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Unsteady MHD flow near a rotating porous disk with uniform suction or injection
Fluid Dynamics Research 23 (1998) 283–290
Unsteady MHD ow near a rotating porous disk with uniform suction or injection Hazem Ali Attia∗ Faculty of Engineering, Cairo University, El-Fayoum Branch, Cairo, Egypt Received 11 December 1996; revised 7 April 1997; accepted 31 July 1997
Abstract In this paper, the MHD ow of an incompressible, viscous and electrically conducting uid above an in nite rotating porous disk is extended to ow starting impulsively from rest. The uid is subjected to an external uniform magnetic eld perpendicular to the plane of the disk. The e ects of uniform suction or injection through the disk on the unsteady MHD ow are also considered. The governing nonlinear partial di erential equations are solved numerically using the nite-di erence approximations with a special treatment of the discontinuity between the initial and boundary conditions. The results of the numerical solution indicate some interesting e ects of the magnetic eld and the suction or injection c 1998 The Japan velocities on the ow and prove that the steady state can be reached via a time-dependent process. Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
1. Introduction Rotating-disk ow is one of the classical problems of uid mechanics which has both theoretical and practical value. The rotating-disk problem was rst formulated by von Karman (1921). He has shown that the partial di erential equations of steady ow of a viscous incompressible uid due to an in nite rotating disk can be reduced to a set of ordinary di erential equations and solved them by an approximate integral method. Later, Cochran (1934) obtained more accurate results by patching two series expansions. Benton (1966) improved Cochran’s solution and extended the hydrodynamic problem to ow starting impulsively from rest. Numerical solutions have been obtained by Rogers and Lance (1962) for the case of rotation at in nity. Stuart (1954) introduced suction through the disk and solved the equations with zero rotation at in nity. Ockendon (1972) used asymptotic methods to determine the solution of the problem for small values of the suction parameter and in the case of rotation at in nity. The e ect of uniform blowing through a rotating porous disk on the
ow induced by this disk was studied by Kuiken (1971). Some interesting e ects of the magnetic ∗
Correspondence address: Faculty of Education for Women, Ibri, P.O. Box 14, Oman, Oman.
c 1998 The Japan Society of Fluid Mechanics and 0169-5983/98/$19.00 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 9 7 ) 0 0 0 3 5 - X
284
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
eld on the steady ow due to the rotation of a disk of in nite or nite extent was examined by El-Mistikawy et al. (1990, 1991). In the present paper, the unsteady MHD laminar ow of a viscous, conducting, and incompressible
uid due to the uniform rotation of an insulating porous disk of in nite extent is studied in the presence of an external uniform magnetic eld directed perpendicular to the disk and uniform suction or injection through the disk. The induced magnetic eld is neglected by assuming that the magnetic Reynolds number Rem 1 (El-Mistikawy et al., 1990). A uniform suction or injection through the disk is considered for the whole range of suction or injection velocities, respectively. The governing non-linear partial di erential equations are integrated numerically using the nitedi erence approximations. Starting the motion from rest leads to a discontinuity between the initial and boundary conditions which results in a numerical oscillation problem. Removal of these oscillations is done by making suitable coordinate transformations. Some interesting e ects of the applied uniform magnetic eld together with uniform suction or injection through the disk on the time development of the unsteady ow are presented and discussed. 2. Basic equations In cylindrical polar coordinates (r; ; z), the components of the ow velocity are (u; v; w) in the direction of increasing (r; ; z) respectively, the pressure is p, the density of the uid is , the kinematic viscosity is , and the electrical conductivity is . The external uniform magnetic eld is applied perpendicular to the plane of the disk and has a constant magnetic ux density B0 which is assumed unaltered by taking Rem 1. Mass transfer to, or from, the uid at the surface of the disk, either by direct injection or suction, is considered over the entire range from large injection velocities to large suction velocities. The continuity and time-dependent Navier–Stokes equations take the form ur +
u + wz = 0; r
(1)
ut + uur −
1 u v2 B02 ur + wuz + u + pr = urr + − 2 + uzz ; r r r
vt + uvr +
v uv B02 vr + wvz + v = vrr + − 2 + vzz ; r r r
(2)
(3)
1 wr + wzz : wt + uwr + wwz + pz = wrr + r
(4)
Appropriate initial and boundary conditions for the ow velocity which is started impulsively from rest, due to the rotation of the disk, into steady rotation with constant angular velocity ! and a uniform ow w0 through the disk, are given by at t = 0;
u = 0;
v = 0;
at z = 0;
u = 0;
w = w0 ;
as z → ∞
u; v → 0:
w = w0 ; v = r!;
(5a) (5b) (5c)
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285
By introducing von Karman transformations, u = r!F();
v = r!G();
z = (=!)1=2 ;
w = (!)1=2 H ();
p = −!P;
where is a non-dimensional distance measured along the axis of rotation and F; G; H , and P are non-dimensional functions of , we de ne the magnetic interaction number by = B02 =!, which represents the ratio between the magnetic force to the uid inertia force. Due to uniform suction or injection, the vertical velocity component H takes a constant non-zero value at = 0. With these de nitions, Eqs. (1)–(5) take the form @H + 2F = 0; @
(6)
@2F @F @F − 2 +H + F 2 − G 2 + F = 0; @t @ @
(7)
@G @G @ 2 G − 2 +H + 2FG + G = 0; @t @ @
(8)
@2H dP @H @H − 2 +H − = 0; @t @ @ d
(9)
with the initial and boundary conditions on the ow velocities given by F(0; ) = 0;
G(0; ) = 0;
F(t; 0) = 0;
G(t; 0) = 1;
F(t; ∞) = 0;
G(t; ∞) = 0;
H (0; ) = a; H (t; 0) = a;
(10a) (10b) (10c)
where “a” is the suction or injection parameter, which takes negative values for suction and positive values for injection. The above system of equations (6)–(8) with the prescribed initial and boundary conditions given by Eq. (10) are sucient to solve for the three components of the ow velocity. Eq. (9) can be used to solve for the pressure distribution if required. 3. The numerical solution Numerical solution for the governing non-linear equations (6)–(8), using the nite-di erence approximations, leads to a numerical oscillation problem resulting from the discontinuity between the initial and boundary conditions (10a) and (10b). A solution for this numerical problem is achieved by using proper coordinate transformation, as suggested by Ames √(1977) for similar problems. Expressing Eqs. (6)–(8) in terms of the modi ed coordinate = =2 t we get √ @H + 4 tF = 0; @
(11)
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H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
@F 1 @2F 1 @F @F − − + F 2 − G 2 + F = 0; + √ H @t 2t @ 4t @2 @ 2 t
(12)
@G 1 @2G 1 @G @G − − + 2FG + G = 0: + √ H 2 @t 2t @ 4t @ 2 t @
(13)
The system of non-linear PDEs (11)–(13) is solved using a marching technique and by applying the Crank–Nicolson implicit method (Ames, 1977). The resulting system of di erence equations has to be solved in the in nite domain 0¡¡∞ and 0¡t¡∞. A nite domain in the -direction can be used instead with chosen large enough to ensure that the solutions are not a ected by imposing the asymptotic conditions at a nite distance. However, due to the suggested coordinate transformation, this nite domain diminishes with the progression of time and greatly a ects the accuracy of the numerical solution. To substitute for this problem, the modi ed Eqs. (11)– (13) are integrated from t = 0 to t = 1. Then, the solution obtained at t = 1 is used as the initial condition for integrating Eqs. (6)–(8) from t = 1 towards the steady state. 4. Results and discussion The time growth of the axial velocity component at in nity H∞ for = 0 is shown in Fig. 1a for di erent values of the suction or injection parameters. Increasing the suction parameter “a” negatively from 0 to −2 leads to a great increment in the axial velocity towards the disk and a decrement in its growth time up to the steady state. For small values of the suction parameter (a = 0 to −0:5), the e ects of the radial ow besides the suction through the disk are to increase the quantity of
uid drawn with time. Two paths are available for the incoming uid, one through the suction holes of the disk and another in the radial direction. Higher values of the suction velocity provide an easier path for the ow through the wall than that in the radial direction and result in an almost time-independent in owing stream towards the disk. For the case of injection, H∞ initially equals the injection velocity and for few radians of the rotation of the disk (1.5 rad for a = 1) it keeps its positive direction. With time progression, the radially ejected uid pumps the ow towards the disk
Fig. 1. (a), (b) Time development of the axial velocity at in nity.
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
287
Fig. 2. (a) The steady-state azimuthal velocity pro le versus “a”. (b) The steady-state radial velocity pro le versus “a”. (c) The steady-state axial velocity pro le versus “a”.
and a crossover point in time that depends on the injection parameter “a” occurs and the axial ow reverses direction till its steady-state value. The greater the injection velocity the more strongly is the in ow opposed and, consequently, the later appearance of the crossover of the axial velocity. Fig. 1b indicates the e ect of the magnetic eld on H∞ for di erent values of the suction or injection parameters with = 1. The magnetic eld leads generally to a reduction in the axial velocity at in nity and its growth time. For the case of uniform suction (a = 0 to −2), the magnetic eld has an apparent e ect on the ow for small values of the suction parameter while its e ect is negligible for large values of “a” (a = −2). In the case of uniform injection (a = 0 to 2), the magnetic eld ( = 1) restrains the azimuthal and radial ows and then supports the out owing stream of injected
ow to stop completely the drawn in ow and prevent the occurrence of the crossover points during time progression. In the presence of the magnetic eld, varying the injection velocity becomes more e ective on the ow than in the non-magnetic case. The magnetic eld has a marked e ect on the
288
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
Fig. 3. (a) The steady-state azimuthal velocity pro le versus “a”. (b) The steady-state radial velocity pro le versus “a”. (c) The steady-state axial velocity pro le versus “a”.
ow for the whole range of the injection velocities. However, its e ect is much lower for small suction velocities and almost neglected for higher suction velocities. Fig. 2 presents the in uence of the axial ow at the disk surface on the steady-state velocity pro les for the case of suction or injection with = 0. Increasing the suction velocity (a = 0 to −2) leads to a rapid decrease in the azimuthal and radial velocity components as shown in Figs. 2a and 2b, while Fig. 2c indicates that the axial ow at in nity towards the disk is larger. Increasing the injection velocity (a = 0 to 2) leads to an increase in the azimuthal and radial ows as shown in Figs. 2a and 2b, while Fig. 2c shows that the axial ow towards the disk is smaller. With increasing injection velocity, the out ow penetrates to greater distances from the disk surface. Consequently, the crossover point between the positive and negative axial velocity is pushed farther outward in the -direction as shown in Fig. 2c. In Fig. 2a, it is seen that the uid injection gives rise to the familiar in ection-point pro les, especially for high values of the injection parameter “a”. Hence, high injection velocities
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
289
are expected to destabilize the laminar ow and lead to transition to turbulence. In Fig. 2b, it is shown that for high values of the injection parameter (a = 2), the radial velocity near the disk (for small values of ) is lower than that for smaller values of “a”. This is due to the fact that, with increasing “a”, the injected ow can sustain axial motion to greater distances from the wall. Then, near the wall, the radial ow which is fed by that axial ow is expected to decrease as the injection parameter increases. The in uence of the magnetic eld, = 1, on the steady-state velocity pro les for di erent values of the suction or injection parameters is shown in Fig. 3. The magnetic eld e ect is to sustain the ow in the azimuthal, radial, and axial directions. It reduces the axial ow towards the disk for small suction velocities and its e ect becomes negligible for larger suction velocities as shown in Fig. 3c. When injection is applied, the magnetic eld reduces the azimuthal and radial ows and, consequently, the injection stream sustains its axial motion towards the disk. Fig. 3c shows the in uence of the magnetic eld in the suppression of the crossover of the axial component of velocity and then reversal of the direction of the axial motion. In Fig. 3a, it is clear that the magnetic eld has a marked e ect in changing the shape of the in ection-point pro les in the case of high injection velocities. Consequently, the magnetic eld works to stabilize the laminar boundary layer and prevents the transition to turbulence. It is shown in Fig. 3b that the reduction in radial velocity near the wall for high values of the injection velocity disappeared in the presence of the magnetic eld. The magnetic eld works to sustain globally the radial ow and the axial ow feeding it. As the injection velocity increases, it becomes the main source for the radial motion, and consequently, leads to an increment in the radial velocity for all distances above the disk. 5. Conclusions In this paper, the e ects of uniform suction or injection on the unsteady MHD ow induced by an in nite rotating porous disk were studied. The results proved that the steady-state solution is approached as the asymptotic development of a time-dependent process and presented some interesting e ects. The most interesting result is the e ect of the magnetic force on the suppression of the crossover of the axial component of velocity in the non-magnetic case with uniform injection. The magnetic eld has a marked e ect on the ow for the whole range of the injection velocities. However, its in uence on the ow with suction decreases greatly by increasing the suction velocity. The magnetic eld has, in general, an apparent e ect on the ow in the case of uniform injection more than that in the case of uniform suction. References Ames, W.F., 1977. Numerical Methods in Partial Di erential Equations. 2nd ed., Academic Press, New York. Benton, E.R., 1966. On the ow due to a rotating disk. J. Fluid Mech. 24 (4), 781– 800. Cochran, W.G., 1934. The ow due to a rotating disk. Proc. Cambridge Philos. Soc. 30 (3) 365 – 375. El-Mistikawy, T.M.A., Attia, H.A., 1990. The rotating disk ow in the presence of strong magnetic eld. Proc. 3rd Int. Congr. of Fluid Mechanics, Cairo, Egypt, vol. 3, 2 – 4 January, pp. 1211–1222. El-Mistikawy, T.M.A., Attia, H.A., Megahed, A.A., 1991. The rotating disk ow in the presence of weak magnetic eld. Proc. 4th Conf. on Theoretical and Applied Mechanics, Cairo, Egypt, 5 – 7 November, pp. 69 – 82.
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H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
Kuiken, H.K., 1971. The e ect of normal blowing on the ow near a rotating disk of in nite extent. J. Fluid Mech. 47 (4), 789 – 798. Lance, G.N., Rogers, M.H., 1962. The axial symmetric ow of a viscous uid between two in nite rotating disks. Proc. Roy. Soc. A 266, 109. Ockendon, H., 1972. An asymptotic solution for steady ow above an in nite rotating disk with suction. Quart. J. Mech. Appl. Math. XXV, 291– 301. Stuart, J.T., 1954. On the e ects of uniform suction on the steady ow due to a rotating disk. Quart. J. Mech. Appl. Math. 7, 446 – 457. von Karman, T., 1921. Uber laminare und turbulente reibung. 1 (4), 233 – 235.
Unsteady MHD ow near a rotating porous disk with uniform suction or injection Hazem Ali Attia∗ Faculty of Engineering, Cairo University, El-Fayoum Branch, Cairo, Egypt Received 11 December 1996; revised 7 April 1997; accepted 31 July 1997
Abstract In this paper, the MHD ow of an incompressible, viscous and electrically conducting uid above an in nite rotating porous disk is extended to ow starting impulsively from rest. The uid is subjected to an external uniform magnetic eld perpendicular to the plane of the disk. The e ects of uniform suction or injection through the disk on the unsteady MHD ow are also considered. The governing nonlinear partial di erential equations are solved numerically using the nite-di erence approximations with a special treatment of the discontinuity between the initial and boundary conditions. The results of the numerical solution indicate some interesting e ects of the magnetic eld and the suction or injection c 1998 The Japan velocities on the ow and prove that the steady state can be reached via a time-dependent process. Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
1. Introduction Rotating-disk ow is one of the classical problems of uid mechanics which has both theoretical and practical value. The rotating-disk problem was rst formulated by von Karman (1921). He has shown that the partial di erential equations of steady ow of a viscous incompressible uid due to an in nite rotating disk can be reduced to a set of ordinary di erential equations and solved them by an approximate integral method. Later, Cochran (1934) obtained more accurate results by patching two series expansions. Benton (1966) improved Cochran’s solution and extended the hydrodynamic problem to ow starting impulsively from rest. Numerical solutions have been obtained by Rogers and Lance (1962) for the case of rotation at in nity. Stuart (1954) introduced suction through the disk and solved the equations with zero rotation at in nity. Ockendon (1972) used asymptotic methods to determine the solution of the problem for small values of the suction parameter and in the case of rotation at in nity. The e ect of uniform blowing through a rotating porous disk on the
ow induced by this disk was studied by Kuiken (1971). Some interesting e ects of the magnetic ∗
Correspondence address: Faculty of Education for Women, Ibri, P.O. Box 14, Oman, Oman.
c 1998 The Japan Society of Fluid Mechanics and 0169-5983/98/$19.00 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 9 7 ) 0 0 0 3 5 - X
284
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
eld on the steady ow due to the rotation of a disk of in nite or nite extent was examined by El-Mistikawy et al. (1990, 1991). In the present paper, the unsteady MHD laminar ow of a viscous, conducting, and incompressible
uid due to the uniform rotation of an insulating porous disk of in nite extent is studied in the presence of an external uniform magnetic eld directed perpendicular to the disk and uniform suction or injection through the disk. The induced magnetic eld is neglected by assuming that the magnetic Reynolds number Rem 1 (El-Mistikawy et al., 1990). A uniform suction or injection through the disk is considered for the whole range of suction or injection velocities, respectively. The governing non-linear partial di erential equations are integrated numerically using the nitedi erence approximations. Starting the motion from rest leads to a discontinuity between the initial and boundary conditions which results in a numerical oscillation problem. Removal of these oscillations is done by making suitable coordinate transformations. Some interesting e ects of the applied uniform magnetic eld together with uniform suction or injection through the disk on the time development of the unsteady ow are presented and discussed. 2. Basic equations In cylindrical polar coordinates (r; ; z), the components of the ow velocity are (u; v; w) in the direction of increasing (r; ; z) respectively, the pressure is p, the density of the uid is , the kinematic viscosity is , and the electrical conductivity is . The external uniform magnetic eld is applied perpendicular to the plane of the disk and has a constant magnetic ux density B0 which is assumed unaltered by taking Rem 1. Mass transfer to, or from, the uid at the surface of the disk, either by direct injection or suction, is considered over the entire range from large injection velocities to large suction velocities. The continuity and time-dependent Navier–Stokes equations take the form ur +
u + wz = 0; r
(1)
ut + uur −
1 u v2 B02 ur + wuz + u + pr = urr + − 2 + uzz ; r r r
vt + uvr +
v uv B02 vr + wvz + v = vrr + − 2 + vzz ; r r r
(2)
(3)
1 wr + wzz : wt + uwr + wwz + pz = wrr + r
(4)
Appropriate initial and boundary conditions for the ow velocity which is started impulsively from rest, due to the rotation of the disk, into steady rotation with constant angular velocity ! and a uniform ow w0 through the disk, are given by at t = 0;
u = 0;
v = 0;
at z = 0;
u = 0;
w = w0 ;
as z → ∞
u; v → 0:
w = w0 ; v = r!;
(5a) (5b) (5c)
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285
By introducing von Karman transformations, u = r!F();
v = r!G();
z = (=!)1=2 ;
w = (!)1=2 H ();
p = −!P;
where is a non-dimensional distance measured along the axis of rotation and F; G; H , and P are non-dimensional functions of , we de ne the magnetic interaction number by = B02 =!, which represents the ratio between the magnetic force to the uid inertia force. Due to uniform suction or injection, the vertical velocity component H takes a constant non-zero value at = 0. With these de nitions, Eqs. (1)–(5) take the form @H + 2F = 0; @
(6)
@2F @F @F − 2 +H + F 2 − G 2 + F = 0; @t @ @
(7)
@G @G @ 2 G − 2 +H + 2FG + G = 0; @t @ @
(8)
@2H dP @H @H − 2 +H − = 0; @t @ @ d
(9)
with the initial and boundary conditions on the ow velocities given by F(0; ) = 0;
G(0; ) = 0;
F(t; 0) = 0;
G(t; 0) = 1;
F(t; ∞) = 0;
G(t; ∞) = 0;
H (0; ) = a; H (t; 0) = a;
(10a) (10b) (10c)
where “a” is the suction or injection parameter, which takes negative values for suction and positive values for injection. The above system of equations (6)–(8) with the prescribed initial and boundary conditions given by Eq. (10) are sucient to solve for the three components of the ow velocity. Eq. (9) can be used to solve for the pressure distribution if required. 3. The numerical solution Numerical solution for the governing non-linear equations (6)–(8), using the nite-di erence approximations, leads to a numerical oscillation problem resulting from the discontinuity between the initial and boundary conditions (10a) and (10b). A solution for this numerical problem is achieved by using proper coordinate transformation, as suggested by Ames √(1977) for similar problems. Expressing Eqs. (6)–(8) in terms of the modi ed coordinate = =2 t we get √ @H + 4 tF = 0; @
(11)
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H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
@F 1 @2F 1 @F @F − − + F 2 − G 2 + F = 0; + √ H @t 2t @ 4t @2 @ 2 t
(12)
@G 1 @2G 1 @G @G − − + 2FG + G = 0: + √ H 2 @t 2t @ 4t @ 2 t @
(13)
The system of non-linear PDEs (11)–(13) is solved using a marching technique and by applying the Crank–Nicolson implicit method (Ames, 1977). The resulting system of di erence equations has to be solved in the in nite domain 0¡¡∞ and 0¡t¡∞. A nite domain in the -direction can be used instead with chosen large enough to ensure that the solutions are not a ected by imposing the asymptotic conditions at a nite distance. However, due to the suggested coordinate transformation, this nite domain diminishes with the progression of time and greatly a ects the accuracy of the numerical solution. To substitute for this problem, the modi ed Eqs. (11)– (13) are integrated from t = 0 to t = 1. Then, the solution obtained at t = 1 is used as the initial condition for integrating Eqs. (6)–(8) from t = 1 towards the steady state. 4. Results and discussion The time growth of the axial velocity component at in nity H∞ for = 0 is shown in Fig. 1a for di erent values of the suction or injection parameters. Increasing the suction parameter “a” negatively from 0 to −2 leads to a great increment in the axial velocity towards the disk and a decrement in its growth time up to the steady state. For small values of the suction parameter (a = 0 to −0:5), the e ects of the radial ow besides the suction through the disk are to increase the quantity of
uid drawn with time. Two paths are available for the incoming uid, one through the suction holes of the disk and another in the radial direction. Higher values of the suction velocity provide an easier path for the ow through the wall than that in the radial direction and result in an almost time-independent in owing stream towards the disk. For the case of injection, H∞ initially equals the injection velocity and for few radians of the rotation of the disk (1.5 rad for a = 1) it keeps its positive direction. With time progression, the radially ejected uid pumps the ow towards the disk
Fig. 1. (a), (b) Time development of the axial velocity at in nity.
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
287
Fig. 2. (a) The steady-state azimuthal velocity pro le versus “a”. (b) The steady-state radial velocity pro le versus “a”. (c) The steady-state axial velocity pro le versus “a”.
and a crossover point in time that depends on the injection parameter “a” occurs and the axial ow reverses direction till its steady-state value. The greater the injection velocity the more strongly is the in ow opposed and, consequently, the later appearance of the crossover of the axial velocity. Fig. 1b indicates the e ect of the magnetic eld on H∞ for di erent values of the suction or injection parameters with = 1. The magnetic eld leads generally to a reduction in the axial velocity at in nity and its growth time. For the case of uniform suction (a = 0 to −2), the magnetic eld has an apparent e ect on the ow for small values of the suction parameter while its e ect is negligible for large values of “a” (a = −2). In the case of uniform injection (a = 0 to 2), the magnetic eld ( = 1) restrains the azimuthal and radial ows and then supports the out owing stream of injected
ow to stop completely the drawn in ow and prevent the occurrence of the crossover points during time progression. In the presence of the magnetic eld, varying the injection velocity becomes more e ective on the ow than in the non-magnetic case. The magnetic eld has a marked e ect on the
288
H.A. Attia / Fluid Dynamics Research 23 (1998) 283–290
Fig. 3. (a) The steady-state azimuthal velocity pro le versus “a”. (b) The steady-state radial velocity pro le versus “a”. (c) The steady-state axial velocity pro le versus “a”.
ow for the whole range of the injection velocities. However, its e ect is much lower for small suction velocities and almost neglected for higher suction velocities. Fig. 2 presents the in uence of the axial ow at the disk surface on the steady-state velocity pro les for the case of suction or injection with = 0. Increasing the suction velocity (a = 0 to −2) leads to a rapid decrease in the azimuthal and radial velocity components as shown in Figs. 2a and 2b, while Fig. 2c indicates that the axial ow at in nity towards the disk is larger. Increasing the injection velocity (a = 0 to 2) leads to an increase in the azimuthal and radial ows as shown in Figs. 2a and 2b, while Fig. 2c shows that the axial ow towards the disk is smaller. With increasing injection velocity, the out ow penetrates to greater distances from the disk surface. Consequently, the crossover point between the positive and negative axial velocity is pushed farther outward in the -direction as shown in Fig. 2c. In Fig. 2a, it is seen that the uid injection gives rise to the familiar in ection-point pro les, especially for high values of the injection parameter “a”. Hence, high injection velocities
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are expected to destabilize the laminar ow and lead to transition to turbulence. In Fig. 2b, it is shown that for high values of the injection parameter (a = 2), the radial velocity near the disk (for small values of ) is lower than that for smaller values of “a”. This is due to the fact that, with increasing “a”, the injected ow can sustain axial motion to greater distances from the wall. Then, near the wall, the radial ow which is fed by that axial ow is expected to decrease as the injection parameter increases. The in uence of the magnetic eld, = 1, on the steady-state velocity pro les for di erent values of the suction or injection parameters is shown in Fig. 3. The magnetic eld e ect is to sustain the ow in the azimuthal, radial, and axial directions. It reduces the axial ow towards the disk for small suction velocities and its e ect becomes negligible for larger suction velocities as shown in Fig. 3c. When injection is applied, the magnetic eld reduces the azimuthal and radial ows and, consequently, the injection stream sustains its axial motion towards the disk. Fig. 3c shows the in uence of the magnetic eld in the suppression of the crossover of the axial component of velocity and then reversal of the direction of the axial motion. In Fig. 3a, it is clear that the magnetic eld has a marked e ect in changing the shape of the in ection-point pro les in the case of high injection velocities. Consequently, the magnetic eld works to stabilize the laminar boundary layer and prevents the transition to turbulence. It is shown in Fig. 3b that the reduction in radial velocity near the wall for high values of the injection velocity disappeared in the presence of the magnetic eld. The magnetic eld works to sustain globally the radial ow and the axial ow feeding it. As the injection velocity increases, it becomes the main source for the radial motion, and consequently, leads to an increment in the radial velocity for all distances above the disk. 5. Conclusions In this paper, the e ects of uniform suction or injection on the unsteady MHD ow induced by an in nite rotating porous disk were studied. The results proved that the steady-state solution is approached as the asymptotic development of a time-dependent process and presented some interesting e ects. The most interesting result is the e ect of the magnetic force on the suppression of the crossover of the axial component of velocity in the non-magnetic case with uniform injection. The magnetic eld has a marked e ect on the ow for the whole range of the injection velocities. However, its in uence on the ow with suction decreases greatly by increasing the suction velocity. The magnetic eld has, in general, an apparent e ect on the ow in the case of uniform injection more than that in the case of uniform suction. References Ames, W.F., 1977. Numerical Methods in Partial Di erential Equations. 2nd ed., Academic Press, New York. Benton, E.R., 1966. On the ow due to a rotating disk. J. Fluid Mech. 24 (4), 781– 800. Cochran, W.G., 1934. The ow due to a rotating disk. Proc. Cambridge Philos. Soc. 30 (3) 365 – 375. El-Mistikawy, T.M.A., Attia, H.A., 1990. The rotating disk ow in the presence of strong magnetic eld. Proc. 3rd Int. Congr. of Fluid Mechanics, Cairo, Egypt, vol. 3, 2 – 4 January, pp. 1211–1222. El-Mistikawy, T.M.A., Attia, H.A., Megahed, A.A., 1991. The rotating disk ow in the presence of weak magnetic eld. Proc. 4th Conf. on Theoretical and Applied Mechanics, Cairo, Egypt, 5 – 7 November, pp. 69 – 82.
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Kuiken, H.K., 1971. The e ect of normal blowing on the ow near a rotating disk of in nite extent. J. Fluid Mech. 47 (4), 789 – 798. Lance, G.N., Rogers, M.H., 1962. The axial symmetric ow of a viscous uid between two in nite rotating disks. Proc. Roy. Soc. A 266, 109. Ockendon, H., 1972. An asymptotic solution for steady ow above an in nite rotating disk with suction. Quart. J. Mech. Appl. Math. XXV, 291– 301. Stuart, J.T., 1954. On the e ects of uniform suction on the steady ow due to a rotating disk. Quart. J. Mech. Appl. Math. 7, 446 – 457. von Karman, T., 1921. Uber laminare und turbulente reibung. 1 (4), 233 – 235.