The Falkner-Skan MHD flow has been investigated by previous researchers[26–30] . Cobble[26] might be the first to consider the MHD effects in the Falkner-Skan flow. Then, the problem was further generalized to a wider scope[27] . Papers published recently solved the associated problem using some latest analytical methods[28–29] . The MHD flow over a moving wedge was studied recently[30] . However, in all the above studies, the solutions were given either numerically or approximately by series. It is very rare for the Falkner-Skan MHD flow to have a closed form analytical solution in the literature. The objective of the current paper is to present an analytical solution in a closed form for the Falkner-Skan MHD flow under the sink flow configuration with m = −1 (β → −∞). This current work is also a generalization of Magyari’s results to the MHD flow[24] .
2
Mathematical formulation
Consider a steady and two-dimensional laminar flow over a continuously moving wall in an electrically conducting fluid, where the free stream velocity is U (x). The sheet moving velocity is Uw = γU (x) with γ being a constant, and the wall mass transfer velocity is vw = 0 assuming an impermeable wall. Because a sink flow configuration is considered in this work, the x-axis runs along the moving surface in the direction opposite to the free stream, and the y-axis is perpendicular to it. A schematic of the flow configuration is shown in Fig. 1.
Fig. 1
Schematic of flow configuration
Analytical solution of magnetohydrodynamic sink flow
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The continuity and momentum equations of the boundary layer are ∂u ∂v + = 0, ∂x ∂y u
(3)
∂u ∂2u ∂u dU (x) σB 2 (x) +v = ν 2 + U (x) + (U (x) − u) ∂x ∂y ∂y dx ρ
(4)
with the BCs u(x, 0) = γU (x),
(5a)
v(x, 0) = 0,
(5b)
u(x, ∞) = U (x),
(5c)
where u and v are the velocity components in the x- and y-directions, respectively. ν is the kinematic viscosity, ρ is the fluid density, and σ is the electrical conductivity of the fluid. A magnetic field with the strength B(x) is applied in the y-direction, and the induced magnetic field is neglected. This group of boundary layer equations is valid for small magnetic Reynolds numbers. According to the definition of the stream function Ψ, we obtain u = ∂Ψ ∂y and v = −1 − ∂Ψ . For the special case with m = −1, namely, the sink flow, we set U (x) = −kx . The ∂x transformation can be used to obtain the similarity equations with the following stream function and similarity variable: √ Ψ = − kνf (η), (6) y k . (7) η= x ν √ η With the above definitions, we obtain u = U (x)f (η) and v = − kν f (η). In order to have x a complete similarity equation, the magnetic field is required to be B(x) = B0 x−1 , where B0 is a constant. Therefore, Eq. (4) is reduced into a similarity equation as follows: f − f 2 + 1 + M (1 − f ) = 0.
(8)
The BCs are f (0) = 0,
(9a)
f (0) = γ,
(9b)
f (∞) = 1,
(9c)
σB 2
where M = ρk0 is the magnetic parameter that shows the effects of the magnetic fields on the flow with M > 0. For M = 0, the equation reduces to the flows past moving boundaries studied by Magyari[24] and Schlichting and Gersten[31] . In order to solve Eq. (8), some transformation manipulations are needed. First, Eq. (8) can be rewritten as follows: M 2 2 M2 − f + Mf + = 0. f + 1 + M + 4 4
(10)
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Ji ZHANG, Tie-gang FANG, and Yong-fang ZHONG
Defining F = f +
M 2 ,
Eq. (10) is then changed to M2 − F 2 = 0. F + 1 + M + 4
(11)
Accordingly, the BCs become F (0) = 0, F (0) = γ +
(12a) M , 2
F (∞) = 1 +
(12b)
M . 2
(12c)
Equation (11) can be further transformed as follows. By defining H(ε) = F a(η) and ε = bη, where both a and b are positive real numbers, we obtain F = aH b, F = ab2 H , and F = ab3 H . Equation (11) becomes H 2
By setting
1+M+ M4 ab3
= 1 and
a b
1+M + + ab3
M2 4
a − H 2 = 0. b
(13)
= 1, we obtain M 12 . a=b= 1+ 2
(14)
Equation (12) is then converted to the classical Falkner-Skan equation for the sink flow H + 1 − H 2 = 0
(15)
with the BCs H(0) = 0, H (0) =
γ+ 1+
(16a) M 2 M 2
,
(16b)
H (∞) = 1.
(16c)
Based on the transformation, we obtain the following relationships between functions H and f : M M M M = abH − = 1+ H − (17) f = F − 2 2 2 2 and M 32 f (0) = ab2 H (0) = 1 + H (0). 2
3
(18)
Analytical solution and discussion
3.1 Wall shear stress f (0) In order to get the wall shear stress, it is straightforward to obtain H (0) = ±
2
2 3 M − 2 γ + M + 2 (γ − 1)2 1+ . 3 2 2 1
3
(19)
Analytical solution of magnetohydrodynamic sink flow
1225
Then, the wall shear stress f (0) is given as follows: 12 2 3 γ + M + 2 (γ − 1)2 . f (0) = ± (20) 3 2 To better illustrate the solution domain, some plots of f (0) versus γ are shown in Fig. 2 for different values of the magnetic parameter.
Fig. 2
Wall stress as function of γ and M
When M = 0, the results reduce to those obtained by Magyari[24]. For M > 0, the solution domain is different from that of M = 0. There are multiple solution branches shown in the plots. For a negative wall stretching parameter, γ < 0, the wall moving velocity is in fact positive and opposite to the free stream velocity. Physically, the wall is stretched out from the sink. For this range of γ, solutions exist only for a certain limit value of γ. Equation (20) requires that for solution existence, γ + 32 M + 2 0, which gives 3 γ − M − 2. 2
(21)
When 1 > γ > − 23 M − 2, there are two solutions as shown by branches A1 and A2 in Fig. 2. branch A1 has a positive wall stress, and branch A2 has a negative wall stress. For γ = − 32 M −2, there is only one solution with no shear stress at the wall, and the boundary layer is in a critical condition for flow separation. When γ < − 23 M − 2, there is no solution. For 0 γ < 1, there still exist two solutions. But in this domain of γ, the wall is either stretched toward the sink (0 < γ < 1) or fixed (γ = 0). With the increase of the magnetic parameter, the solution domain expands to a more negative value of γ. For γ = 1, there is no shear stress, and it is a trivial solution with f (η) = η. For γ > 1, mathematically, there are two solutions for Eq. (20). However, the branch with “+” sign in Eq. (20) has no finite velocity in the flow to match the free stream velocity and has to be discarded in practice[24] . Only the lower branch with “–” sign has a physical meaning. For γ > 1, the wall moves toward the sink faster than the free stream. The wall stress is negative and increases in magnitude for larger values of the magnetic and wall stretching parameters. For each given value of M , there exists one negative critical value of γ to make the wall stress |f (0)| a local maximum. The critical value of γ can be solved from Eq. (20) by setting ∂ 2 3 γ + M + 2 (γ − 1)2 = 0. (22a) ∂γ 3 2 We obtain γcr = −M − 1.
(22b)
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Ji ZHANG, Tie-gang FANG, and Yong-fang ZHONG
3.2 Velocity profile f (η) Equation (15) can be solved to obtain the velocity profiles as 1+ M 1γ+ M 2 2 √ f (η) = 1 + − 2 + 3 tanh η + arctanh 2 31+ 2 − and
M 2
M 2
2 + 3
(branches A1 and A3 )
(23)
1+ M 1γ+ M 2 √ f (η) = 1 + − 2 + 3 tanh2 η − arctanh 2 31+ 2 −
M 2 M 2
(branch A2 ).
M 2 M 2
2 + 3
(24)
To show the flow characteristics, some typical velocity profiles are plotted in Figs. 3–6 for different combinations of the wall stretching and magnetic parameters.
Fig. 3
Velocity profiles for different values of M at lower limit of γ
Fig. 4
Velocity profiles for different values of M at γ = 0 and f (0) > 0
Fig. 5
Velocity profiles for different values of M at γ = 0 and f (0) < 0
Fig. 6
Velocity profiles for different values of γ at M = 2 and f (0) < 0
At the minimum point of γ (γ = − 23 M − 2), the two solution branches become coincident with the wall shear free velocity profiles as shown in Fig. 3. As M increases in this region, a
Analytical solution of magnetohydrodynamic sink flow
1227
larger |γ| value in the negative direction is allowed. Because of a negative stretching parameter, there is a reversal flow in the boundary layer near the wall. A negative velocity here means that the fluid in the vicinity of the wall moves in the positive direction along the x-axis. With the increase of M , the reversal flow becomes stronger near the wall. The flow velocity at γ = − 32 M − 2 becomes
1+ M M 3M 2 √ +3 1+ tanh2 η . (25) f (η) = − 2 + 2 2 2 When −2 − 32 M < γ < 1, there are dual solutions. Two special cases at γ = 0 are depicted in Fig. 4 for f (0) > 0 and Fig. 5 for f (0) < 0. For γ = 0, the wall is fixed with no stretching. For the upper solution branch (branch A1 in Fig. 2) with a positive wall stress, the solution is very similar to the regular Falkan-Skan solutions for m = −1. The velocity monotonically approaches to the free stream as η increases. With the increase of M , the boundary layer thickness becomes thinner. However, for the lower solution branch (branch A2 in Fig. 2) with a negative wall stress, a significant reversal flow occurs. The maximum reversal velocity increases with the increase of the magnetic parameter. However, a larger value of M makes the velocity approach faster to the free stream velocity with a thinner boundary layer. The maximum reversal velocity for branch A2 can be obtained as follows:
3 (26) f (ηmin ) = −2 − M, 2 √ + 23 √ 2 M . It is interesting to note that Eq. (26) does not
1 γ+ M2 M 1+ 2 3 1+ 2 depend on the wall stretching as shown in Fig. 6. Another interesting behavior of the velocity profiles is observed in Fig. 6 as γ → 1. The flow looks like an upside-down “bell” near the wall. With the increase of γ in this range, the velocity profiles are simply shifted or blown away to a further distance from the wall. Each velocity profile has a similar shape. For branch A2 , the reversal flow regions are in the following interval: √ 3M + 4 + 2γ 3M + 4 2 0 < η < arctanh + arctanh (27) 3M + 6 3M + 6 1+ M 2 where ηmin =
arctanh
for −2 − 32 M < γ 0. For 0 < γ < 1, the reversal flow occurs in the interval as √ 3M + 4 + 2γ 3M + 4 2 arctanh − arctanh <η 3M + 6 3M + 6 1+ M 2
√ 3M + 4 + 2γ 3M + 4 2 + arctanh < arctanh . 3M + 6 3M + 6 1+ M 2
As a matter of fact, the velocity profile is symmetric about ηmin =
arctanh
(28) 3M+4+2γ 3M+6
√ 2 1+ M 2
√
in the interval given by Eq. (28) for branch A2 in the range of 0 < γ < 1, and this location is just where the minimum flow velocity or the maximum reversal flow velocity occurs. The relationship of ηmin with the wall stretching parameter is shown in Fig. 7 for different values of M . Different from branch A2 , a reversal flow only exists for branch A1 in the interval as √ 3M + 4 + 2γ 3M + 4 2 − arctanh 0 < η < arctanh (29) 3M + 6 3M + 6 1+ M 2
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Ji ZHANG, Tie-gang FANG, and Yong-fang ZHONG
for −2 − 32 M < γ < 0, and the maximum reversal velocity is γ, which occurs at the wall. When γ > 1, the upper solution becomes physically singular due to the reasons proposed by Magyari[24]. There is only one solution with f (0) < 0, i.e., branch A3 in Fig. 2. As M increases, the velocity approaches to the free stream faster, as shown in Fig. 8. The non-existence of the solutions for γ > 1 with f (0) > 0 can also be illustrated from the solution mathematically as follows. From the solutions given by Eqs. (23) and (24), the term arctanh
M 1 γ+ 2 3 1+ M 2
+
2 3
gives a
complex value for γ > 1 because the argument in the function is larger than 1. It is assumed √ γ+ M that arctanh 13 1+ M2 + 23 = A+Bi with i = −1. For a positive value of the argument function 2 γ+ M 1 1+τ arctanh(τ ) with τ > 1, we obtain A = 2 ln τ −1 with τ = 13 1+ M2 + 23 and B = π2 . It is found 2 √ M M 1+ 2 2 1 γ+ 2 √ that A is a positive number. Then, for the function tanh η+arctanh + 23 , the 3 1+ M 2 2 √ M 1+ 2 π argument is C + 2 i with C = √2 η + A being positive, and the value of the function is a real √ M 1+ 2 √ η− number without any singular points. However, for Eq. (24) with the function tanh2 2 √ 1+ M γ+ M arctanh 13 1+ M2 + 23 , the argument is D − π2 i with D = √2 2 η − A. There exists a singular 2
√ 2 1+ M 2
√ point for D = 0, which occurs at ηs = 12 ln 1+τ τ −1
Fig. 7
Relationship of ηmin with γ for different values of M
.
Fig. 8
Velocity profiles for different values of M at γ > 0 and f (0) < 0
For the special case when γ = 0, it corresponds to the condition with no wall movement. The solution is for the MHD flow in a convergent channel or the sink flow with fixed walls, which is an extension of the solutions given by Pohlhausen[32]. For the MHD sink flow with a fixed wall, the wall stress becomes 4 f (0) = ± M + . (30) 3 √ For a large magnetic parameter, we obtain f (0) → ± M as M → ∞. Accordingly, the velocity profiles become
1+ M 1 M 2 M M 2 √ f (η) = 1 + − 2 + 3 tanh2 + − (31) η + arctanh 2 3 2 + M 3 2 2
Analytical solution of magnetohydrodynamic sink flow
1229
and
f (η) = 1 +
M 2
2
− 2 + 3 tanh
1+ M M 1 M 2 2 √ − η − arctanh + . 3 2 + M 3 2 2
(32)
For a large value of M , the approximated velocity profiles approach to the Heaviside step function for the “+” sign branch (Eq. (31)). In summary, the problem of the MHD sink flow is solved analytically in this work, which is a very rare case for the Falkner-Skan MHD flow with a closed form solution. The solution would not only provide a benchmark flow for numerical code validation, but also enrich the physical insights for the boundary layer theory.
4
Conclusions
In this work, an analytical solution to the famous Falkner-Skan equation for the MHD flow is obtained for a special case, namely, the sink flow with a power index of –1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Solutions exist for both negative and positive wall stretching parameters. With the increase of the magnetic parameter, the negative limit for the wall stretching parameter is expanded. In addition, the magnitude of the wall stress also increases with the magnetic parameter. A wall shear-free solution is obtained at the negative limit of the wall stretching parameter. The magnitude of the wall stress first increases with the wall stretching parameter. After passing a local peak value, it starts to drop with the increase of the wall stretching parameter. Then, it reaches another shear-free trivial solution. Finally, it increases with the wall stretching parameter. Interesting velocity profiles are observed in the results. For certain solution branches, the reversal flow occurs, and the negative velocity overshot (the velocity magnitude higher than 1) near the wall is also observed. These solutions provide a rare case of the Falkner-Skan MHD flow with analytical formulas. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of these important and interesting types of fluid flows.
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