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Unsteady conducting dusty gas flow through a circular pipe in the presence of an applied and induced magnetic field
Mathl. Comput. Modelling Vol. 17, No. 1, pp. 55-64, 1993 Printed in Great Britain. All rights reserved
9695-7177193 $5.96 + 0.96 Copyright@ 1993 Pergamon Press Ltd
UNSTEADY CONDUCTING DUSTY GAS FLOW THROUGH A CIRCULAR PIPE IN THE PRESENCE OF AN APPLIED AND INDUCED MAGNETIC FIELD COLETTE
CALMELET-ELUHU
Department of Physics, Mathematics & Computer Science, Tennessee State University 3596 John A. Merritt Boulevard, Nashville, TN 372041561,
U.S.A.
PHILIP CROOKE Department of Mathematics, Vanderbilt University Nashville, Tennessee 37235, U.S.A.
(Received
October
1991)
Abstract-The unsteady motion of an electrically conducting dusty gas flowing through a circular pipe in the presence of a non-uniform imposed magnetic field is studied. The induced magnetic field is assumed non-negligible. Analytical expressions for the gas and dust velocities and magnetic field are obtained in the case where the dusty gas is set into motion by an arbitrary pressure gradient. The wall of the pipe is assumed non-conducting. The particular case when the pressure decreases exponentially in time is examined.
1. INTRODUCTION Interest in dusty gas or fluid has developed rapidly in recent years. This is due to the use of gas-particle two-phase flow models which arise in various areas of science. These areas include for instance, enviromental pollution, fluidization, combustion in rocket pipes, biology of blood flow, laser velocimetry technology. Among the literature, some scientists were particularly interested on the interactions of Magneto-Hydrodynamic (MHD) flow and dusty fluid flow. Using the model introduced by Saffman [l], and adopting this model to MHD flow, they formulated the conducting dusty gas model. The study of a conducting dusty gas finds many applications. It is of interest in the technology of magnetic separation [2]. Magnetic separators or electromagnetic filters use an external magnetic field to separate the solid particles from the fluid. And recently in Biophysics, the blood flow is considered to be a two-phase flow, where the blood cells form a suspension in a liquid, the blood plasma [3]. In the technology of non-invasive cardiology for the study of the heart rate, the ECG pattern is taken in the presence of external magnetic field which affects the blood flow [4-61. More recently, Ramachandra, Rae and Deshikachar [7] have investigated the effect of magnetic field on the MHD oscillatory flow of blood oxygenation in a channel of varying cross section. Most of these studies [8-121 have examined the flow in the presence of a uniform transverse and applied magnetic field, and neglected the induced magnetic field. However, Pathak and Upadhyay [13] are perhaps the only authors who considered both applied and induced magnetic fields. They studied the unsteady motion of a conducting dusty gas flowing through a circular pipe in the presence of a uniform transverse applied magnetic field. In this paper, we generalize the results of Pathak and Uphadhyay to the case of a non-uniform applied magnetic field. Because of the complexity of the analysis we consider the simplest case where the non-uniform applied magnetic field varies linearly with respect to the length variable. Typeset by AM’l)jX 55
56
C. CALMELET-ELUHU. P. CROOKE 2. THE
BASIC
EQUATIONS
We consider a one-dimensional flow uniform along the z-axis, the axis of the pipe. Initially at rest, the conducting dusty gas is set into motion by an arbitrary pressure gradient, which is a function of time. The flow is influenced by an applied magnetic field which is non-uniform in the zy-plane and normal to the z-axis. An electric current is created and an induced magnetic field is developed along the z-axis. The circular pipe of radius a is assumed to have non-conducting walls. The fluid is assumed to be incompressible, of electrical conductivity u and magnetic permeability /J, and contains non-conducting dust particles. The number density of the particles is assumed constant and equal to Nc. Under Saffman’s formulation and Maxwell’s equations, the basic equations are:
(1) 0.Lo,
(2)
~+(f.v)v+L~), al? -=
(3)
-$v2~+vx(dxri),
at
(4
v.r7=0, VxlLJ’,
(5) (6)
where c, 9 represent the gas and dust particle velocities, I? the total magnetic field (applied and induced), and J’the current density. P is the gas pressure and the constant k is the Stoke’s resistance coefficient for the dust particles of mass m. The constants p and v denote gas density and viscosity, respectively. 3. ANALYSIS In cylindrical functions:
coordinates,
the fluid and dust particle
velocities
are denoted
by the vector-valued
fi(r,0,z,t) = (O,O, u(r, e,t)), V(r,0,z,t)=(O,O,u(r,0,t)). The non-uniform
applied
magnetic
field is assumed
(Ho rcos(28), where Hc is a given constant. g(r,44
Therefore,
to be of the form:
-Ho
rsin(28),
0),
the total
magnetic
field will be of the form:
= (k4++9,
K9(r,%
= (Hcrcos(28),
-Ho
&(r,V)) rsin(28),
H,(r,e,t)),
where the induced magnetic field has only one non-zero component along the z-axis, Hd(r, 8, t). We assume that the uniform pressure gradient has the form: g = KF(t), where K is an arbitrary constant. For some characteristic velocity Uc, we introduce the following non-dimensional variables:
&UC)4 ,-=T a a’ IT=
ka2 f(f)=
xuo
fii-,
U UO
F
v’=_,
V UO
I?*=--,Hz
aHo
Unsteady conducting dusty gas flow
57
In dimensionless form, after dropping the symbol “* 1))and setting H = H,(r, 0,t), the equations of motion (l)-(5) become: 8% “x=
[r yr cos(2#) - H,e sin(2@)] ,
g
-~~t~)+A~+~(u-u)+
(7)
(8) & g where
= AH + &, [F ulr cos(28) - ~$8sin(28)] ,
= kNoo= P vx R m= aU0pa, M=~Hoa
L=kO T mUlJ’
f
J-
&+l,
(91
(Reynolds number), (Magnetic Reynolds number), (Hartmann number).
6,
A comma is used for differentiation, e.g., u,~ = e. Boundary conditions are the same as in [13,14]. Th ere is no fluid slip at the walls of the pipe, hence, we assume that the fluid and dust particle velocity components, u and u, vanish at the boundary. Furthermore, using (6) we have: J r=- ’ H,e, r
Je = -H,,,
J, = 0,
which indicate that H may be considered as a current “stream-function.” Since the walls of the pipe are non-conducting, the boundary P = 1 coincides with the current flow lines along which H is constant. We have assumed H = Hz = 0 outside the boundary, because the applied magnetic field is lying in the ty-plane. Hence, we may assume H = 0 at the walls. Therefore, it is reasonable to consider the following boundary conditions: ~(1, @,t) = u(l,~,~~ = H(l,@,t)
= 0.
(IO)
= H(r,g,O) = 0.
(11)
Moreover, we consider the initial conditions: u(r,B,O) = u(r,8,0)
We will solve this IBVP by finding the exact expressions for the flow variables, U, ZI,H, first in terms of the arbitrary function f(t), and then in the special case of an exponential decreasing in time of the pressure gradient. Since our system of equations is linear, we will employ the Laplace transform. If zl, 6, @ denote the Laplace transforms of u, v, H, respectively, upon transforming (7)-(g), we obtain the following partial differential equations in the transform variables: %pa+rc~(p)-p(~-~)=Au+M2
jj-
In
[r fi,, cos(20) - rl,e sin(20)] ,
p5 = ; (ii - ?3),
(13)
R,,.,pff = AH + R,,, [r ii,, cos(20) - ii,6 sin(2f?)], with the corresponding
boundary
(12)
(14)
conditions:
ii(l,@,p) = @@,f?,p) = 8(1,&p) Here p denotes the transform parameter, and (13), we eliminate is and obtain:
= 0,
p > 0.
(15)
i.e., B(p) = s,‘” e’ pt g(t) dt. Combining Equations (12)
A2-i+ g
[T &,t ~~(2~) - B,e sin(28)] = K f(p) + p s iit
(16)
Arf + c
[Pti,r cos(28) - ii,@sin(2tt)] = R&.,pa,
(17)
C. CALMELET-ELUHU, P. CROOKE
58
where we have set: s = s(p) = & + /3 7 (l/(pr
+ 1)). Next, we introduce
the functions
qqr,ep)=u+J-H+Kf(p) , & PS ’ M
+(r,B,p)=ii--fIH+pst
-
(18)
H(P)
(19)
which can be shown to satisfy (A - PS)~ + M (r d,r cos(28) - d,e sin(28)) cos(2B) - $,e sin(20))
@-~h&kWrrCI,v Letting
A(r, 0) = 3 r2 A(+ PA)
= it4 p I?
(
1- &
>
,
= pii(s - R,,,) + Kf
(20)
cos
28,
w
=
M/2
.
1- + (
1
and hence, AA = 0, one can show:
= (Ad) ewA + 2w ewA (r+,l- cos(20) - +,e sin(20))
+ w2 r2 deWA,
(22)
A($ e--wA ) = (A+) emwA - 2w eVwA (r $J,~co&M) - $,e sin(2B)) + w2 r2 + eVwA, which simplify
(21)
(23)
to [A - (a” + P)]
for an arbitrarily
fixed positive
We next use the Finite
Cosine
(24)
of separation
of variables.
(25)
Suppose
cYr, 0, p) = F(r, p) cos( 2m e),
(28)
1cI(r, 6 p) = G(r, p) cos(2m e),
(27)
integer
m. Substituting
into (24) and (25), respectively,
one finds:
[F(r,p)cos(2m8)eUA]
= bG(r,p)cos(2mO)eWA,
(28)
[G(r,p)cos(2mc9)eSWA]
= bF(r,p)cos(2mO)evWA.
(29)
[A-(a2+B2)] [A-(u2+B2)]
= by5eWA,
(Ic,emwA) = bq5eSWA,
[A - (u2 + P)] where~~=~(ps+pR,),b=~(ps-pR,)andB=wr. We now find 4 and $ using the method
(+e”A)
Transform
J
2r~~~(2ne)exp [i
on (28) and (29), and employ
r2 cos(2e)]
the formula
de = 27fI, (i r”)
,
0
where I,,(5 r”) is the Modified the equations:
Bessel function
of the first kind [15]. This computation
leads to
L{F(r)
[I,+,
(ir2)
+I,-,
(ir’)]}
=bG(r)
[I,+,
(tr’)
+I,-,
(ir’>]
,
(30)
L{G(r)
[I,+,
(ir”)
+I,-,
(ir’)]}
=bF(r)
[I,+n
(ir”)
+I,-,
(zr”)],
(31)
where F(r)
= F(r,p),
G(r) = G(r,p)
and
59
Unsteady conducting dusty gas flow
We apply
the operator
L a second
time on (30) and eliminate
L(G) using
(31). We obtain
[L - b][L + b]K(r) = 0,
(32)
where
L(r) = F(r)
[I,+, (!$r”) + I,-, (ir”)] , 1 d
L-*=$+--zda
Id -_-
L+b=~+rdr Since this ODE for Yn(r) is in factored
form, we first solve the second
order ODE:
= 0.
type [16] and its solution Y,(r)
4n2
pJL+w’r’+F).
[L - b] Y,(r) This ODE is of the Whittaker functions:
4n2
(ps+w’r’+T),
(33)
can be written
= (wf2)ne--%ralF~(f3n,
in terms of hypergeometric
2n + 1,wr2).
and rF1(on,2n+l,wr2) is the confluent Hypergeometricfunction Here, (Y,, = 1/2+n+ps/2M The second linearly independent solution is not incorporated into our solution since 0. Similarly, one can solve the ODE, singularity at r = [L + b] Z,,(r)
= 0,
[16]. it has a
(34)
and find the solution &(r)
= (w r2)n e -(w/2ra)
where
1 7n==+n+2M,
lFl(7”,
ffJ[I,+, (t
r”) + I,_,
it4
P&II
Hence, the solution of (32) is a linear combination Transform of exp[eWA cos(2m e)] yields:
(t
2n + 1,w r2),
w=-.
2
of both solutions.
r”)]
cos(2nO)
Inverting
the Finite
Cosine
= ewA cos(2mQ
II=0
which implies E[A,,
Y,(r)
+ B, Z”(r)] cos(2n8)
= r$(r,B,p) ewA.
(35)
n=O
Starting
with (30) and (31), eliminating
L(F)
[L - bI[L+ bl{G(r) [I,+, one can show, using similar
arguments
E(-1)” n=O Using the boundary
conditions
[A,,Yn(r)
(lo),
and employing
the equation:
(i
(i
r”) + I,-,
r”)] } = 0,
(36)
as above,
- B,J,(r)]cos(2n0)
= t,b(r,O,p)eMWA.
(37)
at r = 1, we have
4(i,e,P) =
tww = F,
(36)
C. CALMELET-ELUHU, P.CROOKE
60
and for any nonnegative
n
integer
A
n1
,=Kf en%& PS [
,
and&=O
(39)
where cn = Substituting
for n = 0,
2,
for n > 0.
for A,, and B, in (35) and (37), one finds
+O”en -Kf
1 C
t#~(r,O,p) = eDwA
n=O
+(r, 0,~)
= ewA
E(
1 n=O We replace obtain:
1,
Y,,(r)
and Y,(l)
by their
PS
In(%) Y,(r) cos(2n 8)
Jw)
_1 nc Kf 1 n ps
respective
a$) y
values,
n
K(r)
1 ,
cos@n 0)
(40) (41)
. 1
and use the identities
(18) and (19) to
(42)
where rl = $(l - r2) - A, r2 = $(l - r2) + A, and A = ir2 cos 28. Next we apply Laplace Transform to (42) and (43) and find:
-L-l
g [ PS
-L-l
First,
we determine
L-‘[Kf/p
1 ’
-1
9 . [ PS
s] using the convolution
theorem
Next, we compute lFl(ch,
2n + 1,~ r2)
ps.1Fl(a,,2n+l,w) Setting
and find
1’
and defining F(p)r
g
lFl(cunr 2n + 1,w r2) = ps1F1(an,2n+ 1 9w)’
the Inverse
Unsteady conducting dusty gas flow
61
Figure 1. Path of complex integration, rm.
we have J3.0(P)l where
= f *
(47)
JmF(P)l,
X+im
L-V(p)1
= &J
We can determine L-‘[F(p)], analytic except at its simple poles pm satisfy
ep’F(P)
X-i00
dP,
Re(p) = A.
using a theorem given by Doetsh poles: po = 0, qo = -(& + /3 T)/&
(48)
[17] applied to F(p), which is 7 and pm, m 2 1. The simple
i+n+&
Pm & + P,:“;l] = &am, (49) [ for any n 2 0 and any m 2 1, where the real negative numbers onm are the zeros of the function variable. Solving (49) for pm, we find 1Fr (a,, 2n + 1, w) in Q,, , considering on an independent the two negative roots: -(&+pT+2MKmT)ffi l+” = > 2&T where
Km = -o*m+n+-
1 2 > 0.
In the complex plane, we consider the path I’m shown in Figure 1, where the semicircles Cm (m 2 1) centered at A, of radius wm, surround the first 2m negative roots & and rk. Using Cauchy’s Theorem, we have:
J
J
eP'F(p)dp+
A+iw, ept
A-iw,
cln
F(P) dp (50)
=
27~
RH
[ept J'(d]p=o
+
I&
[ept
J’(P)]~_,
+ 2 j=l
ml
17:1-E
.
Res [ePtF(p)lpzpj I
C. CALMELET-ELUHU, P. CROOKE
62
Taking the limit as m goes to +oo on both sides of (50) and using (48), we evaluate the residue of ept F(p) at each simple pole and find
P(O) phd -_ 8’(oe
L-l[F(P)l
Q,(o)
+a, p(Pm)
got +
ep,t_
c m=l @(Pm)
+
(51)
We evaluate the derivative of Q(p) at each simple pole and obtain
Q’(0) = Q’(q0)
=
Q’(pm)
(R,+pT)lFl(n+
-(a
+
Pr)
;,2n+
lFl(n
= pm [&(TPm
+
+
Lw),
;,2n
1) + Pr]
+
$
n
(52) (53)
1,~)~
[13’1(an12n
~,w)IQ,,=Q,,,,, d(Pm),
+
(54)
where 4(Pm)
After some computations
=
-2M
l
&(TPm+l)2+pT [
(7Pm
lFl(n
L-‘[F(p)l = lFl(n
I ’
+1)2
using the identities (52)-(54),
Equation (51) becomes
+ +,2n + l,ur2) + f, 2n + 1,w) (55)
1
‘r R,(R?+Pr)
x 1 %+Pr+
exp [-(&i!r)t]}
+ K(r,t),
where l~l(Qnm,
Wn(r,t) = E n=l
A[lFl(a,,2n da,
2n + 1, W ?)(TPm +
l,~)lQn=anm
+ [R,
1)2
eXP(Pm t)
C7Pm + 112 +PT)lKm’
and Km
Using the identity [18]
4,2n+ l,wr2) = L($?2) lFl(n + +,2n+ 1,~) In (5)
lFl(n+
r-2n
exp
[$(r2- l)] ,
(56)
Equation (55) can be simplified: L-l[F(p)]=
“I(?;;’ {&tPr+ ,,,“‘+, 7 )exp
[-(kifr)t]}r-2nexp
[i(r’-l)]
1 kI,(r,t). (57) Therefore, Equation (47) can be written in the following form
Replacing L-‘[Kf/p s] and and in view of the relation
L-l[f
F(p)] by their expressions given in (46) and (58), respectively,
exp [t r2 cos(28)] = E cn I, (i n=O
r”)
cos(2nB),
(59)
Unsteady
Equation
= K~~.I,(~)eYrlrzncas(lnB)T,(r,f), n=O
T,(r,t) =JtW*(frt -c> f(C) 4 =k lJi(Qnm,
X
Equation
Finally,
adding
[J’
0
g
m
(60)
l,wr2)(7pm
+
1)2
[&(~Pm+1)2+P~]
Km
exp(pm (t- c)> f(c)do].
(45) is reduced
u -
2n+
4[1Fl((Yn,2n+1,W)]an=a,, da,
m=l
Similarly,
63
dusty gas flow
(44) becomes:
u+$H
where
conducting
to
H = K E(-1)” n=O
and subtracting
(61)
(60) and (61) yields
u(r, 8, t) = $gcnIn(i)
H(r, 8, t) = s
ewra r”’ cos(2n 0) Tn(r, t).
c,, I,, (i)
[ewrl + (-l)ne”ra]
Zen
I,(i)
r2”cos(2n0)T,(r,t),
[ewrl - (-l)“ewra]
r2n c0s(2n8)Tn(r,t).
(62) (63)
Hence, we have found an analytical expression for the gas velocity and the total magnetic field. We need to find the analytic expression for the dust particle velocity. This expression can be derived from (13), applying the Inverse Laplace Transform and substituting for u given in (62). One finds
v(r,O,t) =
gEcnIn(i)
[ewrl +(-l)ne”ra]
r2”cos(2nt9)Sn(r,t),
n=O
where
&(r,t) =
~tT,(r,dexp [-(?)I 4. SPECIAL
drl.
CASE
We now examine the particular case where the pressure decreases pose j(t) = e-(XalRe)t, where A is a given real constant. Substituting respective expressions for U, V, H given in (49)-(51), we obtain u(r,
8, t) = +zcnIn($)
[ewrl +(-l)ne”ra]
KR+O” H(r, 8, t) = *ghL(y) = v(r,
8, t) = g Zen
I, ($)
(64)
r2”c0s(2nO)~(r,i!),
[ewrl -(-l)newra]
[ewrl + (-1)”
exponentially in time. Supfor j(C), 0 2 (’ 5 t, in the
r2”cos(2n8)~(r,t),
ewra] r2n cos(2n0)
Sz(r,t),
C. CALMELET-ELUHU, P. CROOKE
64
where TnO(r,t)
=
$fJ m=l
(7pm + 1)2 (epmt - e-(Xa/Re)t) &I
[&
(%l + 1)2
+ P Tl (p”
lFl(%m,
&
+ E)
lFl(a,,, S%,t)
= ;cl
$n [&:;mm++:;
’
1
cPrn + 2)
+ PT]
&
ePmf
2n + 1, + r2)
[&(an,2n+
1,
+‘>I Qn=a’,, ’
2n + I, 7 r2)
[&((Y,
I 2n + 1, $:>I Qn= anm
e-‘/T
,-@/R.)t
’ (Pm + +)
-
(Pm + $)
(E
-
$)
+
(p”
+ +?)
(g
_ ;)
I
5. CONCLUSION We have been able to obtain an exact solution of the IBVP in terms of the flow parameters and the pressure gradient responsible for the motion of the conducting dusty gas. In particular, the solution is valid for any value of the Hartmann, Reynolds and Magnetic Reynolds numbers for which the Saffman’s formulation holds. REFERENCES 1. P.G. S&man, On the stability of laminar flow of a dusty gas, .I. Fluid Mech. 13 (l), 120-129 (1962). 2. Y.A. Liu, Ed., Proceedings of Ihe International Conference on Industrial Applications of Magnetic Separation, Franklin Pierce College, Rindge, New Hampshire, July 30-August 4, 1978, IEEE, New York, (1979). 3. S.I. Rubinow, Mathematical Problems in the Biological Sciencea, Regional Conference Series in Applied Mathematics, Vol. VII, SIAM, Philadelphia, PA, (1973). 4. E.M. Korchevski and L.S. Marochnik, Magnetohydrc+dynamic version of movement of blood, Biophysics 10, 411-413 (1965). 5. V.K. Sud, P.K. Suri and R.K. Mishra, Laminar flow of blood in an elastic tube in the presence of magnetic field, Sludia Biophyaica 69, 175-186 (1978). 6. V.A. Vardanyan, Effect of a magnetic field on blood flow, Biophysics 18, 515-521 (1973). 7. Ft. Bamachandra and K.S. Deshikachar, MHD oscillatory flow of blood through channels of variable cross section, Inl. J. Eng. Sci. 24 (lo), 1615-1628 (1986). 8. M.C. Baral, Plane parallel flow of a conducting dusty gas, J. Phys. Sot. of Japan 25 (6), 1701-1702 (1968). 9. L. Debnath and U. Basu, Unsteady slip flow in an electrically conducting two-phase fluid under transverse magnetic fields, Nuov. Cim. 28B (2), 349-362 (1975). 10. D.K. Guha, Unsteady flow of a viscous conducting dusty gas between two coaxial cylinders, Indian J. Techn. 19, 261-266 (1981). 11. P. Mitra and P. Bhattacharyya, On the flow of an electrically conducting incompressible dusty fluid near an accelerated plate in presence of another fixed parallel plate, J. Techn. India 26 (l), 13-24 (1981). 12. H.C. Sharma and M. Gupta, MHD flow of a dusty gas through a hexagonal channel, Current Sci. 53 (22), 1195-1197 (1984). 13. R.S. Path& and B.N. Upadhyay, Unsteady dusty fluid flow through a circular pipe under transverse magnetic field, J. Indian Acad. Math. 4 (l), 39-47 (1982). 14. J.A. Shercliff, Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Camb. Phil. Sot. 49, 136-144 (1953). 15. N.W. MacLachlan, Bearel Fzlnctions for Engineera, Clarendon Press, Oxford, p. 164, (1934). Function, Springer Tracts in Natural Phil., 15, Springer-Verlag, 16. H. Buchholz, The Confluent Hypergeomelric Heidelberg, p. 10, (1969). to the Theory and Application of the Lap/ace Transformation, Springer-VerIeg, 17. G. Doetsch, Introduclion Heidelberg, p. 162, (1974). Fun&ions, Vol. 1, McGraw-Hill Book Co., New York, p. 265, (1953). 18. A. Erddblyi, Higher Transcendenta/
9695-7177193 $5.96 + 0.96 Copyright@ 1993 Pergamon Press Ltd
UNSTEADY CONDUCTING DUSTY GAS FLOW THROUGH A CIRCULAR PIPE IN THE PRESENCE OF AN APPLIED AND INDUCED MAGNETIC FIELD COLETTE
CALMELET-ELUHU
Department of Physics, Mathematics & Computer Science, Tennessee State University 3596 John A. Merritt Boulevard, Nashville, TN 372041561,
U.S.A.
PHILIP CROOKE Department of Mathematics, Vanderbilt University Nashville, Tennessee 37235, U.S.A.
(Received
October
1991)
Abstract-The unsteady motion of an electrically conducting dusty gas flowing through a circular pipe in the presence of a non-uniform imposed magnetic field is studied. The induced magnetic field is assumed non-negligible. Analytical expressions for the gas and dust velocities and magnetic field are obtained in the case where the dusty gas is set into motion by an arbitrary pressure gradient. The wall of the pipe is assumed non-conducting. The particular case when the pressure decreases exponentially in time is examined.
1. INTRODUCTION Interest in dusty gas or fluid has developed rapidly in recent years. This is due to the use of gas-particle two-phase flow models which arise in various areas of science. These areas include for instance, enviromental pollution, fluidization, combustion in rocket pipes, biology of blood flow, laser velocimetry technology. Among the literature, some scientists were particularly interested on the interactions of Magneto-Hydrodynamic (MHD) flow and dusty fluid flow. Using the model introduced by Saffman [l], and adopting this model to MHD flow, they formulated the conducting dusty gas model. The study of a conducting dusty gas finds many applications. It is of interest in the technology of magnetic separation [2]. Magnetic separators or electromagnetic filters use an external magnetic field to separate the solid particles from the fluid. And recently in Biophysics, the blood flow is considered to be a two-phase flow, where the blood cells form a suspension in a liquid, the blood plasma [3]. In the technology of non-invasive cardiology for the study of the heart rate, the ECG pattern is taken in the presence of external magnetic field which affects the blood flow [4-61. More recently, Ramachandra, Rae and Deshikachar [7] have investigated the effect of magnetic field on the MHD oscillatory flow of blood oxygenation in a channel of varying cross section. Most of these studies [8-121 have examined the flow in the presence of a uniform transverse and applied magnetic field, and neglected the induced magnetic field. However, Pathak and Upadhyay [13] are perhaps the only authors who considered both applied and induced magnetic fields. They studied the unsteady motion of a conducting dusty gas flowing through a circular pipe in the presence of a uniform transverse applied magnetic field. In this paper, we generalize the results of Pathak and Uphadhyay to the case of a non-uniform applied magnetic field. Because of the complexity of the analysis we consider the simplest case where the non-uniform applied magnetic field varies linearly with respect to the length variable. Typeset by AM’l)jX 55
56
C. CALMELET-ELUHU. P. CROOKE 2. THE
BASIC
EQUATIONS
We consider a one-dimensional flow uniform along the z-axis, the axis of the pipe. Initially at rest, the conducting dusty gas is set into motion by an arbitrary pressure gradient, which is a function of time. The flow is influenced by an applied magnetic field which is non-uniform in the zy-plane and normal to the z-axis. An electric current is created and an induced magnetic field is developed along the z-axis. The circular pipe of radius a is assumed to have non-conducting walls. The fluid is assumed to be incompressible, of electrical conductivity u and magnetic permeability /J, and contains non-conducting dust particles. The number density of the particles is assumed constant and equal to Nc. Under Saffman’s formulation and Maxwell’s equations, the basic equations are:
(1) 0.Lo,
(2)
~+(f.v)v+L~), al? -=
(3)
-$v2~+vx(dxri),
at
(4
v.r7=0, VxlLJ’,
(5) (6)
where c, 9 represent the gas and dust particle velocities, I? the total magnetic field (applied and induced), and J’the current density. P is the gas pressure and the constant k is the Stoke’s resistance coefficient for the dust particles of mass m. The constants p and v denote gas density and viscosity, respectively. 3. ANALYSIS In cylindrical functions:
coordinates,
the fluid and dust particle
velocities
are denoted
by the vector-valued
fi(r,0,z,t) = (O,O, u(r, e,t)), V(r,0,z,t)=(O,O,u(r,0,t)). The non-uniform
applied
magnetic
field is assumed
(Ho rcos(28), where Hc is a given constant. g(r,44
Therefore,
to be of the form:
-Ho
rsin(28),
0),
the total
magnetic
field will be of the form:
= (k4++9,
K9(r,%
= (Hcrcos(28),
-Ho
&(r,V)) rsin(28),
H,(r,e,t)),
where the induced magnetic field has only one non-zero component along the z-axis, Hd(r, 8, t). We assume that the uniform pressure gradient has the form: g = KF(t), where K is an arbitrary constant. For some characteristic velocity Uc, we introduce the following non-dimensional variables:
&UC)4 ,-=T a a’ IT=
ka2 f(f)=
xuo
fii-,
U UO
F
v’=_,
V UO
I?*=--,Hz
aHo
Unsteady conducting dusty gas flow
57
In dimensionless form, after dropping the symbol “* 1))and setting H = H,(r, 0,t), the equations of motion (l)-(5) become: 8% “x=
[r yr cos(2#) - H,e sin(2@)] ,
g
-~~t~)+A~+~(u-u)+
(7)
(8) & g where
= AH + &, [F ulr cos(28) - ~$8sin(28)] ,
= kNoo= P vx R m= aU0pa, M=~Hoa
L=kO T mUlJ’
f
J-
&+l,
(91
(Reynolds number), (Magnetic Reynolds number), (Hartmann number).
6,
A comma is used for differentiation, e.g., u,~ = e. Boundary conditions are the same as in [13,14]. Th ere is no fluid slip at the walls of the pipe, hence, we assume that the fluid and dust particle velocity components, u and u, vanish at the boundary. Furthermore, using (6) we have: J r=- ’ H,e, r
Je = -H,,,
J, = 0,
which indicate that H may be considered as a current “stream-function.” Since the walls of the pipe are non-conducting, the boundary P = 1 coincides with the current flow lines along which H is constant. We have assumed H = Hz = 0 outside the boundary, because the applied magnetic field is lying in the ty-plane. Hence, we may assume H = 0 at the walls. Therefore, it is reasonable to consider the following boundary conditions: ~(1, @,t) = u(l,~,~~ = H(l,@,t)
= 0.
(IO)
= H(r,g,O) = 0.
(11)
Moreover, we consider the initial conditions: u(r,B,O) = u(r,8,0)
We will solve this IBVP by finding the exact expressions for the flow variables, U, ZI,H, first in terms of the arbitrary function f(t), and then in the special case of an exponential decreasing in time of the pressure gradient. Since our system of equations is linear, we will employ the Laplace transform. If zl, 6, @ denote the Laplace transforms of u, v, H, respectively, upon transforming (7)-(g), we obtain the following partial differential equations in the transform variables: %pa+rc~(p)-p(~-~)=Au+M2
jj-
In
[r fi,, cos(20) - rl,e sin(20)] ,
p5 = ; (ii - ?3),
(13)
R,,.,pff = AH + R,,, [r ii,, cos(20) - ii,6 sin(2f?)], with the corresponding
boundary
(12)
(14)
conditions:
ii(l,@,p) = @@,f?,p) = 8(1,&p) Here p denotes the transform parameter, and (13), we eliminate is and obtain:
= 0,
p > 0.
(15)
i.e., B(p) = s,‘” e’ pt g(t) dt. Combining Equations (12)
A2-i+ g
[T &,t ~~(2~) - B,e sin(28)] = K f(p) + p s iit
(16)
Arf + c
[Pti,r cos(28) - ii,@sin(2tt)] = R&.,pa,
(17)
C. CALMELET-ELUHU, P. CROOKE
58
where we have set: s = s(p) = & + /3 7 (l/(pr
+ 1)). Next, we introduce
the functions
qqr,ep)=u+J-H+Kf(p) , & PS ’ M
+(r,B,p)=ii--fIH+pst
-
(18)
H(P)
(19)
which can be shown to satisfy (A - PS)~ + M (r d,r cos(28) - d,e sin(28)) cos(2B) - $,e sin(20))
@-~h&kWrrCI,v Letting
A(r, 0) = 3 r2 A(+ PA)
= it4 p I?
(
1- &
>
,
= pii(s - R,,,) + Kf
(20)
cos
28,
w
=
M/2
.
1- + (
1
and hence, AA = 0, one can show:
= (Ad) ewA + 2w ewA (r+,l- cos(20) - +,e sin(20))
+ w2 r2 deWA,
(22)
A($ e--wA ) = (A+) emwA - 2w eVwA (r $J,~co&M) - $,e sin(2B)) + w2 r2 + eVwA, which simplify
(21)
(23)
to [A - (a” + P)]
for an arbitrarily
fixed positive
We next use the Finite
Cosine
(24)
of separation
of variables.
(25)
Suppose
cYr, 0, p) = F(r, p) cos( 2m e),
(28)
1cI(r, 6 p) = G(r, p) cos(2m e),
(27)
integer
m. Substituting
into (24) and (25), respectively,
one finds:
[F(r,p)cos(2m8)eUA]
= bG(r,p)cos(2mO)eWA,
(28)
[G(r,p)cos(2mc9)eSWA]
= bF(r,p)cos(2mO)evWA.
(29)
[A-(a2+B2)] [A-(u2+B2)]
= by5eWA,
(Ic,emwA) = bq5eSWA,
[A - (u2 + P)] where~~=~(ps+pR,),b=~(ps-pR,)andB=wr. We now find 4 and $ using the method
(+e”A)
Transform
J
2r~~~(2ne)exp [i
on (28) and (29), and employ
r2 cos(2e)]
the formula
de = 27fI, (i r”)
,
0
where I,,(5 r”) is the Modified the equations:
Bessel function
of the first kind [15]. This computation
leads to
L{F(r)
[I,+,
(ir2)
+I,-,
(ir’)]}
=bG(r)
[I,+,
(tr’)
+I,-,
(ir’>]
,
(30)
L{G(r)
[I,+,
(ir”)
+I,-,
(ir’)]}
=bF(r)
[I,+n
(ir”)
+I,-,
(zr”)],
(31)
where F(r)
= F(r,p),
G(r) = G(r,p)
and
59
Unsteady conducting dusty gas flow
We apply
the operator
L a second
time on (30) and eliminate
L(G) using
(31). We obtain
[L - b][L + b]K(r) = 0,
(32)
where
L(r) = F(r)
[I,+, (!$r”) + I,-, (ir”)] , 1 d
L-*=$+--zda
Id -_-
L+b=~+rdr Since this ODE for Yn(r) is in factored
form, we first solve the second
order ODE:
= 0.
type [16] and its solution Y,(r)
4n2
pJL+w’r’+F).
[L - b] Y,(r) This ODE is of the Whittaker functions:
4n2
(ps+w’r’+T),
(33)
can be written
= (wf2)ne--%ralF~(f3n,
in terms of hypergeometric
2n + 1,wr2).
and rF1(on,2n+l,wr2) is the confluent Hypergeometricfunction Here, (Y,, = 1/2+n+ps/2M The second linearly independent solution is not incorporated into our solution since 0. Similarly, one can solve the ODE, singularity at r = [L + b] Z,,(r)
= 0,
[16]. it has a
(34)
and find the solution &(r)
= (w r2)n e -(w/2ra)
where
1 7n==+n+2M,
lFl(7”,
ffJ[I,+, (t
r”) + I,_,
it4
P&II
Hence, the solution of (32) is a linear combination Transform of exp[eWA cos(2m e)] yields:
(t
2n + 1,w r2),
w=-.
2
of both solutions.
r”)]
cos(2nO)
Inverting
the Finite
Cosine
= ewA cos(2mQ
II=0
which implies E[A,,
Y,(r)
+ B, Z”(r)] cos(2n8)
= r$(r,B,p) ewA.
(35)
n=O
Starting
with (30) and (31), eliminating
L(F)
[L - bI[L+ bl{G(r) [I,+, one can show, using similar
arguments
E(-1)” n=O Using the boundary
conditions
[A,,Yn(r)
(lo),
and employing
the equation:
(i
(i
r”) + I,-,
r”)] } = 0,
(36)
as above,
- B,J,(r)]cos(2n0)
= t,b(r,O,p)eMWA.
(37)
at r = 1, we have
4(i,e,P) =
tww = F,
(36)
C. CALMELET-ELUHU, P.CROOKE
60
and for any nonnegative
n
integer
A
n1
,=Kf en%& PS [
,
and&=O
(39)
where cn = Substituting
for n = 0,
2,
for n > 0.
for A,, and B, in (35) and (37), one finds
+O”en -Kf
1 C
t#~(r,O,p) = eDwA
n=O
+(r, 0,~)
= ewA
E(
1 n=O We replace obtain:
1,
Y,,(r)
and Y,(l)
by their
PS
In(%) Y,(r) cos(2n 8)
Jw)
_1 nc Kf 1 n ps
respective
a$) y
values,
n
K(r)
1 ,
cos@n 0)
(40) (41)
. 1
and use the identities
(18) and (19) to
(42)
where rl = $(l - r2) - A, r2 = $(l - r2) + A, and A = ir2 cos 28. Next we apply Laplace Transform to (42) and (43) and find:
-L-l
g [ PS
-L-l
First,
we determine
L-‘[Kf/p
1 ’
-1
9 . [ PS
s] using the convolution
theorem
Next, we compute lFl(ch,
2n + 1,~ r2)
ps.1Fl(a,,2n+l,w) Setting
and find
1’
and defining F(p)r
g
lFl(cunr 2n + 1,w r2) = ps1F1(an,2n+ 1 9w)’
the Inverse
Unsteady conducting dusty gas flow
61
Figure 1. Path of complex integration, rm.
we have J3.0(P)l where
= f *
(47)
JmF(P)l,
X+im
L-V(p)1
= &J
We can determine L-‘[F(p)], analytic except at its simple poles pm satisfy
ep’F(P)
X-i00
dP,
Re(p) = A.
using a theorem given by Doetsh poles: po = 0, qo = -(& + /3 T)/&
(48)
[17] applied to F(p), which is 7 and pm, m 2 1. The simple
i+n+&
Pm & + P,:“;l] = &am, (49) [ for any n 2 0 and any m 2 1, where the real negative numbers onm are the zeros of the function variable. Solving (49) for pm, we find 1Fr (a,, 2n + 1, w) in Q,, , considering on an independent the two negative roots: -(&+pT+2MKmT)ffi l+” = > 2&T where
Km = -o*m+n+-
1 2 > 0.
In the complex plane, we consider the path I’m shown in Figure 1, where the semicircles Cm (m 2 1) centered at A, of radius wm, surround the first 2m negative roots & and rk. Using Cauchy’s Theorem, we have:
J
J
eP'F(p)dp+
A+iw, ept
A-iw,
cln
F(P) dp (50)
=
27~
RH
[ept J'(d]p=o
+
I&
[ept
J’(P)]~_,
+ 2 j=l
ml
17:1-E
.
Res [ePtF(p)lpzpj I
C. CALMELET-ELUHU, P. CROOKE
62
Taking the limit as m goes to +oo on both sides of (50) and using (48), we evaluate the residue of ept F(p) at each simple pole and find
P(O) phd -_ 8’(oe
L-l[F(P)l
Q,(o)
+a, p(Pm)
got +
ep,t_
c m=l @(Pm)
+
(51)
We evaluate the derivative of Q(p) at each simple pole and obtain
Q’(0) = Q’(q0)
=
Q’(pm)
(R,+pT)lFl(n+
-(a
+
Pr)
;,2n+
lFl(n
= pm [&(TPm
+
+
Lw),
;,2n
1) + Pr]
+
$
n
(52) (53)
1,~)~
[13’1(an12n
~,w)IQ,,=Q,,,,, d(Pm),
+
(54)
where 4(Pm)
After some computations
=
-2M
l
&(TPm+l)2+pT [
(7Pm
lFl(n
L-‘[F(p)l = lFl(n
I ’
+1)2
using the identities (52)-(54),
Equation (51) becomes
+ +,2n + l,ur2) + f, 2n + 1,w) (55)
1
‘r R,(R?+Pr)
x 1 %+Pr+
exp [-(&i!r)t]}
+ K(r,t),
where l~l(Qnm,
Wn(r,t) = E n=l
A[lFl(a,,2n da,
2n + 1, W ?)(TPm +
l,~)lQn=anm
+ [R,
1)2
eXP(Pm t)
C7Pm + 112 +PT)lKm’
and Km
Using the identity [18]
4,2n+ l,wr2) = L($?2) lFl(n + +,2n+ 1,~) In (5)
lFl(n+
r-2n
exp
[$(r2- l)] ,
(56)
Equation (55) can be simplified: L-l[F(p)]=
“I(?;;’ {&tPr+ ,,,“‘+, 7 )exp
[-(kifr)t]}r-2nexp
[i(r’-l)]
1 kI,(r,t). (57) Therefore, Equation (47) can be written in the following form
Replacing L-‘[Kf/p s] and and in view of the relation
L-l[f
F(p)] by their expressions given in (46) and (58), respectively,
exp [t r2 cos(28)] = E cn I, (i n=O
r”)
cos(2nB),
(59)
Unsteady
Equation
= K~~.I,(~)eYrlrzncas(lnB)T,(r,f), n=O
T,(r,t) =JtW*(frt -c> f(C) 4 =k lJi(Qnm,
X
Equation
Finally,
adding
[J’
0
g
m
(60)
l,wr2)(7pm
+
1)2
[&(~Pm+1)2+P~]
Km
exp(pm (t- c)> f(c)do].
(45) is reduced
u -
2n+
4[1Fl((Yn,2n+1,W)]an=a,, da,
m=l
Similarly,
63
dusty gas flow
(44) becomes:
u+$H
where
conducting
to
H = K E(-1)” n=O
and subtracting
(61)
(60) and (61) yields
u(r, 8, t) = $gcnIn(i)
H(r, 8, t) = s
ewra r”’ cos(2n 0) Tn(r, t).
c,, I,, (i)
[ewrl + (-l)ne”ra]
Zen
I,(i)
r2”cos(2n0)T,(r,t),
[ewrl - (-l)“ewra]
r2n c0s(2n8)Tn(r,t).
(62) (63)
Hence, we have found an analytical expression for the gas velocity and the total magnetic field. We need to find the analytic expression for the dust particle velocity. This expression can be derived from (13), applying the Inverse Laplace Transform and substituting for u given in (62). One finds
v(r,O,t) =
gEcnIn(i)
[ewrl +(-l)ne”ra]
r2”cos(2nt9)Sn(r,t),
n=O
where
&(r,t) =
~tT,(r,dexp [-(?)I 4. SPECIAL
drl.
CASE
We now examine the particular case where the pressure decreases pose j(t) = e-(XalRe)t, where A is a given real constant. Substituting respective expressions for U, V, H given in (49)-(51), we obtain u(r,
8, t) = +zcnIn($)
[ewrl +(-l)ne”ra]
KR+O” H(r, 8, t) = *ghL(y) = v(r,
8, t) = g Zen
I, ($)
(64)
r2”c0s(2nO)~(r,i!),
[ewrl -(-l)newra]
[ewrl + (-1)”
exponentially in time. Supfor j(C), 0 2 (’ 5 t, in the
r2”cos(2n8)~(r,t),
ewra] r2n cos(2n0)
Sz(r,t),
C. CALMELET-ELUHU, P. CROOKE
64
where TnO(r,t)
=
$fJ m=l
(7pm + 1)2 (epmt - e-(Xa/Re)t) &I
[&
(%l + 1)2
+ P Tl (p”
lFl(%m,
&
+ E)
lFl(a,,, S%,t)
= ;cl
$n [&:;mm++:;
’
1
cPrn + 2)
+ PT]
&
ePmf
2n + 1, + r2)
[&(an,2n+
1,
+‘>I Qn=a’,, ’
2n + I, 7 r2)
[&((Y,
I 2n + 1, $:>I Qn= anm
e-‘/T
,-@/R.)t
’ (Pm + +)
-
(Pm + $)
(E
-
$)
+
(p”
+ +?)
(g
_ ;)
I
5. CONCLUSION We have been able to obtain an exact solution of the IBVP in terms of the flow parameters and the pressure gradient responsible for the motion of the conducting dusty gas. In particular, the solution is valid for any value of the Hartmann, Reynolds and Magnetic Reynolds numbers for which the Saffman’s formulation holds. REFERENCES 1. P.G. S&man, On the stability of laminar flow of a dusty gas, .I. Fluid Mech. 13 (l), 120-129 (1962). 2. Y.A. Liu, Ed., Proceedings of Ihe International Conference on Industrial Applications of Magnetic Separation, Franklin Pierce College, Rindge, New Hampshire, July 30-August 4, 1978, IEEE, New York, (1979). 3. S.I. Rubinow, Mathematical Problems in the Biological Sciencea, Regional Conference Series in Applied Mathematics, Vol. VII, SIAM, Philadelphia, PA, (1973). 4. E.M. Korchevski and L.S. Marochnik, Magnetohydrc+dynamic version of movement of blood, Biophysics 10, 411-413 (1965). 5. V.K. Sud, P.K. Suri and R.K. Mishra, Laminar flow of blood in an elastic tube in the presence of magnetic field, Sludia Biophyaica 69, 175-186 (1978). 6. V.A. Vardanyan, Effect of a magnetic field on blood flow, Biophysics 18, 515-521 (1973). 7. Ft. Bamachandra and K.S. Deshikachar, MHD oscillatory flow of blood through channels of variable cross section, Inl. J. Eng. Sci. 24 (lo), 1615-1628 (1986). 8. M.C. Baral, Plane parallel flow of a conducting dusty gas, J. Phys. Sot. of Japan 25 (6), 1701-1702 (1968). 9. L. Debnath and U. Basu, Unsteady slip flow in an electrically conducting two-phase fluid under transverse magnetic fields, Nuov. Cim. 28B (2), 349-362 (1975). 10. D.K. Guha, Unsteady flow of a viscous conducting dusty gas between two coaxial cylinders, Indian J. Techn. 19, 261-266 (1981). 11. P. Mitra and P. Bhattacharyya, On the flow of an electrically conducting incompressible dusty fluid near an accelerated plate in presence of another fixed parallel plate, J. Techn. India 26 (l), 13-24 (1981). 12. H.C. Sharma and M. Gupta, MHD flow of a dusty gas through a hexagonal channel, Current Sci. 53 (22), 1195-1197 (1984). 13. R.S. Path& and B.N. Upadhyay, Unsteady dusty fluid flow through a circular pipe under transverse magnetic field, J. Indian Acad. Math. 4 (l), 39-47 (1982). 14. J.A. Shercliff, Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Camb. Phil. Sot. 49, 136-144 (1953). 15. N.W. MacLachlan, Bearel Fzlnctions for Engineera, Clarendon Press, Oxford, p. 164, (1934). Function, Springer Tracts in Natural Phil., 15, Springer-Verlag, 16. H. Buchholz, The Confluent Hypergeomelric Heidelberg, p. 10, (1969). to the Theory and Application of the Lap/ace Transformation, Springer-VerIeg, 17. G. Doetsch, Introduclion Heidelberg, p. 162, (1974). Fun&ions, Vol. 1, McGraw-Hill Book Co., New York, p. 265, (1953). 18. A. Erddblyi, Higher Transcendenta/