NSM solution for unsteady MHD Couette flow of a dusty conducting fluid with variable viscosity and electric conductivity

Applied Mathematical Modelling 35 (2011) 303–316

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NSM solution for unsteady MHD Couette flow of a dusty conducting fluid with variable viscosity and electric conductivity P. Eguía a, J. Zueco b,*, E. Granada a, D. Patiño a a b

Department of Mechanics, Thermal Machines and Motors and Fluids, University of Vigo, 36310 Vigo, Spain ETSII, Department of Engineering Thermal and Fluids, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain

a r t i c l e

i n f o

Article history: Received 9 March 2009 Received in revised form 26 May 2010 Accepted 4 June 2010 Available online 10 June 2010 Keywords: Dusty fluid Couette flow Magnetohydrodynamics Heat transfer Transient state Network simulation method

a b s t r a c t The effects of dependence on temperature of the viscosity and electric conductivity, Reynolds number and particle concentration on the unsteady MHD flow and heat transfer of a dusty, electrically conducting fluid between parallel plates in the presence of an external uniform magnetic field have been investigated using the network simulation method (NSM) and the electric circuit simulation program Pspice. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field perpendicular is applied to the plates. We solved the steady-state and transient problems of flow and heat transfer for both the fluid and dust particles. With this method, only discretization of the spatial co-ordinates is necessary, while time remains as a real continuous variable. Velocity and temperature are studied for different values of the viscosity and magnetic field parameters and for different particle concentration and upper wall velocity. Ó 2010 Published by Elsevier Inc.

1. Introduction There have been numerous theoretical and experimental studies of heat and mass transfer of fluids in channels. These studies have many applications in physical systems, such as, chemical reactors, nuclear reactors, combustion systems, pneumatic transport etc. In some of these applications, the fluid may contain suspended dust particles. In this paper, the procedure called network simulation method (NSM) is applied for the numerical solution of the unsteady Couette flow and heat conduction of an electrically conducting, viscous, incompressible dusty fluid. The hydrodynamic flow of dusty fluids has been studied by a number of authors [1–5]. Later investigations studied the influence of magnetic fields on the flow of electrically conducting dusty fluids [6–10]. To achieve more accurate prediction of the flow and heat transfer it is necessary to take into account the variation of the physical properties with temperature, especially the variation of fluid viscosity with temperature [11] and the effects of an external uniform magnetic field. Zueco [12] studied the effect of temperature-dependent specifics and thermal conductivity in unsteady free convection processes with mass transfer. Klemp et al. [13] studied the effect of temperature-dependent viscosity on the entrance flow in a channel in the hydrodynamic case. Attia and Kotb [14] studied steady MHD fully developed flow and heat transfer between two parallel plates with temperature-dependent viscosity. Sharma and Varshney [15] investigated the effects of thermal dispersion and viscous dissipation on the unsteady flow of a viscous incompressible dusty gas through a hexagonal channel of uniform cross section under the influence of a magnetic field and time-dependent pressure gradient. Attia [16] studied the unsteady Couette flow and heat transfer of a dusty conducting fluid between two parallel plates with temperature-dependent viscosity and thermal conductivity with an exponential decaying pressure gradient and an external uniform magnetic field.

* Corresponding author. Tel.: +34 968 32 59 89; fax: +34 968 32 59 99. E-mail address: [email protected] (J. Zueco). 0307-904X/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.apm.2010.06.005

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Nomenclature a B0 C Cs Ci Ec G Gi Ha j L0 mp N p Pr R Ri t0 t T0 T u U v V

v0 x y y0 y Z0 Z

viscosity parameter magnetic field specific heat capacity of the fluid specific heat capacity of particles capacitor at i nodal point Eckert number particle mass dimensionless current source at i nodal point Hartmann number electric current dimensionless temperature relaxation time particles mass number of dust particles per volume pressure Prandtl number particle concentration resistor at i nodal point time dimensionless time fluid temperature dimensionless fluid temperature fluid velocity dimensionless fluid velocity particles velocity dimensionless particles velocity y component of the fluid velocity axial co-ordinate dimensionless vertical co-ordinate vertical co-ordinate dimensionless vertical co-ordinate particles temperature dimensionless particles temperature

Greek symbols a pressure gradient cT temperature relaxation time q density of clean fluid r electric conductivity Dy vertical thickness of the cell l dynamic viscosity Subscripts i associated with i nodal point i, i  D, i + D associated to the center, left and right position on the cell

This paper studies the effect of variable viscosity, electrical conductivity, particle concentration and upper wall velocity on the unsteady laminar flow of an electrically conducting, viscous, incompressible dusty fluid and heat transfer between parallel non-conducting plates. The fluid is flowing between two infinite plates maintained at two constant but different temperatures. An external uniform magnetic field is applied perpendicular to the plates. The spatial co-ordinates of the governing equations are discretized while time remains as a continuous variable, transforming the partial differential equations into a set of ordinary equations (ODEs) which can be described by an elemental network cell. Various studies have been carried out using NSM. For example, Zueco [17] employed it to study the radiation and viscous dissipation effects on MHD unsteady free convection over a vertical porous plate. Zueco [18] obtained a solution for Burgers’ equation by means of the NSM. 2. Physical and mathematical model The dusty fluid is assumed to be flowing between two infinite horizontal plates located at the y0 = ±h planes, as shown in Fig. 1.

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305

Fig. 1. The geometry of the problem.

The dusty particles are assumed to be uniformly distributed throughout the fluid. The two plates are assumed to be electrically non-conducting and kept at two constant temperatures: T1 for the lower plate and T2 for the upper plate with T2 > T1. A constant pressure gradient is applied in the x-direction and the upper plate is moving with a constant velocity U0 while the lower plate is kept stationary. A uniform magnetic field B0 is applied in the positive y-direction. By assuming a very small magnetic Reynolds number the induced magnetic field is neglected. The fluid motion starts from rest at t = 0, and the no-slip condition at the plates implies that the fluid and dust particles velocities have neither a z nor an x-component at y0 = ±h. The initial temperatures of the fluid and dust particles are assumed to be equal to T1 and the fluid viscosity is assumed to vary exponentially with temperature. The electrical conductivity varies lineally with temperature. Since the plates are infinite in the x and z-directions, the physical variables are invariant in these directions. The flow of the fluid is governed by the Navier–Stokes equation

q

@u dp @ ¼ þ 0 @t0 dx @y



l

 @u  rðTÞB20 u  KNðu  mÞ; @y0

ð1Þ

where q is the density of clean fluid, l is the viscosity of clean fluid, u is the velocity of fluid, v is the velocity of dust particles, r is the electric conductivity, p is the pressure acting on the fluid, N is the number of dust particles per unit volume. The motion of the dust particles is governed by Newton’s second law via

mp

@m ¼ KNðu  mÞ: @t 0

ð2Þ

The initial conditions are given by

t0 ¼ 0;

u ¼ m ¼ 0:

ð3Þ

For t0 > 0 the no-slip condition implies that

y0 ¼ h;

u ¼ m ¼ 0;

ð4Þ

y0 ¼ þh;

u ¼ m ¼ U0 :

ð5Þ

The two energy equations required to describe the temperature distributions for both the fluid and dust particles and are respectively given by

 2 qp C s 0 @T 0 @2T 0 @u ¼ k þ l þ rB20 u2 þ ðZ  T 0 Þ; @y0 @t 0 @y02 cT @z0 1 ¼  ðz0  T 0 Þ; @t 0 cT

qc

ð6Þ ð7Þ

where T0 is the temperature of the fluid, Z0 is the temperature of the particles, C is the specific heat capacity of the fluid at constant pressure, Cs is the specific heat capacity of the particles, k is the thermal conductivity of the fluid and cT is the temperature relaxation time.

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The initial and boundary conditions on the temperature fields are given as

t0 6 0; t0 > 0;

T 0 ¼ Z 0 ¼ T 01 ; y0 ¼ h; T 0 ¼ Z 0 ¼ T 01 ;

ð8Þ ð9Þ

t0 > 0;

y0 ¼ þh;

T 0 ¼ Z 0 ¼ T 02 :

ð10Þ

The viscosity of the fluid is assumed to vary exponentially with temperature [13,14]. The function f1(T0 ) takes the form 0 0 f1 ðT 0 Þ ¼ ea1 ðT T 1 Þ , where the parameter a1 has the dimension of T0 1 and such that at T 0 ¼ T 01 ; f ðT 01 Þ ¼ 1 and then l = l0. Also the electric conductivity is assumed to vary with a linear dependence with temperature in the form f2 ðT 0 Þ ¼ 1 þ b1  ðT 02  T 01 ÞT 0 . Writing the equations in dimensionless form

@U @ 2 U 1 @f1 ðTÞ @U Ha2 R ¼ a þ f1 ðTÞ  2 þ   f2 ðTÞU  ðU  VÞ; @t @y Re @y @y Re Re @V G ¼ ðU  VÞ; @t t  0; U ¼ V ¼ 0; t > 0; y ¼ 1; U ¼ V ¼ 0; t > 0; y ¼ þ1; U ¼ 1; V ¼ 0;  2 @T 1 @ 2 T Ec @U Ec 2R ¼  f ðZ  TÞ; þ ðTÞ þ f2 ðTÞ Ha2 U 2 þ 1 @t Re  Pr @y2 Re @y Re 3Re  Pr @Z ¼ L0 ðZ  TÞ; @t t  0; T ¼ Z ¼ 0; t > 0; y ¼ 1; T ¼ Z ¼ 0; t > 0; y ¼ þ1; T ¼ Z ¼ 1;

ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ ð18Þ ð19Þ ð20Þ

where f1(T) = eaT where a is the viscosity parameter; f2(T) = 1 + bT where b the electronic conductivity parameter; y = y0 /h, dimensionless vertical co-ordinate; U = u/U0, fluid velocity non-dimensional; V = v/U0, particle velocity non-dimensional; t = t0 U0/h, time non-dimensional; T ¼ ðT 0  T 01 Þ=ðT 02  T 01 Þ, fluid temperature non-dimensional; Z ¼ ðZ 0  T 01 Þ=ðT 02  T 01 Þ, particle 2 temperature non-dimensional; Ha2 ¼ rB20 h =l0 , Ha is the Hartmann number; Re = qU0h/l0 is the Reynolds number; R = KNh2/l0 represents the particle concentration; G = mpl0/qh2K represents the particle mass; Pr = l0C/k is the Prandtl number; Ec ¼ U 20 =ðCðT 2  T 1 ÞÞ is the Eckert number; Lo = qh2/l0cT represents the temperature relaxation time. 3. Numerical procedure: network simulation method By means of the network model of Figs. 2–5, with the appropriate numerical values for the parameters of the system and using the computation program Pspice [19], the temporal response for a MHD dusty fluid with temperature-dependent

Fig. 2. Nomenclature and network model of the fluid velocity U.

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Fig. 3. Nomenclature and network model of the particles velocity V.

Fig. 4. Nomenclature and network model of the fluid temperature T.

Fig. 5. Nomenclature and network model of the particles temperature Z.

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physical properties problem can easily be obtained. NSM is based on the classical thermoelectric analogy between thermal and electrical variables. The ability of NSM to incorporate any kind of non-linearity into the model (due to boundary conditions, phase-change processes, temperature dependencies of the thermal properties etc.) distinguishes NSM from the analogies generally expounded in monographs. The first phase of NSM is the formulation of a set of finite-differential equations, one for each control volume, obtained by spatial discretization of the transformed transport equations. Electrical and thermal systems are analogous if they are formulated in the same domain with similar equations and identical boundary conditions. It is necessary to extend the electrical analogy when momentum and energy equations are treated. In this analogy, the variable voltage is equivalent to velocities (u, v) and temperature (T), while the electric current j is equivalent to the velocity and temperature fluxes. The NSM has been applied successfully to a great variety of linear and non-linear transport problems [12,17,18,20]. NSM simulates the behavior of unsteady electric circuits by means of resistors, capacitors and non-linear devices that seek to resemble thermal systems governed by unsteady linear or non-linear equations. This method permits the solution for both transient and steady-state problems at the same time, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. A spatial discretization is performed using the following expressions:

@f1 @ðeaT Þ eaT iþDy  eaT iDy ¼ ¼ ; @y @y Dy @u U iþDy  U iDy ¼ ; @y Dy @ 2 u ðU iDy  U i Þ  ðU i  U iþDy Þ ¼ ; @y2 Dy2 =2 @ m V iþDy  V iDy ¼ ; @y Dy 2 @ m ðV iDy  V i Þ  ðV i  V iþDy Þ ¼ ; @y2 Dy2 =2   @f2 ðTÞ 1 þ bT iþDy  1  bT iDy bðT iþDy  T iDy Þ ¼ ¼ ; @y Dy Dy @T T iþDy  T iDy ¼ ; @y Dy @ 2 T ðT iDy  T i Þ  ðT i  T iþDy Þ ¼ ; @y2 Dy2 =2 @Z Z iþDy  Z iDy ¼ ; @y Dy 2 @ Z ðZ iDy  Z i Þ  ðZ i  Z iþDy Þ ¼ : @y2 Dy2 =2

ð21Þ ð22Þ ð23Þ ð24Þ ð25Þ ð26Þ ð27Þ ð28Þ ð29Þ ð30Þ

Four circuits are developed representing each non-dimensional equation. Using the above expressions, the dimensionless governing equations take the following form for the Momentum balance:

 aT  dU i e iþDy  eaT iDy U iþDy  U iDy Ha2 1 ðU iDy  U i Þ  ðU i  U iþDy Þ a þ ð1 þ bT i ÞU i  eaT i  Re dt Dy Dy Re Dy2 =2 R þ ðU i  V i Þ ¼ 0: Re

ð31Þ

The above equation defines the following currents:

U iDy  U i ; Dy2 =2 U i  U iþDy ; jU;iþD ¼ Dy2 =2 dU i ; jUC;i ¼ dt ðHa2 þ RÞ ; jUR;i ¼ U i Re  aT  iþ D y e  eaT iDy U iþDy  U iDy Ha2 ðU iDy  U i Þ  ðU i  U iþDy Þ R þ V i: jUG;i ¼ a þ  bT i U i þ ð1 þ eaT i Þ  D y2 Re Dy Dy Re

jU;iD ¼

ð32:aÞ ð32:bÞ ð32:cÞ ð32:dÞ ð32:eÞ

2

With this currents Eq. (31) can be written in the form of Kirchhoff’s law,

J U;iþD  jU;iD þ jUC;i þ jUR;i  jUG;i ¼ 0;

ð33Þ

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where jU,i+D and jU,iD are the currents that leave and enter the cell for the fluid velocity and are implemented by means of two resistors of value Dy2/2. Current jUC,i is implemented by means of a capacitor of capacity the unity. Current jUR,i is implemented by means of a resistor of value Re/(Ha2 + R). One voltage control current generator is necessary to implement jUG,i. This current relates the fluid velocity U with particle velocity V and the fluid temperature T. Connecting the above five elements and in agreement with Kirchhoff’s current law, the basic electrical circuit of Fig. 2 can be obtained, corresponding to any cell i. Connecting N basic circuits in series gives the network representing the one-dimensional medium in which the velocity U is the voltage in the center of the cell i. The next step is to include the initial and boundary conditions in the network model. The velocity V of particles is governed by the following expression:

dV i 1  ðU i  V i Þ ¼ 0: G dt

ð34Þ

Three currents are derived from Eq. (34)

dV i ; dt Vi jVR;i ¼ ; G Ui jVG;i ¼ : G

jVC;i ¼

ð35:aÞ ð35:bÞ ð35:cÞ

Again, Eq. (34) can be written in the form of Kirchhoff’s law

J VC;i þ jVR;i  jVG;i ¼ 0:

ð36Þ

Current jVC,i represents the variation of particle velocity V with time and is implemented with a capacitor of capacitance unity. A resistor of value G is used to represent jVR,i and a voltage control current generator relates the particle velocity V with the fluid velocity U as follows. The energy equation is presented below

  dT i 1 ðT iDy  T i Þ  ðT i  T iþDy Þ Ec  eaT i Ec 2R ðU iþDy  U iDy Þ2  ð1 þ bT i ÞHa2 U 2i  ðZ i  T i Þ ¼ 0:   2 Re 3Re  Pr dt Re  Pr Dy =2 Re  Dy2

ð37Þ

Again, five currents are defined and implemented as follows:

T iDy  T i ; Re  Pr  Dy2 =2 T i  T iþDy ¼ ; Re  Pr  Dy2 =2

jT;iD ¼

ð38:aÞ

jT;iþD

ð38:bÞ

jTC;i ¼

dT i ; dt

2R ; 3Re  Pr aT i Ec  e Ec 2R Zi : ¼ ðU iþDy  U iDy Þ2 þ ð1 þ bT i ÞHa2 U 2i þ Re 3Re  Pr Re  Dy2

ð38:cÞ

jTR;i ¼ T i

ð38:dÞ

jTG;i

ð38:eÞ

The energy Eq. (37) is then written in the form of Kirchhoff’s law,

J T;iþD  jT;iD þ jTC;i þ jTR;i  jTG;i ¼ 0;

ð39Þ

where jT,i+D and jT,iD are the currents that leave and enter the cell for the fluid temperature and are implemented by means of two resistors of value RePrDy2/2. Current jTC,i is implemented by means of a capacitor of capacity unity. Current jTR,i is implemented by means of a resistor of value 3RePr/2R. One voltage control current generator is necessary to implement jTG,i . This current relates the fluid temperature T with the fluid velocity U and particle temperature Z. Particle temperature Z, in the discrete form, is represented by the following equation:

dZ i þ L0 ðZ i  T i Þ ¼ 0: dt

ð40Þ

The currents for this network are defined and implemented by 40.a, 40.b, 40.c and represented in Fig. 5

jZC;i ¼

ð40:aÞ

jZR;i

dZ i ; dt ¼ L0 Z i ;

ð40:bÞ

jZG;i ¼ L0 T i :

ð40:cÞ

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Eq. (40) in the form of Kirchhoff’s law is then

J ZC;i þ jZR;i  jZG;i ¼ 0:

ð41Þ

Current jZC,i represents the variation of particle temperature Z with time and is implemented with a capacitor of capacitance unity. A resistor of value 1/L0 is used to represent jZR,i and a voltage control current generator relates the particle temperature Z with the fluid temperature T calculating L0Ti. To implement the boundary conditions in the four circuits, it is necessary to use constant voltage sources of values 0 and 1 to represent null velocity at y = h, maximum velocity at y = +h, null temperature at y = h and maximum temperature at y = +h. 4. Results and discussion In the following discussion selected parameters are given the following fixed values:

G ¼ 0:8;

a ¼ 5; Pr ¼ 1; Ec ¼ 0:2 and L0 ¼ 0:7:

4.1. Variation of the Reynolds number ‘‘Re” Figs. 6 and 7 indicate the variations with time of the velocities U and V at the center of the channel (y = 0) and the variations with time of the temperatures T and Z at the center of the channel (y = 0) for different values of Re with constant values for a, b, Ha and R. With higher Reynolds number a higher value of the velocity U0 is obtained. As expected, higher values

Fig. 6. The evolution of U (continuous) and V (dotted) for different values of Re (a = 0, b = 0, Ha = 1, R = 0.8).

Fig. 7. The evolution of T (continuous) and Z (dotted) for different values of Re (a = 0, b = 0, Ha = 1, R = 0.8).

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in the velocity of the upper wall results in higher velocities of the fluid and the particles. Also, a higher velocity gradient increments the Joule dissipation and the temperatures T and Z. The steady state profile of velocities and temperatures are presented in Figs. 8 and 9. Higher Reynolds number results in higher velocities and temperatures for all y. 4.2. Variation of the particle concentration parameter ‘‘R” Small values of R indicate a poor presence of mass particles in the fluid. The steady state velocities and temperatures are reached in a smaller time. This is shown in Figs. 10 and 11. 4.3. Variation of the viscosity parameter ‘‘a” The effect of the viscosity parameter a on the unsteady velocities and temperatures is represented on Figs. 12 and 13. The source of the dust particles velocity is the fluid velocity and that’s the reason why U reaches the steady state more quickly than V. It’s also shown that the influence of a is negligible for some time and increases as time develops. Increasing parameter a, the viscous forces are reduced faster and the values of velocities increase. As a result of this, the time required to approach the steady state is also increased at the center of the channel. The distribution of the steady velocities and temperatures are shown in Figs. 14 and 15. Negative values of a are equivalent to a variable injection normal for the plates and positive values of a are equivalent to a variable suction normal for the plates. Higher values of a increase the values of T for all y due to the effect of the Joule heating. The steady-state results for

Fig. 8. Distribution of velocity U for different values of Re (a = 0, b = 0, Ha = 1, R = 0.8).

Fig. 9. Distribution of temperature T for different values of Re (a = 0, b = 0, Ha = 1, R = 0.8).

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Fig. 10. The evolution of U (continuous) and V (dotted) for different values of R (a = 0, b = 0, Ha = 1, Re = 1).

Fig. 11. The evolution of T (continuous) and Z (dotted) for different values of R (a = 0, b = 0, Ha = 1, Re = 1).

Fig. 12. The evolution of U (continuous) and V (dotted) for different values of a (b = 0, Ha = 0, Re = 1, R = 0.8).

the temperature of the fluid T (Fig. 15) are compared with those obtained by Attia [21]. The excellent degree of agreement of the present results with those of this author using finite difference methods can be appreciated.

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Fig. 13. The evolution of T (continuous) and Z (dotted) for different values of a (b = 0, Ha = 0, Re = 1, R = 0.8).

Fig. 14. Distribution of velocity U for different values of a (b = 0, Ha = 1, Re = 1, R = 0.8).

Fig. 15. Distribution of temperature T for different values of a (b = 0, Ha = 1, Re = 1, R = 0.8).

4.4. Variation of the conductivity parameter ‘‘b” Fig. 16 shows that increasing b decreases the velocities of the fluid and the particles because of the increasing magnetic forces. Fig. 17 shows that the effect of this parameter on the value of T at the center of the channel (y = 0) is small because of

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Fig. 16. The evolution of U (continuous) and V (dotted) for different values of b (a = 0, Ha = 1, Re = 1, R = 0.8).

Fig. 17. The evolution of T (continuous) and Z (dotted) for different values of b (a = 0, Ha = 1, Re = 1, R = 0.8).

the sum of two effects: on the one hand, increments in b reduce the velocities and the Joule dissipation, and on the other, increments in b result in higher electric conductivity and higher Joule dissipation. Fig. 18 presents the steady state velocities of the fluid for all y with different values of b. The peak of velocity is nearer the upper wall when b is smaller. The steady-state results for the velocity of the fluid U are compared with those obtained by Attia [21]. Excellent agreement is observed between the NSM solutions and those of Attia [21].

Fig. 18. Distribution of velocity U for different values of b (a = 0, Ha = 1, Re = 1, R = 0.8).

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P. Eguía et al. / Applied Mathematical Modelling 35 (2011) 303–316 Table 1 Variation of U with time for different values of Ec (a = 0.5, b = 0.5, Ha = 1, Re = 1 and R = 0.8).

t=1 t=2 t=3 t=5 t = 10

Ec = 0.01

Ec = 0.1

Ec = 0.25

Ec = 0.5

1.8030 2.0000 2.0710 2.1090 2.1150

1.8010 1.9920 2.0590 2.0930 2.0970

1.7940 1.9700 2.0260 2.0490 2.0490

1.7780 1.9100 1.9380 1.9390 1.9300

Table 2 Variation of T with time for different values of Ec (a = 0.5, b = 0.5, Ha = 1, Re = 1 and R = 0.8).

t=1 t=2 t=3 t=5 t = 10

Ec = 0.01

Ec = 0.1

Ec = 0.25

Ec = 0.5

0.3991 0.4671 0.4949 0.5193 0.5306

0.5278 0.6796 0.7452 0.7993 0.8226

0.7525 1.0590 1.1920 1.2980 1.3410

1.1560 1.7480 1.9920 2.1680 2.2310

Table 3 Variation of U with time for different values of Ec (a = 0.5, b = 0.5, Ha = 0, Re = 1 and R = 0.8).

t=1 t=2 t=3 t=5 t = 10

Ec = 0.01

Ec = 0.1

Ec = 0.25

Ec = 0.5

2.5570 3.1450 3.4060 3.6000 3.6660

2.5800 3.2370 3.5660 3.8570 3.9970

2.6170 3.3910 3.8540 4.3870 4.8550

2.6750 3.6450 4.3740 5.5790 8.3410

Table 4 Variation of T with time for different values of Ec (a = 0.5, b = 0.5, Ha = 0, Re = 1 and R = 0.8).

t=1 t=2 t=3 t=5 t = 10

Ec = 0.01

Ec = 0.1

Ec = 0.25

Ec = 0.5

0.3922 0.4600 0.4893 0.5158 0.5285

0.4529 0.6018 0.6849 0.7701 0.8200

0.5511 0.8382 1.0278 1.2634 1.4799

0.7074 1.2274 1.6304 2.2749 3.5265

4.5. Variation of the Eckert number ‘‘Ec” Eckert number represents the relation between the kinetic energy and the enthalpy difference of the fluid at the upper and lower wall. The unsteady velocity and temperature of fluid and particles are presented in Tables 1 and 2 with different values of Ec. Here, Ha = 1 and the effects of the damping magnetic forces induce higher values of Ec that reduce U in y = 0. On the other hand, the temperatures T increases with Ec because of the Joule dissipation. It’s also shown that times to approach the steady state velocity and temperature are reduced for higher Ec. When Ha = 0 the damping effect disappears and the steady state velocities and temperatures grow when Ec grows as shown in Tables 3 and 4. The time to approach the steady state conditions also grows significantly. 5. Conclusions In this work we have demonstrated some of the possibilities that NSM has for analysing unsteady Couette flow of dusty fluids with variable physical properties. More precisely, we have shown how simple it is to introduce the temperaturedependence of the system’s coefficients into the network model, which is especially useful when hard non-linearities are present in the equations of the process. This temperature-dependency introduces a strong degree of non-linearity in the mathematical model, which does not pose a real problem in designing the network. The network simulation method (NSM) computational technique has been described in detail and employed to solve the partial differential equations for a wide range of the governing thermophysical parameters. The main results of the investigation may be briefly summarized as follows:

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– Increasing the viscosity parameter a increases velocities U, V and temperatures T and Z for all values of y. The time needed to approach the steady state also increases. – A higher value for the electrical conductivity parameter b results on lower velocities of the fluid and particles. The temperatures at the center of the channel remain almost constant with the changes of b. – The application of a higher velocity of the upper wall (represented by a higher Reynolds number) increases the values of the velocities and temperatures and the time needed to approach the steady state. – A higher particle concentration reduces the velocities and temperatures and the time needed to approach those values. – Variations with the Eckert number depend on the application of the magnetic field represented by the Hartmann number Ha. When Ha = 1, higher Ec numbers induce lower velocities (U and V) and higher temperatures (T and Z). If Ha = 0, also the velocities and temperatures grow when Ec grows. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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