Mechanics, Materials Science & Engineering, September 2016
ISSN 2412-5954
Fundamental Solutions for Micropolar Fluids with Two-Temperature M. Zakaria1, 2, a 1 Mathematics department, Faculty of Science, Al-Baha University, Al-Baha, P.O. Box 1988, Kingdom of Saudi Arabia 2
Mathematics department, Faculty of Education, El-Shatby Alexandria University, Alexandria, 21526, Egypt
a
zakariandm@yahoo.com DOI 10.13140/RG.2.2.28685.95201
Keywords: micropolar, two-temperature, low-Reynolds-number ABSTRACT. New fundamental solutions for steady two temperature micropolar fluids are derived in explicit form for two- and three dimensional steady unbounded Stokes and Oseen flows due to a point force, a point couple, and time harmonic sources including the two-dimensional micropolar Stokeslet, the two- and three-dimensional micropolar Stokes couplet, the three-dimensional micropolar Oseenlet, and the three-dimensional micropolar Oseen couplet. These fundamental solutions do not exist in Newtonian flow due to the absence of microrotation velocity field. The flow due to these singularities is useful for understanding and studying microscale flows. As an application, the drag coefficients for a solid sphere or a circular cylinder that translates in a lowReynolds-number micropolar flow are determined and compared with those corresponding to Newtonian flow.
1. Introduction. In the theory of micropolar fluids [1], rigid particles contained in a small volume element can rotate about the centroid of the volume element. An independent microrotation vector describes the rotation. Micropolar fluids can support body couples and exhibit microrotational effects. The theory of micropolar fluids has shown promise for predicting fluid behaviour at microscale. Papautsky et al. [2] found that a numerical model for water flow in microchannels based on the theory of micropolar fluids gave better predictions of experimental results than those obtained using the Navier Stokes equation. Micropolar fluids can model anisotropic fluids, liquid crystals with rigid molecules, magnetic fluids, clouds with dust, muddy fluids, and some biological fluids [3]. In view of their potential application in microscale fluid mechanics and non-Newtonian fluid mechanics, it is worth exploring new fundamental solutions.
NOMENCLATURE del operator; V
velocity vector of the fluid;
P
pressure; viscosity; coupling viscosity coefficient or vortex viscosity;
r
F
constant vector;
xi
coordinates system, i
U
free stream;
Ir
micro-inertia;
cr,cm
two angular viscosities;
microrotation vector; q
heat flux vector;
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Srinivas and Kothandapani [4] studied the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. Chen [5] performed a magnetohydrodynamic mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption. Many authors have extended the above model to the flow of micropolar fluid past an infinite horizontal moving plate, in the presence of a magnetic under different physical situations (see for instance [6 10]).
o
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relaxation time;
k thermal conductivity; CE
specific heat at constant pressure; density of the fluid;
T
dynamical temperature of the fluid;
To
reference temperature chosen;
Q
strength of the applied heat source per unit mass; the conductive temperature;
The physical mechanisms of heat, mass, and a a > 0 two-temperature parameter momentum transport in small-scale units may differ significantly from those in micro-scale equipment [11, 12]. Fundamental and applied investigations of micro-scale phenomena in fluid mechanics are motivated by developments in the areas of biological molecular machinery, atherogenesis, microcirculation, and microfluidics. At scales larger than a micron, the fluid can be treated as a continuum, and the flow is governed by the Navier Stokes equation. The continuum model assumes that the properties of the material vary continuously throughout the flow domain. In Newtonian continuum mechanics, the fluid is modeled as a dense aggregate of particles, possessing mass, and translational velocity. However, the field equation, such as the Navier Stokes equation, does not account for the rotational effects of the fluid micro-constituents. In the theory of micropolar fluids [13], rigid particles contained in a small volume element can rotate about the centroid of the volume element. The rotation is described by an independent micro-rotation vector. Micropolar fluids can support body couples and exhibit microrotational effects. The theory of micropolar fluids has shown promise for predicting fluid behaviour at microscale. Papautsky et al. [11] found that a numerical model for water 2. The mathematical model. Consider a point force in an unbounded, quiescent, incompressible micropolar fluid. Without loss of generality, the point force is placed at the origin, and the free-stream velocity U is taken to be (U , 0, 0). Based on the Oseen approximation, the governing equations reduce to [11] The equations of motion have the form
(1)
(2)
I rU
x1
2
r
V
2
cr
cm
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,
(3)
Mechanics, Materials Science & Engineering, September 2016
ISSN 2412-5954
The equation of energy is given by:
(4)
The entropy may be written in terms of temperature as follows
(5)
-[14]
(6)
(7)
By eliminating between (4) and (5) and using (6) we get the equation of heat conduction for the linear theory as follows
(8)
The pressure, p, translational velocity, V |x in an unbounded flow, as follows
3. The formulation of the problem in the Fourier transform domain The Fourier transform f
of a function
is the transformed variable of x i
is the imaginary unit, i=
in the n-dimensional complex Fourier is defined as:
x=
1 x1
nxn
. The divergence of (2) yields MMSE Journal. Open Access www.mmse.xyz
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(9)
which states that p is harmonic everywhere except at the pole. To solve (9) for p, we take the Fourier transform, finding
Taking the inverse Fourier transform, we find
(10)
Stokes and Oseen flows due to a point force have the same pressure field, regardless of whether the fluid is Newtonian or micropolar. From (3), we have
(11)
Taking the curl of (2) gives
(12)
Substituting (11) in (12), we derive a partial differential vector equation containing only one
(13)
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where
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,
To solve partial differential equations of higher order, such as (13), we may factorize the higher-order partial differential operator into products of lower order [4]. This method was used by Olmstead and Majumdar [5]. Formally, it is proposed that
where L
is a fourth-order partial differential operator;
A1, A2, B1 and B2
are constants.
While the method of factorization is attractive, a certain relationship between the parameters must exist for L to admit the desired factorization. To factorize the differential operator in (16), the following must be true:
(14)
Consequently, it is required that
We have five equations and only four unknowns. To expedite the solution, the value of B1 is taken to be zero since B1B2 = 0. Therefore,
Then, from A1B2 + B1A2 = a3,
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Consequently,
However, A1 + A2
a1, or
which gives
(15)
Hence, the partial differential operator in (14) can be factorized as follows:
This allows (13) to be rewritten as
(16)
where 2no =
/ and 2mo =
+
r
). The above factorization is valid under the physical
transform.
which gives MMSE Journal. Open Access www.mmse.xyz
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(17)
The inverse Fourier transform gives
which is the micropolar Oseenlet of
(18)
where
and
is the
satisfied. As U
(19)
Equation (19) gives the microrotation velocity in the presence of a point force. The curl of (3) gives the curl of the curl of V as
which can be written as MMSE Journal. Open Access www.mmse.xyz
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transform, we find
Substituting (14) and (16) in the above equation leads to
The inverse Fourier transform of this expression yields u in the form
We find the translational velocity u is given by the micropolar Oseenlet of V
F
1
e
o
1
e
o
x1 t
x1
V F
x1
e not K o ( no s ) e o 2 U
x1 t
e no ( t
e mot
e mot K o (
o
s)
ns
dt , n
2,
(20) x1 t
s)
e 4 sU
o
x1 t
os
1
dt , n
3.
In the limit r , the translational velocity and microrotation velocity fields decouple. Then, mo no o , and the expression (19) of V simplifies to o
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Mechanics, Materials Science & Engineering, September 2016
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(21)
We see that the Newtonian Oseenlet is recovered. In the limit U in (19) becomes the micropolar Stokeslet of V,
V
(22)
Finally, By eliminating transform.
between (7) and (8) for T, we obtain it is convenient to take the Fourier
(23)
It is convenient to take the Fourier transform.
(24)
The inverse Fourier transform gives
which is the dynamical temperature of the fluid T
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Mechanics, Materials Science & Engineering, September 2016
ISSN 2412-5954
(25)
where
and
Now, by eliminating theory as follows
between (7) and (8) we get the equation of heat conduction for the linear
(26)
Taking the Fourier transform, we find.
(27)
The inverse Fourier transform gives
which is the conductive temperature of the fluid
(28)
Finally, we find the translational velocity u is given by the micropolar Oseenlet of V The solution of
for a micropolar fluid is much more complicated than that for a Newtonian fluid. MMSE Journal. Open Access www.mmse.xyz
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4. Drag on a translating solid sphere in a micropolar viscous flow Consider the flow produced by a solid sphere of radius R translating with velocity U in an ambient micropolar fluid of infinite expanse. The flow due to the sphere may be obtained in terms of a point force and a potential dipole, both placed at the center of the sphere, as in the case of Stokes flow [15, 16]. Hence, the velocity is given by
(25)
where B is the vectorial strength of the potential dipole; is the Reynolds number, assumed to be small. Requiring the boundary condition V = U at r = R to be satisfied on the surface of the sphere, yields two algebraic equations for the coefficients F and B,
whose solution is
The drag comes exclusively from the point force. The dimensionless drag coefficient is
Putting
r=
0, we recover the result for the classical viscous flow
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Summary. New fundamental solutions for micropolar fluids with two temperature have been derived in explicit form. The problem of two- and three-dimensional, steady, unbounded Stokes and Oseen flows of a micropolar fluid due to a point force and a point couple has been considered. The new fundamental solutions for Stokes and Oseen flows are the two-dimensional micropolar Stokeslet, given by (17), (20), (24) and (27) , the three-dimensional micropolar Oseenlet, given by (18) and (19). These fundamental solutions are possible due to the existence of microrotation velocity fields in micropolar fluids. The fundamental solutions can generate further fundamental solutions by successive differentiation with respect to the singular point [15,16]. These fundamental solutions for micropolar fluids can be used as the basic building blocks to construct new solutions of microscale flow problems by employing the boundary-integral method or the singularity method. It was demonstrated that these fundamental solutions can be used to calculate the drag coefficients for a translating solid sphere and circular cylinder, respectively, in a micropolar fluid at low Reynolds numbers. The drag coefficients in a micropolar fluid are greater than those in a Newtonian fluid by the factor + r . References [1] I. Papautsky, J. Brazzle, T. Ameel, A. B. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory. Sensors Actuators A Physical (1999) 73(1-2):101 108, DOI 10.1016/S09244247(98)00261-1 [2] J. J. Shu , Microscale heat transfer in a free jet against a plane surface. Superlattices Microstruct (2004) 35: 645 656, DOI 10.1016/j.spmi.2003.12.005 [3] A. C. Eringen, Microcontinuum field theories II: fluent Media. Springer-Verlag, New York, Inc (2001). [4] S. Srinivas, M. Kothandapani: The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, App. Math. Comp. 213 (2009) 197 208, DOI 10.1016/j.amc.2009.02.054 [5] Chien-Hsin: Magneto-hydrodynamic mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption, Int. J. Non-Linear Mech. 44 (2009) 596 603, DOI 10.1016/j.ijnonlinmec.2009.02.004 [6] S. Srinivas, M. Kothandapani: The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Appl. Math. Comp.213 (2009) 197 208, DOI 10.1016/j.amc.2009.02.054 [7] R. C. Chaudhary, Abhay Kumar Jha: Effects of chemical reactions on MHD micropolar fluid flow past a vertical plate in slip-flow regime, Appl. Math. Mech. -Engl. Ed., 2008, 29(9):1179 1194, DOI 10.1007/s10483-008-0907-x [8] Youn J. Kim: Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium, Transport in Porous Media 56: 17 37, 2004, DOI 10.1023/B:TIPM.0000018420.72016.9d [9] A. Ishak , R. Nazar , I. Pop: MHD boundary-layer flow of a micropolar fluid past a wedge with variable wall temperature, Acta Mech 196, 75 86 (2008), 10.1007/s00707-007-0499-8 [10] A. Ishak, R. Nazar, I. Pop: Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet, Meccanica (2008) 43: 411 418, DOI 10.1007/s11012-007-9103-5 [11] Papautsky I, Brazzle J, Ameel T, Frazier AB: Laminar fluid behavior in microchannels using micropolar fluid theory. Sensors Actuators A Physical 73(1 2):101 108 (1999) MMSE Journal. Open Access www.mmse.xyz
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[12] Shu J-J: Microscale heat transfer in a free jet against a plane surface. Superlattices Microstruct 35(3 6):645 656 (2004) [13] Eringen AC: Microcontinuum field theories II: fluent Media. Springer- Verlag, New York, Inc (2001) [14] Zwillinger D: Handbook of differential equations. Academic Press (1998) [15] Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press [16] Kohr M, Pop I (2004) Viscous incompressible flow for low Reynolds numbers. WIT Press Cite the paper M. Zakaria (2016). Fundamental Solutions for Micropolar Fluids with TwoTemperature. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.28685.95201
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