On a certain boundary value problem arising in shrinking sheet flows

Applied Mathematics and Computation 217 (2010) 4086–4093

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On a certain boundary value problem arising in shrinking sheet flows Rafael Cortell Departamento de Física Aplicada, Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Valencia, 46071 Valencia, Spain

a r t i c l e

i n f o

Keywords: Laminar boundary layer Mass suction Shrinking surfaces Numerical solution

a b s t r a c t This paper presents an analysis of the boundary value problem resulting from the magnetohydrodynamic (MHD) viscous flow influenced by a shrinking sheet with suction for the cases of two-dimensional (m = 1) and axisymmetric (m = 2) shrinking. The influences of the parameter m as well as the effects of suction parameter s and Hartmann number M2 on similar entrainment velocity f(1) and flow characteristics are studied. To this purpose, the resulting nonlinear ordinary differential equation is solved numerically using the 4th order Runge–Kutta method in combination with a shooting procedure. The obtained results elucidate reliability and efficiency of the technique from which interesting features between the skin friction coefficient f00 (0) and the entrainment velocity f(1) as function of the mass transfer parameter s can also be obtained. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Boundary layer behaviour over a moving continuous solid surface is a relevant type of flow which is present in many industrial processes such as manufacture and drawing of plastics and rubber sheets, processing of sheet-like materials in paper production, cooling of metallic sheets and manufacture of metal and polymer solid cylinders [1] and crystal growing just to name a few. In virtually all such processing operations, both the flow kinematics and the rate of cooling play an important role about the quality of the final product. Sakiadis [2] initiated the study of the boundary layer flow over a continuous solid surface moving with constant speed. Crane [3] was the first one who studied the stretching problem taking into account the fluid flow over a linearly stretched surface. The elastic sheet moves in its own plane with a velocity varying linearly with the distance from a fixed point. Later, Gupta and Gupta [4] extended the Crane’s work [3] to include suction or blowing. After these pioneering works, analyses to the flow and heat transfer over a linearly stretching surface have intensely been made in recent years [5–11]. Besides that, Vajravelu [12] studied the flow due to non-linear stretching sheets and later the associated heat transfer problem has been devised by Cortell [13–15], and also the flow and heat transfer of a micropolar fluid over a non-linearly stretching sheet were analyzed by Hayat and his co-workers [16]. Very recently, the effects of transpiration for the flow past a non-linear stretching sheet with variable external velocity were studied by Afzal [17]. Unlike the linear/non-linear stretching sheet problem, little work has been made on boundary layer flow and heat transfer induced by a shrinking surface. Probably, this peculiar research’s line starts from Wang [18] who studied the unsteady shrinking film solution and Miklavcic and Wang [19] who concluded that the solution for shrinking sheets may not be unique at certain suction rates. MHD stagnation flow also was considered, Wang [20], Rahimpour et al. [21]. Moreover, various interesting aspects have been studied by Fang and his co-workers [22–24]. The problem of unsteady shrinking sheet in a rotating fluid with mass transfer has recently been devised by Ali et al. [25]. The existence of the solution of the governing field equations arising in shrinking sheet flows was studied by Akyildiz and Siginer [26]. Other studies which belong to these classes of problems were recently done by Noor et al. [27], Sajid and Hayat [28]. In [28], for the largest value of the mass E-mail address: rcortell@fis.upv.es 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.10.024

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R. Cortell / Applied Mathematics and Computation 217 (2010) 4086–4093

suction parameter s, the velocity profile f 0 (g) appears to exhibit a maximum in the interval [0, g1[. Later, the existence of this spurious maximum was discussed by Fernández [29]. The present paper gathers accurate numerical results for the abovementioned model by considering, at least, five-decimal accuracy as the criterion for convergence, and in accordance with Fernández [29], the commented maximum is not appeared. The momentum transfer characteristics heavily depend on the suction parameter s, the m parameter and the Hartman parameter M2. 2. Mathematical formulation We consider a steady, three-dimensional laminar MHD viscous flow over a shrinking sheet in an electrically conducting fluid under a constant transverse magnetic field of strength B0 applied in the z-direction; also, the induced magnetic field is neglected and there is no electric field influences. Let (u, v, w) be the velocity components along the (x, y, z) directions, respectively. The boundary-layer equations governing the flow are then (see [25] for example)

@u @ v @w þ ¼ 0; þ @x @y @z ! @u @u @u 1 @p @2u @2u @2u rB2  0 u; þv þw ¼ þt þ þ u 2 2 2 @x @y @z @x @y @z q @x q ! 2 2 2 @v @v @v 1 @p @ v @ v @ v rB2 þt þ 2 þ 2  0 v; u þv þw ¼ 2 @x @y @z q @y @x @y @z q ! @w @w @w 1 @p @2w @2w @2w þ 2 þ 2 ; þv þw ¼ þt u @x @y @z @x2 @y @z q @z

ð1Þ ð2Þ ð3Þ ð4Þ

where r and t are the electrical conductivity and the kinematic viscosity, respectively. p is the fluid pressure, q is the fluid density; also, the group of Navier–Stokes Eqs. (1)–(4) is valid for small magnetic Reynolds number. The boundary conditions on the shrinking surface at z = 0 are,

u ¼ ax;

v ¼ aðm  1Þy;

w ¼ W

ð5:1Þ

and far away from the surface,

v ! 0 as z ! 1:

u ! 0;

ð5:2Þ

where a (>0) and W are the shrinking constant and the suction velocity, respectively. Here we assume m = 1 for the case in which the sheet shrinks in x-direction alone (two-dimensional) and m = 2 for the case in which the sheet shrinks axisymmetrically. The similarity variable is defined as:

g¼

rffiffiffi a z

ð6Þ

t

and we also introduce the dimensionless variable f related on velocity components as 0

u ¼ axf ðgÞ;

v ¼ aðm  1Þyf 0 ðgÞ;

pffiffiffiffiffiffi w ¼  atmf ðgÞ;

ð7Þ

where primes indicate the differentiation with respect to g. Then, on substitution into Eqs. (2) and (3), we obtain the nonlinear ordinary differential equation 00

f 000  M2 f 0  ðf 0 Þ2 þ mff ¼ 0:

ð8Þ p

Realize that the continuity equation, Eq. (1) is automatically satisfied and Eq. (4) becomes q ¼ t to be solved subject to the following boundary conditions:

f ¼ s;

f 0 ¼ 1 at g ¼ 0;



w2 2

þ constant. Eq. (8) is

ð9:1Þ

f 0 ! 0 as g ! 1: 2 pffiffiffiffi is the suction parameter, M ¼ Here, s ¼ mW at

@w @z

ð9:2Þ rB20 qa

is the Hartmann number and f(1) = limg?1f(g) is the entrainment velocity.

3. Numerical procedure Here, our momentum transfer problems governed by Eqs. (8) and (9) need numerical integration. For the purposes of that numerical solution of the ordinary differential equation, Eq. (8), one can easily write it as the equivalent first-order system:

w01 ¼ w2 ; w02 ¼ w3 ; w03 ¼ M 2 w2 þ w22  mw1 w3 ;

ð10Þ

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R. Cortell / Applied Mathematics and Computation 217 (2010) 4086–4093 2

where w1 ¼ f ðgÞ; w2 ¼ ddfg ¼ f 0 ðgÞ; w3 ¼ ddg2f ¼ f 00 ðgÞ, and in accordance with boundary condition (9.1) we get

w1 ð0Þ ¼ s;

w2 ð0Þ ¼ 1:

ð11Þ

Using numerical methods of integration and disregarding temporarily the boundary condition (9.2), a family of solutions of  2  Eqs. (10), (11) can be obtained for arbitrarily chosen values of ddg2f ¼ f 00 ð0Þi0: We tentatively assume that a special value g¼0 of f 00 (0)(the single missed value at g = 0 which is guessed in our iterative procedure) yields a solution for which f 0 (g) vanishes at a certain g = g1 and simultaneously satisfies the additional conditions at infinity:

  2  df    ¼ 0; d f ¼ 0 at g ¼ g : 1 dg dg2

ð12Þ

We estimate f 00 (0) and integrate Eqs. (10) and (11) as an initial value problem by the Runge–Kutta method of fourth order with the additional conditions (12). These conditions (12) at infinity correct unphysical behaviours of the solution, and seem to play an important role in this class of boundary problems. It is obvious that when Eq. (12) are satisfied, hence the velocities tend to zero at infinity in an asymptotical fashion as must be required from boundary-layer theory. Runge–Kutta numerical method of fourth order by applying to an initial value problem (IVP) usually necessitates a guess for the limited integral region (i.e., the g1value) instead of infinity for numerical integration, but our iterative shooting procedure does not need this because it acts only onto f 00 (0) (the missed skin friction coefficient) controlling, at the same time, the additional conditions (12). On the other hand, it is well-known that there is a marked influence of the selected g1 value onto f 00 (0), and then for each numerical solution in the iterative process the size of the integration domain (i.e., the g1value) is obtained (no fixed before calculation as usually is made) as a natural part of the solution, and there is no necessity to select the extent of the integration domain before calculation. Therefore, our governing momentum transfer equation can be solved (numerically) by marching freely from the origin as it also is common for boundary layers involving interesting problems in the area of fluid dynamics, Cortell [8,13–15]. This innovative way of integration prevents us from unphysical behaviours of the solution and provides high accuracy of the results. 3.1. Results and discussions The numerical computation for momentum transfer is carried out for different values of the parameters m = 1, 2; suction parameter s and Hartmann number M2 using the numerical procedure discussed in Section 3. Numerical results for the stated problem at M = 2, m = 2 and s = 1.8 are listed in Table 1. It can be observed from Table 1 that both velocity and velocitygradient distributions are monotonically and simultaneously tend to zero as the distance increases from the boundary and also one can see how the boundary conditions at infinity, Eq. (12), are totally satisfied. Further, for each set of parameters entering the problem, our procedure also gives the corresponding value of the entrainment velocity f(1). Also, in order to

Table 1 Some numerical Runge–Kutta results for which were used two different step sizes of Dg = 0.1, 0.01 at M = m = 2, s = 1.8. The reader clearly finds that f (1) = 1.558986. M

m

s

Dg

g

f (g)

 f 0 (g)

f 00 (g)

2

2

1.8

0.1

2

2

1.8

0.01

0.0 0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.8 1.664502 1.605372 1.579160 1.567992 1.562959 1.559503 1.559059 1.559004 1.559002 1.559009 1.8 1.664467 1.605343 1.579397 1.567980 1.562951 1.559498 1.559053 1.558995 1.558986 1.558986 1.558986

1.0 0.4344555 0.1904132 0.0837363 0.0368815 0.0162549 0.0020959 0.0002662 0.0000271 0.0000083 0.0000203 1.0 0.4344086 0.1902935 0.0836610 0.0368392 0.0162331 0.0020944 0.0002703 0.0000349 0.0000045 0.0000006 0.0000001

4.20423 1.80007 0.78367 0.34366 0.15118 0.06659 0.00860 0.00111 0.00015 0.00003 0.00002 4.204113245 1.799429 0.783169 0.342240 0.150998 0.066500 0.008577 0.001107 0.000143 0.000018 0.000002 0.000000

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Fig. 1. Graphical display for several values of s when M = 2 and m = 2 (axisymmetric shrinking). (a) f; (b) f 0 and (c) f 00 .

verify the effect of the integration step size Dg, the code was run with several step sizes, and in each case, the numerical results exhibited excellent consistency; therefore, a grid sensitivity analysis is undertaken, as shown in Table 1. Very recently, the shapes of the velocity profiles f 0 (g) have been subject to some controversy. In a recent work [28], when suction is strong (i.e., s = 1.8), the velocity profiles show an ‘‘overshooting’’ shape, increasing from the prescribed value f 0 (1) = 0 at the outer edge of the boundary layer to a maximum at a certain distance from the wall in the interval [0, g1] (a little overshoot above the free stream occurs). Later, Fernández [29] discussed this behaviour and established that the

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Fig. 2. Graphical display for several values of s when M = 2 and m = 1(two-dimensional shrinking). (a) f; (b) f 0 and (c) f 00 (curves as in Fig. 1c).

above-mentioned excessive maximum is not appeared. Moreover, in this ‘‘overshooting’’ range, f 0 (g) will be positive and this is not possible (see [26]) for the stated problem. The later is strengthened in this work by displaying Figs. 1, 2 (see also Table 1) from which one easily can see that, at M = 2, neither for m = 2 nor m = 1 the velocities exhibit a maximum. On the contrary, velocity distributions are always negative in the integration domain [0, g1], f 0 (g) and f 00 (g) tend simultaneously to zero at infinity in an asymptotical fashion and f(g) tends to a constant value f(1) at the outer edge of the boundary layer; moreover, as was expected for the flow induced by a shrinking wall, the entrainment velocity f(1) becomes negative when suction is low (i.e., s = 0.2). It is also immediately seen from Fig. 1a and Fig. 2a that for non-zero magnetic field, namely M = 2, a higher magnitude of the entrainment velocity f(1) is found for the case m = 2 as compared the other one (m = 1). One can

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R. Cortell / Applied Mathematics and Computation 217 (2010) 4086–4093 Table 2 Some momentum transfer characteristics for m = 2 (axisymmetric shrinking), m = 1 (two-dimensional shrinking) and several values of s at M = 2. Parentheses indicate earlier results from open literature. M

m

s

f 00 (0)

f(1)

2

2

2

1

1.8 1 0.2 1.8 1 0.2

4.20411325 (4.20411340) [29] 2.89160454 (2.89160) [28] 1.84287373 2.85192169 2.30277550 (2.30277) [28] 1.83493537

1.558986 0.645171 0.370584 1.449361 0.565805 0.344954

Table 3 Some momentum transfer characteristics for m = 1, 2 (two-dimensional and axisymmetric shrinking, respectively) and pffiffiffi several values of s at M ¼ 0:5; 5. Parentheses indicate earlier results from open literature (see [30]). M

m

s

f 00 (0)

f(1)

0.5

1

pffiffiffi 5

2

1.75 1.8 1.85 1.9 2

1.0 (1.0) 1.1449427 (1.1449) 1.25 (1.25) 1.3405083 (1.3405) 4.7461389 (4.7461)

0.75 0.926566 1.05 1.153989 1.787122

3

6.5387146 (6.5387)

2.846200

Table 4 Magnetic field effects on momentum transfer characteristics for m = 1 (two-dimensional shrinking) and s = 1.8. m

s

M

f 00 (0)

f(1)

1

1.8

0.5 2

1.14494275 2.85192169

0.9265655 1.4493610

Fig. 3. Velocity profiles for various values of s and M at m = 1 (two-dimensional shrinking).

also graphically observe that the integration domain shrinks with increasing values of the parameter s and then the magnitude of f(1) increases. In other words, the momentum boundary-layer thicknesses depress with suction parameter s, and further the velocity boundary-layer thickness for the axisymmetrically case (m = 2) are always smaller than those of the two-dimensional case (m = 1). pffiffiffiffiffiffi From third Eq. (7) we found w1 ¼  atmf ð1Þ, and then quantities of physical interest such as the skin friction coeffi00 cient f (0) and the entrainment velocity f(1) which is related to w1 and the amount of fluid dragged by the shrinking wall are gathered in Table 2 for the case of non-zero magnetic field, namely M = 2. From this Table, one immediately observes a very good agreement between present results and the earlier ones from open literature. Irrespective of m and taking negative values of f(1) in absolute sense, one can easily see that a stronger suction produces an augment in both the magnitudes of f 00 (0) andf(1). Also, the later are greater for m = 2 than for m = 1.

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R. Cortell / Applied Mathematics and Computation 217 (2010) 4086–4093

The effect of Hartmann number on velocity profiles are shown in Tables 3, 4 and the accompanying Fig. 3. It can be seen from numerical results (Table 4) that the skin friction coefficient f 00 (0) and the entrainment velocity f(1) increase with increasing M. From these behaviours and from Fig. 3 it can be also seen that the combined effect of increasing values of the parameters s (suction parameter) and M (Hartman number) is to increase the velocity profiles and then the f 0 (g) profiles become closer to the surface. Moreover, for fixed M, the effect of increasing suction is to boost the entrainment velocity f(1). 4. Discussions and conclusions This paper analyzes numerically the problem of steady flow induced by a shrinking sheet with suction. The similarity transformations are employed to transform the partial boundary layer equations into an ordinary differential equation. Numerical solutions for momentum transfer are obtained by employing a shooting 4th order Runge–Kutta algorithm. Results obtained are compared quantitatively with the previously published results and the agreements are found to be very good. A strong accent has been placed on the role played by the augmented boundary condition at infinity f 00 (g1) = 0 from which very accurate numerical results can be reached. In this manner, all our velocity profiles tend to zero at infinity asymptotically. Furthermore, and for each studied case, the special values of the parameters entering the stated problem fix the integration domain, and the latter is obtained as a natural part of our numerical approach (see Table 1), and there is no necessity to select the g1 value before calculations. On the contrary, if one enforces the far field conditions by fixing (before calculations) an inadequate (small) finite integration domain (i.e., the g1 value), the accuracy of the results could be severely contaminated and then the profiles do not approach the horizontal axis asymptotically and intersect it. Reviewed influences on both the shapes of the velocity profiles and the shrinking flow characteristics (enlarged to the similar entrainment velocity f(1)) of the parameters s, m and M were examined. From our numerical results the following conclusions may be drawn: 1. The velocity profiles f 0 (g) are monotonically increasing, negative, and bounded. 2. The dimensionless stream function f, its derivatives and the similar entrainment velocity f(1) are affected with the mass suction parameter s. As shown, for fixed M, both the velocity profile and f(1) increase with the mass suction parameter. 3. The boundary layer thicknesses for the axisymmetrically case (m = 2) are always smaller than those of the two-dimensional case (m = 1). 4. The effect of increasing M is to increase both the similar entrainment velocity f(1) and the skin friction coefficient f 00 (0).

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