Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 1, pp. 127-133, Warsaw 2016
DOI: 10.15632/jtam-pl.54.1.127

VORTICITY TRANSPORT ANALYSIS IN MAGNETIC

VISCOELASTIC FLUID

Pardeep Kumar, Hari Mohan

Department of Mathematics, ICDEOL, Himachal Pradesh University, Shimla, India

e-mail: pkdureja@gmail.com

Gamal Hoshoudy

Department of Applied Mathematics, Faculty of Science, South Valley University, Kena, Egypt

e-mail: g hoshoudy@yahoo.com

Results of investigation on the transport of vorticity in Rivlin-Ericksen viscoelastic fluid
in the presence of suspended magnetic particles is presented here. Equations governing the
transport of vorticity in Rivlin-Ericksen viscoelastic fluid in the presence of suspendedma-
gnetic particles are obtained fromthe equations ofmagnetic fluid flow.From these equations
it follows that the transport of solid vorticity is coupled with the transport of fluid vortici-
ty. Further, we find that because of thermokinetic process, fluid vorticity may exist in the
absence of solid vorticity, but when fluid vorticity is zero, then solid vorticity is necessarily
zero. A two-dimensional case is also studied and found that the fluid vorticity is indirectly
influenced by the temperature and the magnetic field gradient.

Keywords: Rivlin-Ericksen viscoelastic fluid, suspendedmagnetic particles, vorticity

1. Introduction

Amagneto-rheological fluid contains particles of magnetic materials mixed in a liquid that acts
as a carrier. Under normal conditions, the material behaves like a viscous fluid. When it is
exposed to a magnetic field, the particles inside align and it responds to the field, exhibiting
magnetized behaviour. There are a number of uses for magnetic fluids, ranging from medicine
to industrial manufacturing. It is, therefore, a two-phase system consisting of solid and liquid
phases. We assume that the liquid phase is non-magnetic in nature and magnetic force acts
only on the magnetic particles. Thus, the magnetic force changes the velocity of the magnetic
particles. Consequently, the dragging force acting on the carrier liquid is changed and thus the
flow of carrier liquid is also influenced by the magnetic force. Because of the relative velocity
between the solid and liquid particles, the net effect of the particles suspended in the fluid is
extra dragging force acting on the system. Taking this force into consideration, Saffman (1962)
proposed the equations of the flow of suspension of non-magnetic particles. These equations
were subsequentlymodified to describe the flow of magnetic fluid with themagnetic body force
µ0M∇H taken into account by Wagh (1991). Wagh and Jawandhia (1996) have studied the
transport of vorticity in amagnetic fluid.Yan andKoplik (2009) have studied the transport and
sedimentation of suspended particles in inertial pressure-driven flow.
In all the above studies, the fluidwas considered asNewtonian, butmany industrially impor-

tant fluids (moltenplastics, polymers, pulpsand foods) exhibit anon-Newtonianfluidbehaviour.
Many common materials (paints and plastics) and more exotic ones (silicic magma, saturated
soils, and the Earth’s lithosphere) behave as viscoelastic fluids. With the growing importance
of non-Newtonian materials in various manufacturing and processing industries, considerable
effort has been directed towards understanding their flow. The stability of a horizontal layer of



128 P. Kumar et al.

Maxwell’s viscoelastic fluid heated frombelow has been investigated byVest andArpaci (1969).
The nature of the instability and other factors can affect viscoelastic fluids differently thanNew-
tonian fluids. For example, Bhatia and Steiner (1972) have considered the effect of a uniform
rotation on the thermal instability of Maxwellian viscoelastic fluid, where rotation is found to
have a destabilizing effect. This is in contrast to the thermal instability of a Newtonian fluid
where rotation has a stabilizing effect. The thermal instability of anOldroydian viscoelastic fluid
acted on by a uniform rotation has been studied by Sharma (1976). There aremany viscoelastic
fluids that cannot be characterized by Maxwell’s or Oldroyd’s constitutive relations. One such
fluid is Rivlin-Ericksen viscoelastic fluid, having relevance and importance in geophysical fluid
dynamics, chemical technology andpetroleum industry.Rivlin andEricksen (1955) have studied
the stress-deformation relaxation for isotropicmaterials.Garg et al. (1994) have studied the drag
on a sphere oscillating in conducting dustyRivlin-Ericksen viscielastic fluid.Thermal instability
in Rivlin-Ericksen viscoelastic fluid in the presence of rotation and magnetic field, separately,
has been investigated by Sharma and Kumar (1996, 1997b). Sharma and Kumar (1997a) have
studied the hydromagnetic stability of two Rivlin-Ericksen viscoelastic superposed conducting
fluids and the analysis has been carried out, for two highly viscous fluids of equal kinematic
viscosities and equal kinematic viscoelasticities wherein it was found that the stability criterion
is independent of the effects of viscosity and viscoelasticity and is dependent on the orientation
and magnitude of the magnetic field. The stability of two superposed Rivlin-Ericksen visco-
elastic fluids in the presence of suspended particles has been considered by Kumar and Singh
(2006). Kumar et al. (2007) have studied the hydrodynamic and hydromagnetic stability of two
stratified Rivlin-Ericksen viscoelastic superposed fluids.

Keeping in mind the importance of viscoelastic fluids in modern technology and industries,
the present paper attempts to study the transport of vorticity in magnetic Rivlin-Ericksen
viscoelastic fluid-particle mixtures by using the equations proposed by Wagh and Jawandhia
(1996).

2. Basic assumptions and magnetic body force

The particles of magnetic material are much larger than the molecules of the carrier liquid.
Accordingly, we consider the limit of a microscopic volume element in which the fluid can be
assumed to be a continuous medium and the magnetic particles must be treated as discrete
entities. If we consider a cell of magnetic fluid containing a larger number of magnetic particles,
then we must consider the microrotation of the cell in addition to its translations as a point
mass.Wemust, therefore, assign average velocity qd and the average angular velocity ω to the
cell. But, here as an approximation, we neglect the effect of microrotation. We also assume the
following:

(i) The free current density J is negligible, and J×B is insignificant.

(ii) Themagnetic field is curl free i.e.∇×H=0.

(iii) The liquid compressibility is unimportant in many practical situations. Hence, the con-
tribution due to magnetic friction can be neglected. The remaining force of the magnetic
field is referred as magnetization force.

(iv) All time-dependent magnetization effects in the fluid (such as hysteresis) are negligible,
and the magnetization M is collinear withH.

From electromagnetic theory, the force per unit volume (inMKS units) on a piece ofmagne-
tizedmaterial ofmagnetization M (i.e. dipolemoment per unit volume) in the field ofmagnetic
intensity H is µ0(M ·∇)H, where µ0 is the free space permeability.



Vorticity transport analysis in magnetic viscoelastic fluid 129

Here we note that

∇·a=
∂a1
∂x
+
∂a2
∂y
+
∂a3
∂z

a ·∇= a1
∂

∂x
+a2

∂

∂y
+a3

∂

∂z

where a= a1i+a2j+a3k.
Using assumption (iv), we obtain

µ0(M ·∇)H=
µ0M

H
(H ·∇)H (2.1)

whereM = |M| andH = |H|.
But by assumption (ii), we have

(H ·∇)H=
1

2
∇(H ·H)−H× (∇×H)=

1

2
∇(H ·H) (2.2)

Hence

µ0(M ·∇)H=
(µ0M

H

)1

2
∇(H ·H)=µ0M∇H

Themagnetic body force therefore becomes (Rosensweig, 1997)

fm =µ0M∇H (2.3)

3. Derivation of the equations governing the vorticity transport in a magnetic

Rivlin-Ericksen viscoelastic fluid

Let Γij, τij, eij, δij, ui, xi, p, µ and µ
′ denote the stress tensor, shear stress tensor, rate-of-

strain tensor, Kronecker delta, velocity vector, position vector, isotropic pressure, viscosity and
viscoelasticity, respectively. The constitutive relations for the Rivlin-Ericksen viscoelastic fluid
(Rivlin and Ericksen, 1955; Sharma and Kumar, 1997a) are

Γij =−pδij + τij τij =2
(

µ+µ′
∂

∂t

)

eij eij =
1

2

(∂ui
∂xj
+
∂uj
∂xi

)

To describe the flow of a magnetic fluid by including the body force µ0M∇H acting on the
suspendedmagnetic particles,Wagh (1991)modified the Saffman’s equations for flow of suspen-
sion. The equations for the flow of suspendedmagnetic particles and the flow of Rivlin-Ericksen
viscoelastic fluid in which magnetic particles are suspended are therefore written as

mN
(∂V

∂t
+(V ·∇)V

)

=mNg+µ0M∇H+KN(u−V)

ρ
(∂u

∂t
+(u ·∇)u

)

=−∇P +ρg+
(

µ+µ′
∂

∂t

)

∇2u+KN(V−u)

(3.1)

where P, ρ, u(ux,uy,uz), g(0,0,−g), V(l,r,s), m and N(x,t) respectively denote the pressure
minus hydrostatic pressure, density, velocity of fluid particles, gravity force, velocity of solid
particles, particle mass and particle number density. Moreover, x = (x,y,z), and K = 6πµη,
where η is the particle radius, is the Stokes’ drag coefficient.
If we assume that the particle has a uniform spherical shape and that the velocity relative

to the fluid is small, then in the equations of motion for the viscoelastic fluid, because of the
presence of suspended particles, an additional force term appears proportional to the velocity
difference between the suspended particles and the fluid. Since the force exerted by the fluid



130 P. Kumar et al.

on the suspended particles is equal and opposite to that exerted by the particles on the fluid,
an additional force term equal in magnitude but opposite in sign appears in the equations of
motion for the suspended particles. We neglect the buoyancy force on the particles. This force
is proportional to the quotient of ρ and the particle density, and an analysis in the case of
free-free boundary conditions (no tangential stresses) shows that its small stabilizing effect is
negligible. We also assume that the distances between particles are quite large compared with
their diameter, and we therefore also ignore particle interactions.
By making use of the Lagrange’s vector identities

(qd ·∇)qd =
1

2
∇q2d−qd×Ω (q ·∇)q=

1

2
∇q2−q×Ω1 (3.2)

Equations (3.1) become

mN
[∂V

∂t
− (V×Ω)

]

=−∇mNgz−
1

2
mN∇V2+µ0M∇H+KN(u−V)

ρ
[∂u

∂t
− (u×Ω1)

]

=−∇P −∇ρgz−
1

2
ρ∇u2+

(

µ+µ′
∂

∂t

)

∇2u+KN(V−u)

(3.3)

whereΩ=∇×V andΩ1 =∇×u are solid vorticity and fluid vorticity.
Taking the curl of these equations and recalling that the curl of a gradient is identically equal

to zero, we obtain

mN
[∂Ω

∂t
− (∇×V×Ω)

]

=µ0∇×M∇H+KN(Ω1−Ω)

ρ
[∂Ω1
∂t
− (∇×u×Ω1)

]

=
(

µ+µ′
∂

∂t

)

∇2Ω1+KN(Ω−Ω1)

(3.4)

By making use of the vector identities

∇× (V×Ω)= (Ω ·∇)V− (V ·∇)Ω+V∇·Ω−Ω∇·V=(Ω ·∇)V− (V ·∇)Ω

∇× (u×Ω1)= (Ω1 ·∇)u− (u ·∇)Ω1+u∇·Ω1−Ω1∇·u=(Ω1 ·∇)u− (u ·∇)Ω1

(3.5)

Equations (3.4) become

mN
DΩ

Dt
=µ0∇×M∇H+mN(Ω ·∇)V+KN(Ω1−Ω)

DΩ1
Dt
=
(

ν+ν′
∂

∂t

)

∇2Ω1+(Ω1 ·∇)u+
KN

ρ
(Ω−Ω1)

(3.6)

where ν and ν′ are kinematic viscosity and kinematic viscoelasticity, respectively and
D
Dt
≡ ∂
∂t
+(V ·∇) is the convective derivative.

In equation (3.6)1

∇× (M∇H)= (∇M×∇H)+(M∇×∇H) (3.7)

Since the curl of the gradient is zero, the last term in equation (3.7) is zero. Also since
M =M(H,T).
Therefore

∇M =
(∂M

∂H

)

∇H+
(∂M

∂T

)

∇T (3.8)

By making use of (3.8), equation (3.7) becomes

∇× (M∇H)=
(∂M

∂H

)

∇H×∇H+
(∂M

∂T

)

∇T ×∇H (3.9)



Vorticity transport analysis in magnetic viscoelastic fluid 131

The first term on the right hand side of this equation is clearly zero, hence we get

∇× (M∇H)=
(∂M

∂T

)

∇T ×∇H (3.10)

Substituting this expression in equation (3.6)1, we obtain

mN
DΩ

Dt
=µ0

(∂M

∂T

)

∇T ×∇H+mN(Ω ·∇)V+KN(Ω1−Ω) (3.11)

Here (3.6)2 and (3.11) are the equations governing the transport of vorticity inmagnetic Rivlin-
Ericksen viscoelastic fluid-particle mixtures.

In equation (3.11), the first term in the right-hand side i.e. µ0(∂M/∂T)∇T ×∇H describes
the production of vorticity due to thermo-kinetic processes. The last term KN(Ω1−Ω) gives
the change in solid vorticity on account of the exchange of vorticity between the liquid and solid.

It follows from equations (3.6)2 and (3.11) that the transport of solid vorticity Ω is coupled
with the transport of fluid vorticity Ω1.

From equation (3.11), we see that if solid vorticity Ω is zero, then the fluid vorticity Ω1 is
not-zero and it is given by

Ω1 =−
µ0
KN

(∂M

∂T

)

∇T ×∇H (3.12)

This implies that due to thermo-kinetic processes, fluid vorticity can exist in the absence of solid
vorticity.

From equation (3.6)2, we find that ifΩ1 is zero, thenΩ is also zero. This implies that when
fluid vorticity is zero, then solid vorticity is necessarily zero.

In the absence of suspendedmagnetic particles,N is zero andmagnetizationM is also zero.
Then, equation (3.11) is identically satisfied and equation (3.6)2 reduces to

DΩ1
Dt
=
(

ν+ν′
∂

∂t

)

∇2Ω1+(Ω1 ·∇)q (3.13)

This equation is thevorticity transport equation.The last termonthe righthand sideof equation
(3.13) represents the rate at which Ω1 varies for a given particle, when the vortex lines move
with the fluid, the strengths of the vortices remaining constant. The first term represents the
rate of dissipation of vorticity through friction (resistance) and rate of change of vorticity due
to fluid viscoelasticity.

3.1. Two-dimensional case

Here we consider the two-dimensional case:

Let

V= vx(x,y)i+vy(x,y)j u=ux(x,y)i+uy(x,y)j (3.14)

where components vx, vy and ux, uy are functions of x, y and t, then

Ω=Ωzk Ω1 =Ω1zk (3.15)

In two-dimensional case, equation (3.11) becomes

DΩz
Dt
=
µ0
mN

(∂M

∂T

)(∂T

∂x

∂H

∂y
−
∂H

∂x

∂T

∂y

)

+
K

m
(Ω1z −Ωz) (3.16)



132 P. Kumar et al.

and equation (3.6)2 similarly becomes

DΩ1z
Dt
= ν∇2(Ω1z)+ν

′
∂

∂t
∇2(Ω1z)+

KN

ρ
(Ωz −Ω1z) (3.17)

since it can be easily verified that

(Ω ·∇)V=0 (Ω1 ·∇)u=0 (3.18)

The first term on the right hand side of equation (3.17) is the change of fluid vorticity due to
internal friction (resistance). The second term is the rate of change of fluid vorticity due to fluid
viscoelasticity and the third term is change in fluid vorticity due to the exchange of vorticity
between solid and liquid. Equation (3.17) does not explicitly involve the term representing
change of vorticity due to magnetic field gradient and/or temperature gradient. But equation
(3.16) shows that solid vorticity Ωz depends on these factors. Hence, it follows that the fluid
vorticity is indirectly influenced by the temperature and the magnetic field gradient.
In the absence of magnetic particles, N is zero andmagnetization M is also zero. Equation

(3.16) is therefore satisfied identically, and equation (3.17) reduces to the classical equation for
the transport of fluid vorticity. If we consider a suspension of non-magnetic particles instead of
amagnetic fluid, then the corresponding equation for the transport of vorticitymay be obtained
by setting M equal to zero in the equations governing the transport of vorticity in magnetic
fluids. If magnetization M of the magnetic particles is independent of temperature, then the
first term of equations (3.11) and (3.16) vanishes, and the equations governing the transport of
vorticity in a magnetic fluid become the same as those governing the transport of vorticity in
non-magnetic suspensions.
If the temperature gradient ∇T vanishes or if the magnetic field gradient ∇H vanishes or

if∇T is parallel to∇H, then also the first term in (3.11) and (3.16) vanishes.We thus see that
the transport of vorticity in a magnetic fluid is also the same as the transport of vorticity in
non-magnetic suspension.

Acknowledgements

The authors are grateful to the learned referee for his useful technical comments and valuable sugge-

stions, which led to a significant improvement of the paper.

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Manuscript received December 24, 2014; accepted for print July 13, 2015