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UNSTEADY FLOW INDUCED BY VARIABLE SUCTION ON A POROUS DISK ROTATING ECCENTRICALLY
WITH A FLUID AT INFINITY
A. K. GHOSH, S. PAUL, and LOKENATH DEBNATH Received 10 June 2001 and in revised form 15 November 2001
We study the effect of variable suction or blowing on the flow of an incompressible viscous fluid due to noncoaxial rotations of a porous disk and a fluid at infinity. The inquiries are made about the components of fluid velocity and the shear stress at the disk. It is found that the effect of uniform suction or blowing on the flow is enhanced in the presence of variable suction or blowing.
2000 Mathematics Subject Classification: 76S05.
1. Introduction. Thornley [5] has studied the unsteady flow developed in an in- compressible viscous fluid due to nontorsional oscillations of an infinite rigid plate when both the fluid and the plate are in a state of solid body rotation. A similar prob- lem of magnetohydrodynamic Ekman layer over an infinite rigid nonconducting plate was examined by Gupta [3]. On the other hand, the flow due to noncoaxial rotations of a disk and a fluid at infinity was initiated by Berker [1]. Subsequently, Erdogan [2]
constructed solutions of the problem of steady flow due to eccentrically rotations of a porous disk and a fluid at infinity with the same angular velocity both for the cases of uniform suction and blowing at the disk. Of late, Kasiviswanathan and Rao [4] ob- tained an exact solution for the unsteady flow due to noncoaxial rotations of a porous disk oscillating in its own plane and a fluid at infinity. In the present paper, the flow due to noncoaxial rotations of a porous disk subjected to variable suction or blowing and a fluid at infinity has been investigated. Analytical solutions are obtained both for the components of fluid velocity and the components of shear stress at the disk.
Quantitative evolution of the results are also made with a view to examine the effects of variable suction and blowing on the flow. It is found that the effects of uniform suc- tion or blowing on the flow field enhances in presence of variable suction or blowing at the disk.
2. Formulation of the problem. We consider the flow due to a porous disk lying in the xy-plane rotating about the z-axis perpendicular to the disk with uniform angular velocityΩ. The fluid at infinity rotates with the same angular velocityΩabout an axis parallel to thez-axis passing through the point(x, y). The unsteady motion is established in the fluid due to variable suction at the disk. For this motion, the velocity field has the form
u= −Ωy+g(z, t), v=Ωx−f (z, t), w= −V (t), (2.1)
whereV (t) >0 represents the suction velocity which satisfies the equation of conti- nuity. Introducing (2.1) in Navier-Stokes equations, we get
v∂2g
∂z2+V (t)∂g
∂z−∂g
∂t −Ωf
= −Ω2x+1 ρ
∂p
∂x, (2.2)
v∂2f
∂z2+V (t)∂f
∂z−∂f
∂t +Ωg
=Ω2y+1 ρ
∂p
∂x, (2.3)
∂V (t)
∂t =1 ρ
∂p
∂z. (2.4)
We now suppose that the suction velocity normal to the disk oscillates in magnitude and not in direction about a nonzero mean given by
V (t)=W0
1+εAeiσ t
, (2.5)
whereW0is a positive constant;ε >0 is small andAis a real positive constant such thatεA1.
From (2.4) and (2.5), we find that∂p/∂zis small and hence can be neglected. This shows thatpis independent ofz.
Eliminatingpfrom (2.2) and (2.3) by differentiating with respect tozand combining them, we get
v∂3U
∂z3+V (t)∂2U
∂z2−∂2U
∂z∂t−iΩ∂U
∂z =0, (2.6)
whereU=f+ig.
Since no unsteady motion other than suction is imposed on the disk, we must have the boundary conditions forU (z, t)as
U (z, t)=0 atz=0, U (z, t)=Ω x1+iy1
atz= ∞, t >0. (2.7)
In addition to these, we assume that the solutions are bounded at infinity.
Again, from (2.5), we assumed that
U (z, t)=F0(z)+εF1(z)eiσ t. (2.8)
Substituting (2.8) and (2.5) in (2.3), comparing harmonic terms and neglecting co- efficient ofε2, we get
vd3F0
dz3 +W0
d2F0
dz2 −iΩdF0
dz =0, (2.9)
vd3F1
dz3 +W0
d2F1
dz2 −i(Ω+σ )dF1
dz = −W0Ad2F0
dz2, (2.10)
with
F0(0)=0, F1(0)=0, F0(∞)=Ω
x1+iy1
, F1(∞)=0. (2.11)
3. Solution of the problem. We introduceξ=
Ω/2vz, S=W0/2√
Ωv, andn= 1+σ /Ωin (2.10) and (2.11) to obtain
d3F0
dξ3 +2 2Sd2F0
dξ2 −2idF0
dξ =0, (3.1)
d3F1
dξ3 +2 2Sd2F1
dξ2 −2in2dF1
dξ = −23/2εAd2F0
dξ2, (3.2)
subject to conditions (2.11).
On solving (3.1) and (3.2) subject to (2.11), we get f
Ω=x1
1−e−α0ξcosβ0ξ
−y1e−α0ξsinβ0ξ +ε23/2AS
P2+Q2
x1L−y1M
e−α0ξcos
β0ξ−σ t
−e−α1ξcos
β1ξ−σ t +
y1L+x1M
e−α0ξsin
β0ξ−σ t
−e−α1ξsin
β1ξ−σ t , (3.3)
g Ω=y1
1−e−α0ξcosβ0ξ
+x1e−α0ξsinβ0ξ
+ε23/2AS P2+Q2
y1L+x1M
e−α0ξcos
β0ξ−σ t
−e−α1ξcos
β1ξ−σ t
−
x1L−y1M
e−α0ξsin
β0ξ−σ t
−e−α1ξsin
β1ξ−σ t , (3.4)
whereα0=√
2S+γ0,γ0=[√
S4+1+S2]1/2,β0=[√
S4+1−S2]1/2,α1=√
2S+γ1,γ1= [√
S4+n4+S2]1/2,β1=[√
S4+n4−S2]1/2,L=α0P−β0Q,M=β0P−α0Q,P=(α1− α0)[α1+α0−2√
2S]−(β21−β20),Q=(β1−β0)[α1+α0−2√
2S]−(α1−α0)(β1+β0).
In particular, whenA=0, the general results (3.3) and (3.4) reduce to f
Ω=x1
1−e−α0ξcosβ0ξ
−y1e−α0ξsinβ0ξ, (3.5) g
Ω=y1
1−e−α0ξsinβ0ξ
+x1e−α0ξsinβ0ξ. (3.6) These results coincide with the nonoscillating part of the results [4, (11), (12)] and describe the flow in absence of variable suction at the disk.
Again, on puttingx1=0 andy1=lin (3.3) and (3.4), and replacing−f bygandg byf, we get
f Ωl=
1−e−α0ξcosβ0ξ
+ε23/2AS P2+Q2
L
e−α0ξcos
β0ξ−σ t
−e−α1ξcos
β1ξ−σ t +M
e−α0ξsin
β0ξ−σ t
−e−α1ξsin
β1ξ−σ t , (3.7)
g
Ωl=e−α0ξsinβ0ξ+ε23/2AS P2+Q2
M
e−α0ξcos
β0ξ−σ t
−e−α1ξcos
β1ξ−σ t
−L
e−α0ξsin
β0ξ−σ t
−e−α1ξsin
β1ξ−σ t , (3.8)
which are exactly the same as those given in [2] whenA=0. Thus, the effect of variable suction at the disk introduces a transient part depending onε,A, andσ superposed on the steady solution corresponding to uniform suction at the disk. The case ofS=0 corresponds to impermeable case and recovers the solution for steady Ekman layer on the disk. For the flow very near to the porous disk, we have, from (3.7) and (3.8),
f
Ωl=α0ξ+ε23/2AS P2+Q2
L
α1−α0
cosσ t− β1−β0
sinσ t
−M β1−β0
cosσ t+ α1−α0
sinσ t ,
(3.9)
g
Ωl=β0ξ+ε23/2AS P2+Q2
M
α1−α0
cosσ t− β1−β0
sinσ t
−L β1−β0
cosσ t+ α1−α0
sinσ t .
(3.10)
Consequently, the inclination of the fluid velocity vector toy-axis nearz=0 becomes
θ=tan−1 C
D
, (3.11)
where C = β0(P2+Q2)+ε23/2AS[M{(α1−α0)cosσ t−(β1−β0)sinσ t} −L{(β1− β0)cosσ t+(α1−α0)sinσ t}], D=α0(P2+Q2)+ε23/2AS[L{(α1−α0)cosσ t−(β1− β0)sinσ t} −M{(β1−β0)cosσ t+(α1−α0)sinσ t}]. WhenS=0, θ=45◦ and when S≠0 butA=0,θ=tan−1(β0/α0) <45◦.
In the caseσ t=π /2 andS≠0,A≠0, the inclination of the fluid velocity toy-axis nearz=0 will be
tan−1 β0
P2+Q2
−ε23/2AS L
α1−α0
+M β1−β0
α0
P2+Q2
−ε23/2AS M
α1−α0 +L
β1−β0
(3.12)
which indicates a further reduction in the value of the inclination of the fluid velocity compared with its value in the case of uniform suction.
The variations off (ξ)andg(ξ)corresponding to (3.7) and (3.8) for various values of the suction parameterS, the magnitude of fluctuation of suction velocityA, and the frequency of fluctuation of suction velocitynare illustrated in Figures 3.1,3.2, 3.3,3.4,3.5, and3.6.
For the case of blowing,S <0 and the components of fluid velocity in presence of variable blowing at the disk can be obtained easily from (3.7) on replacingS by−λ, whereλ >0. The results in the case of blowing are represented in Figures3.7,3.8,3.9, 3.10,3.11, and3.12.
0 0.2 0.4 0.6 0.8 1.0 Fluid velocity
1.0 2.0 3.0 4.0 5.0
ξ
1.00.5 0.0
g/Ωl f /Ωl
1.0 0.5 0.0
ε=0.1 A=0.0 n=√
1.5
Figure3.1. Variations off /Ωlandg/Ωlfor different values of suction pa- rameterSin absence of variable suctionA.
0 0.2 0.4 0.6 0.8 1.0 Fluid velocity 1.0
2.0 3.0 4.0 5.0
ξ
1.0 0.5 0.0 g/Ωl
f /Ωl
0.51.0 0.0
ε=0.1 A=2.0 n=√
1.5
Figure3.2. Variations off /Ωlandg/Ωlfor different values of suction pa- rameterSin presence of variable suctionA.
0 0.2 0.4 0.6 0.8 1.0 Fluid velocity 1.0
2.0 3.0 4.0 5.0
ξ
1.0 0.5 0.0
g/Ωl f /Ωl
1.0 0.5 0.0
ε=0.1 A=8.0 n=√
1.5
Figure3.3. Variations off /Ωlandg/Ωlfor different values of suction pa- rameterSin presence of variable suctionA.
0 0.2 0.4 0.6 0.8 1.0 Fluid velocity 1.0
2.0 3.0 4.0 5.0
ξ
8 20
g/Ωl f /Ωl
8 02
ε=0.1 S=0.5 n=√
1.5
Figure3.4. Variations off /Ωlandg/Ωlfor different values ofA, the mag- nitude of fluctuations in suction velocity.
0 0.2 0.4 0.6 0.8 1.0 Fluid velocity 1.0
2.0 3.0 4.0 5.0
ξ
8 2 0
g/Ωl f /Ωl
8 2 0
ε=0.1 S=1.0 n=√
1.5
Figure3.5. Variations off /Ωlandg/Ωlfor different values ofA, the mag- nitude of fluctuation in suction velocity.
0 0.2 0.4 0.6 0.8 1.0 1.2 Fluid velocity
1.0 2.0 3.0 4.0 5.0
ξ
n=3n=1.225
g/Ωl f /Ωl
n=1.225 n=3
ε=0.1 A=8.0 S=1.0
Figure3.6.Variations off /Ωlandg/Ωlwith frequencynof the fluctuation in suction velocity.
0 0.2 0.4 0.6 0.8 1.0 Fluid velocity 1.0
2.0 3.0 4.0 5.0
ξ
1.0
0.5 0.0 1.0 0.5
0.0
ε=0.1 A=0.0 n=√
1.5
Figure3.7. Variations off /Ωlandg/Ωlfor different values of the blowing parameter and in absence ofA, the magnitude of the fluctuation in blowing velocity.
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Fluid velocity
1.0 2.0 3.0 4.0 5.0
ξ
0.0 0.5
1.0
g/Ωl f /Ωl
0.0 0.5 1.0
ε=0.1 A=2.0 n=√
1.5
Figure3.8. Variations off /Ωlandg/Ωlfor different values of the blowing parameter and in presence ofA, the magnitude of the fluctuation in blowing velocity.
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Fluid velocity
1.0 2.0 3.0 4.0 5.0
ξ
0.0 0.5
1.0
g/Ωl f /Ωl
0.0 0.5
1.0
A=8.0 ε=0.1 n=√
1.5
Figure3.9. Variations off /Ωlandg/Ωlfor different values of the blowing parameter and in presence ofA, the magnitude of fluctuation in blowing velocity.
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Fluid velocity 1.0
2.0 3.0 4.0 5.0
ξ
0 2 8
g/Ωl f /Ωl
0 2 8
ε=0.1 λ=0.5 n=√
1.5
Figure3.10. Variations off /Ωlandg/Ωlfor different values ofA, the mag- nitude of fluctuations in blowing velocity.
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Fluid velocity
1.0 2.0 3.0 4.0 5.0
ξ 0 2 8
g/Ωl
f /Ωl
0 2 8
ε=0.1 S=1.0 n=√
1.5
Figure3.11. Variations off /Ωlandg/Ωlfor different values ofA, the mag- nitude of fluctuations in blowing velocity.
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Fluid velocity
1.0 2.0 3.0 4.0 5.0
ξ
n=3 n=1.225 n=3
g/Ωl f /Ωl
n=1.225
ε=0.1 A=8.0 S=1.0
Figure3.12. Variation off /Ωlandg/Ωlwithn, the frequency of fluctua- tions of the blowing velocity.
Finally, the components of the shear stress at the diskz=0 corresponding to the fluid velocity given by (3.7) and (3.8) can be obtained as
Px0+iPy0= τx0+iτy0
µΩ3ρl2/21/2
= ∂
∂ξ f
Ωl
+i ∂
∂ξ g
Ωl
ξ=0
=α0+εAS23/2 P2+Q2
L α1−α0
−M
β1−β0 cosσ t
− M
α1−α0
+L
β1−β0 sinσ t +i
β0+εAS23/2 P2+Q2
L β1−β0
+M
α1−α0 cosσ t +
L
α1−α0
−M
β1−β0 sinσ t
,
(3.13)
which, whenσ t=π /2, yields Px0+iPy0=α0−R
M α1−α0
+L
β1−β0 +i β0+R
L α1−α0
−M
β1−β0 , (3.14) whereR=εAS23/2/P2pQ2.
The components of shear stress at the disk in the presence of variable blowing can also be found similarly.
References
[1] R. Berker,Intégration des équations du mouvement d’un fluide visqueux incompressible, Handbuch der Physik, Vol. VIII/2, Springer, Berlin, 1963, pp. 1–384 (French).
[2] M. E. Erdogan,Flow due to eccentrically rotating a porous disk and a fluid at infinity, ASME J. Appl. Mech.43(1976), 203–204.
[3] A. S. Gupta,Magnetohydrodynamic Ekman layer, Acta Mech.13(1972), 155–160.
[4] S. R. Kasiviswanathan and A. R. Rao,An unsteady flow due to eccentrically rotating porous disk and a fluid at infinity, Int. J. Engg. Sci.25(1987), 1419–1425.
[5] C. Thornley,On Stokes and Rayleigh layers in a rotating system, Quart. J. Mech. Appl. Math.
21(1968), 451–461.
A. K. Ghosh and S. Paul: Department of Mathematics, Jadavpur University, Calcutta 700 032, India
L. Debnath: Department of Mathematics, University of Texas-Pan American,1201W.
University Drive, Edinburg, TX78539, USA E-mail address:[email protected]
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