Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

International Journal of Non-Linear Mechanics 46 (2011) 938–941

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Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field F. Labropulu  Luther College—Mathematics, University of Regina, Regina, SK, Canada S4S 0A2

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 December 2008 Accepted 5 April 2011 Available online 15 April 2011

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y¼ 0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann’s number. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Unsteady Newtonian Magnetic field Oscillations

1. Introduction The flow of an incompressible viscous fluid over a moving plate has its importance in many industrial applications. The extrusion of plastic sheets, fabrication of adhesive tapes and application of coating layers onto rigid substrates are some of the examples. If a magnetic field is present, viscous flows due to a moving plate in an electro-magnetic field, i.e. magnetohydrodymanic (MHD) flows, are relevant to many engineering applications such as petroleum engineering, chemical engineering, MHD power generators, MHD pumps, heat exchangers and metallurgy industry. Examples of MHD flows in the metallurgy industry include the cooling of continuous strips and filaments drawn through a quiescent fluid and the purification of molten metals from non-metallic inclusions. In the history of fluid dynamics, considerable attention has been given to the study of two-dimensional stagnation-point flow. Hiemenz [1] derived an exact solution of the steady flow of a Newtonian fluid impinging orthogonally on an infinite plate. Stuart [2], Tamada [3] and Dorrepaal [4] independently investigated the solutions of a stagnation-point flow when the fluid impinges on the plate obliquely. The flow over a moving infinite plate is essentially a two-dimensional stagnation-point flow. This consists of a class of flows in the vicinity of a stagnation line that results from a two-dimensional flow impinging on a surface at right angles and flowing thereafter symmetrically about the stagnation-point. Furthermore, the unsteady or time dependent

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viscous flow near a stagnation-point has also been widely investigated. Glauert [5] and Rott [6] studied the stagnation-point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane. The Hiemenz flow of a Newtonian fluid in the presence of a magnetic field was first considered by Na [7] and later on by Ariel [8]. In this paper, we consider the unsteady two-dimensional flow of a viscous incompressible fluid impinging on an infinite plate in the presence of a magnetic field. The plate is assumed to make harmonic oscillations in its own plane. The magnetic field is assumed to be transverse or perpendicular everywhere in the flow field. With the use of a transformation, the governing equations are reduced to a system of ordinary differential equations. Solutions for small frequencies of the oscillations are obtained using a finite difference technique. Solutions for large frequencies are found using a series expansion.

2. Flow equations The two-dimensional flow of a viscous incompressible fluid in the presence of a magnetic field is governed by @u @v þ ¼0 @x @y

ð1Þ

@u @u @u 1 @p sB0 2 þu þv þ ¼ nr u u @t @x @y r @x r

ð2Þ

@v @v @v 1 @p þu þv þ ¼ nr2 v @t @x @y r @y

ð3Þ

F. Labropulu / International Journal of Non-Linear Mechanics 46 (2011) 938–941

where u ¼ uðx,y,tÞ, v ¼ vðx,y,tÞ are the velocity components, p ¼ pðx,y,tÞ is the pressure, n ¼ m=r is the kinematic viscosity, s is the electrical conductivity and B0 is the magnetic field. It is assumed that sB0 5 1, so that it is possible to neglect the effect of the induced magnetic field. The continuity Eq. (1) implies the existence of a streamfunction c ¼ cðx,y,tÞ such that u¼

@c , @y

v¼

@c @x

ð4Þ

Substitution of Eq. (4) in Eqs. (2) and (3) and elimination of pressure from the resulting equations using pxy ¼ pyx yields @ @ðc, r2 cÞ sB0 @2 c ðr2 cÞ nr4 c þ ¼0 @t @ðx,yÞ r @y2

ð5Þ

Having obtained a solution of Eq. (5), the velocity components are given by (4) and the pressure can be found by integrating Eqs. (2) and (3).

939

and G000 þ FG00 F 0 G0 MG0  G0 ð0Þ ¼ 1,

io 0 G ¼0 c

G0 ð1Þ ¼ 0

ð15Þ

When M¼0, the solution of system (14) corresponds to the well-known Hiemenz flow [1]. System (14) is solved numerically using the shooting method with a finite difference technique for different values of M. We found that for M ¼ 0, F 00 ð0Þ ¼ 1:23259 which is in good agreement with the Hiemenz solution [1] and Glauert’s solution [5]. Numerical values of F 00 ð0Þ for different values of M are given in Table 1 and they are in good agreement with the values obtained by Ariel [8]. Fig. 1 shows the profiles of F 0 ðZÞ for various values of M. Letting fðZÞ ¼ G0 ðZÞ, then system (15) becomes

f00 þF f0 F 0 fMf

io f¼0 c

fð0Þ ¼ 1, fð1Þ ¼ 0 3. Solutions We consider the two-dimensional flow of an incompressible fluid against an infinite plate normal to the flow. We assume that the plate is making harmonic oscillations on its own plane and its velocity in the x-direction is aeiot where a and o are constants. The boundary conditions are then given by @c ¼ aeiot , @y @c ¼ cx @y

@c ¼0 @x

at y ¼ 0

ð6Þ

ð16Þ

The only parameter in Eq. (16) is the frequency ratio o=c. In the following subsections, series solutions will be developed, valid for small and large values of o=c respectively. 3.1. Small values of o=c First, consider the case where o ¼ 0, which implies that the plate velocity has the constant value a. Letting f ¼ f0 , then system (16) gives

f000 þF f00 F 0 f0 Mf0 ¼ 0 as y-1

ð7Þ

f0 ð0Þ ¼ 1, f0 ð1Þ ¼ 0

ð17Þ

Following Glauert [5], we assume that

c ¼ cxf ðyÞ þ aeiot gðyÞ

ð8Þ

Table 1 Numerical values of F 00 ð0Þ, f00 ð0Þ, f01 ð0Þ and f02 ð0Þ for different values of M.

The boundary conditions take the form f ð0Þ ¼ f 0 ð0Þ ¼ 0, f 0 ð1Þ ¼ 1,

g 0 ð0Þ ¼ 1

g 0 ð1Þ ¼ 0

ð9Þ

Using Eq. (8) in (5), we obtain

nf ðivÞ þcff 000 cf 0 f 00 

sB0 00 f ¼0 r

ng ðivÞ þcfg 000 cf 00 g 0 iog 00 

sB0 00 g ¼0 r

M

F 00 ð0Þ

f00 ð0Þ

f01 ð0Þ

f02 ð0Þ

0.0 0.5 1.0 2.0 2.25 3.0

1.23259 1.41976 1.58533 1.87353 1.93895 2.11589

 0.811318  1.05648  1.2615  1.60113  1.67603  1.88287

 0.49307  0.40999  0.357149  0.29228  0.280805  0.253213

0.0945276 0.0945488 0.0582149 0.0400542 0.020476 0.015276

ð10Þ

Non-dimensionalizing system (10) and the boundary conditions (9) using rffiffiffi rffiffiffi rffiffiffi c n n Z¼ ð11Þ y, f ðyÞ ¼ FðZÞ, gðyÞ ¼ GðZÞ n c c we get F ðivÞ þ FF 000 F 0 F 00 MF 00 ¼ 0

ð12Þ

and GðivÞ þFG000 F 00 G0 MG00 

io 00 G ¼0 c

ð13Þ

where M ¼ sB0 =rc is the Hartmann’s number. Integrating Eqs. (12) and (13) once with respect to Z and using the conditions at infinity, we have F 000 þ FF 00 F 02 MF 0 ¼ 1M Fð0Þ ¼ 0,

F 0 ð0Þ ¼ 0,

F 0 ð1Þ ¼ 1

ð14Þ

Fig. 1. Variation of F 0 ðZÞ with M.

940

F. Labropulu / International Journal of Non-Linear Mechanics 46 (2011) 938–941

Fig. 2. Variation of f0 ðZÞ with M. Fig. 3. Variation of f1 ðZÞ with M.

This system is solved numerically using a shooting method and it is found that for M ¼0, f00 ð0Þ ¼ 0:811318 which is in good agreement with the value obtained by Glauert [5]. Numerical values of f00 ð0Þ for different values of M are shown in Table 1. Fig. 2 depicts the profiles of f0 ðZÞ for various values of M. For small but non-zero values of o=c, we let   2 1  X io n io io fðZÞ ¼ fn ðZÞ ¼ f0 ðZÞ þ f1 ðZÞ þ f2 ðZÞ þ    c c c n¼0 ð18Þ Substituting (18) into (16), we get for n Z 1

f00n þ F f0n F 0 fn Mfn ¼ fn1 fn ð0Þ ¼ 0, fn ð1Þ ¼ 0

ð19Þ

Numerical integration of system (19) for n ¼1 using a finite difference technique gives for M¼0, f01 ð0Þ ¼ 0:49307 which is in good agreement with Glauert’s value [5]. Numerical values of f01 ð0Þ for different values of M are shown in Table 1. Fig. 3 shows the profiles of f1 ðZÞ for various values of M. Numerical integration of system (19) for n ¼2 using a finite difference technique gives for M¼0, f02 ð0Þ ¼ 0:0945488 which is in good agreement with Glauert’s value [5]. Numerical values of f02 ð0Þ for different values of M are shown in Table 1. Fig. 4 depicts the profiles of f2 ðZÞ for various values of M. The oscillating component of the shear stress on the wall is given by rffiffiffiffiffi   t12 cn iot 0 io 0 ¼ f ð0Þ þ f ð0Þ ð20Þ e 0 c 1 ra2 a2

Fig. 4. Variation of f2 ðZÞ with M.

fð0Þ ¼ 1, fð1Þ ¼ 0

ð23Þ

The expansion for F(Y) near the wall Y ¼0 is FðZÞ ¼

1 2 2 1 1 Aa Y  ð1þ MÞa3 Y 3 þ MAa4 Y 4 2 6 24 1 ðA2 M2 MÞa5 Y 5 þ    þ 120

ð24Þ

where f00 ð0Þ and f01 ð0Þ are given in Table 1 for different values of M. When M ¼0, the value of the shear stress on the wall is in good agreement with the value obtained by Glauert [5].

which is valid for small Hartmann’s number. In this expansion, A ¼ F 00 ð0Þ. Since for large values of o=c the parameter a is small, we let

3.2. Large values of o=c

f¼

1 X

an fn ðYÞ ¼ f0 ðYÞ þ af1 ðYÞ þ a2 f2 ðYÞ þ   

ð25Þ

n¼0

Solutions for large values of o=c have been obtained by Labropulu [9] but for completion they are also outlined here. When o=c is large, we let rffiffiffiffiffiffi rffiffiffiffiffiffi io io Z¼ y ð21Þ Y¼ c n pffiffiffiffiffiffiffiffiffiffiffi Letting io=c ¼ a, then d=dZ ¼ ð1=aÞðd=dYÞ and system (16) takes the form   d2 f df dF  þ a F f ð22Þ fa2 M f ¼ 0 2 dY dY dY

The boundary conditions (23) become

f0 ð0Þ ¼ 1, fn ð0Þ ¼ 0 if n Z1, fn ð1Þ ¼ 0 for all n

ð26Þ

Substituting Eq. (25) in (22) and equating the coefficients of different powers of a to zero, we find that the boundary-value problem for f0 ðYÞ is d2 f0 dY

2

f0 ¼ 0,

f0 ð0Þ ¼ 1, f0 ð1Þ ¼ 0

with solution f0 ðYÞ ¼ expðYÞ.

ð27Þ

F. Labropulu / International Journal of Non-Linear Mechanics 46 (2011) 938–941

The second equation gives that f1 is zero. The next four equations for f2 ðYÞ, f3 ðYÞ, f4 ðYÞ and f5 ðYÞ are d2 f2 dY

2

d2 f3 dY

2

d2 f4 dY

2

d2 f5 dY

2

f2 ¼ M f0 1 df0 þ AY f0 f3 ¼ M f1  AY 2 2 dY f4 ¼ M f2 þ

1 df0 1 2 df1 1 ð1 þ mÞY 3  AY þ AY f1  ð1þ MÞY 2 f0 6 2 2 dY dY

1 df2 1 df1 1 df0 MAY 4 þ ð1þ MÞY 3  þ AY f2 f5 ¼ M f3  AY 2 2 6 24 dY dY dY 1 1  ð1 þ MÞY 2 f1 þ MAY 3 f0 2 6

ð28Þ

Solving these equations and using the boundary conditions, we obtain M Y Ye 2   1 3 3 2 3 Y þ Y þ Y f3 ðYÞ ¼ AeY 12 8 8

f2 ðYÞ ¼ 

   3 M2 3 M2 2 ð1þ MÞ þ ð1 þMÞ þ Yþ Y 16 16 2 2  M þ 1 3 M þ1 4 Y þ Y þ 8 48   1 5 1 4 1 3 9 2 9 Y þ Y þ Y þ Y þ Y f5 ðYÞ ¼ MAeY  240 96 8 32 32

f4 ðYÞ ¼ eY

941

to many applications in metallurgy, such as the cooling of continuous strips and filaments drawn though a quiescent fluid. Another interesting application lies in the purification of molten metals from non-metallic inclusions by the application of a magnetic field [10]. Results for this flow are obtained for various values of the Hartmann’s number M. At the higher frequencies, the perturbation is a shear layer, exactly as on a plate oscillating in a fluid at rest. Fig. 1 shows the variation of F 0 ðZÞ for various Hartmann’s number M. The effect of the Hartmann’s number is to increase the velocity F 0 ðZÞ near the wall as it increases. This velocity increase is due to the presence of Lorentz force with a non-vanishing electric filed, which results in acceleration of the flow. Table 1 shows that F 00 ð0Þ is increasing with increasing M. Fig. 2 shows that f0 ðZÞ decreases near the wall as M is increasing. Fig. 3 shows that f1 ðZÞ increases near the wall as M is increasing and Fig. 4 shows that f2 ðZÞ decreases near the wall as M is increasing. Also, from Table 1, F 00 ð0Þ increases with the magnetic parameter M. The reason for this behaviour is that the magnetic field B0 induces a force along the surface which supports the motion. As a result, the velocity along the surface increases everywhere and hence the shear stress on the wall increases with increasing Hartmann’s number.



If M ¼0, results obtained by Glauert [5] are recovered. The oscillating component of the shear stress on the wall is given by rffiffiffiffiffi   t12 cn iot 1 M a 3Aa2 þ þ    ð29Þ ¼  þ e a 2 8 ra2 a2

4. Conclusions The unsteady stagnation-point flow of a viscous fluid in the presence of a magnetic field is examined. Such flows are relevant

References [1] K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytech. J. 326 (1911) 321. [2] J.T. Stuart, The viscous flow near a stagnation point when the external flow has uniform vorticity, J. Aerosp. Sci. 26 (1959) 124. [3] K.J. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn 46 (1979) 310. [4] J.M. Dorrepaal, An exact solution of the Navier–Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions, J. Fluid Mech. 163 (1986) 141. [5] M.B. Glauert, The laminar boundary layer on oscillating plates and cylinders, J. Fluid Mech. 1 (1956) 97–110. [6] N. Rott, Unsteady viscous flow in the vicinity of a stagnation point, Q. Appl. Math. 13 (1956) 444–451. [7] T.Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press, New York, 1979. [8] P.D. Ariel, Hiemenz flow in hydromagnetics, Acta Mech. 103 (1994) 31–43. [9] F. Labropulu, Unsteady MHD stagnation-point flow, IASME Trans. 2 (7) (2005) 1166–1170. [10] A.D. Barinberg, A.B. Kapusta, B.V. Chekin, Magn. Gidrodin. 11 (1975) 111 (English translation).