Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer

Physica B 405 (2010) 4188–4194

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Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer Magdy A. Ezzat Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

a r t i c l e in f o

a b s t r a c t

Article history: Received 14 May 2010 Received in revised form 5 July 2010 Accepted 6 July 2010

In this work, a new mathematical model of thermoelectric MHD theory has been constructed in the context of a new consideration of heat conduction with fractional order. This model is applied to Stokes’ first problem for a viscoelastic fluid with heat sources. Laplace transforms and state-space techniques [1] will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameter on all the studied fields. & 2010 Elsevier B.V. All rights reserved.

Keywords: Thermoelectric MHD Viscoelastic fluid Fourier’s law Modified Riemann–Liouville fractional derivative New fractional Taylor’s series Stokes’ first problem State space approach Laplace transforms Numerical method

1. Introduction Although the tools of fractional calculus have been available and applicable to various fields of study, the investigation of the theory of fractional differential equations has only been started quite recently [2]. The differential equations involving Riemann– Liouville differential operators of fractional order 0 o a o1 appear to be important in modelling several physical phenomena [3] and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations. Fractional calculus has been used successfully to modify many existing models of physical processes. The first application of fractional derivatives was given by Abel who applied fractional calculus in the solution of an integral equation that arises in the formulation of the tautochrone problem. One can state that the whole theory of fractional derivatives and integrals was established in the 2nd half of the 19th century. Caputo and Mainardi [4,5] and Caputo [6] found good agreement with experimental results when using fractional derivatives for description of

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viscoelastic materials and established the connection between fractional derivatives and the theory of linear viscoelasticity. Some applications of fractional calculus to various problems of mechanics of viscoelastic fluid are reviewed in the literature [7–11]. In most of these investigations the effect of the thermal state in the fluid flow is not taken into account. In recent years, the study of viscoelastic fluid flow is an important type of flow occurring in several engineering processes. Such processes are wire drawing, glass fiber and paper production, crystal growing, drawing of plastic sheets, among which we also cite many applications in petroleum in drilling, manufacturing of foods and slurry transporting [12,13]. In the current work, a new formula of heat conduction equation with the time fractional derivative of order a has been derived in the context of generalized thermoelectric MHD theory. The governing coupled equations in the frame of the boundary layer are applied to Stocks’ first problem of an electro-conducting viscoelastic flow over an infinite plate with heat source distribution in the presence of a transverse magnetic field normal to the plate. Laplace transforms and state space approach techniques are used to get the solution. Laplace transforms are obtained using the complex inversion formula of the transform together with Fourier expansion techniques. The effects of various physical parameters on various momentum and heat transfer characteristics are discussed in detail and represented graphically.

M.A. Ezzat / Physica B 405 (2010) 4188–4194

2. Derivation of the governing heat transfer equation Thermoelectric devices have many attractive features compared with the conventional fluid-based refrigerators and power generation technologies, such as long life, no moving part, no noise, easy maintenance and high reliability. However, their use has been limited by the relatively low performance of present thermoelectric materials. The efficiency of a thermoelectric material is related to the so-called dimensionless thermoelectric figure-of-merit ZT. The thermoelectric figure-of-merit provides a measure of the quality of such materials for applications and is defined in Ref. [14] 2

ZT ¼

s0 S k

T

ð1Þ

where S is the Seebeck coefficient, s0 the electrical conductivity, T the absolute temperature and k is the thermal conductivity. The best thermoelectric materials that are currently in devices have a value of ZTc1. A related effect (the Peltier effect) was discovered a few years later by Peltier, who observed that if an electrical current is passed through the junction of two dissimilar materials, heat is either absorbed or rejected at the junction depending on the direction of the current. This effect is due to the difference in Fermi energies of the two materials. The absolute temperature T, the Seebeck coefficient S and the Peltier coefficient P are related by the first Thomson relation as in Ref. [15]

P ¼ ST

ð2Þ

The classical heat conduction equation has the property that the heat pulses propagate at infinite speed. Much attention was recently paid to the modification of the classical heat conduction equation, so that the pulses propagate at finite speed. Mathematically speaking, this modification changes the governing partial differential equation from parabolic to hyperbolic type. Cattaneo [16] was the first to offer an explicit mathematical correction of the propagation speed defect inherent in Fourier’s heat conduction law. Cattaneo’s theory allows for the existence of thermal waves, which propagate at finite speeds. Starting from Maxwell’s idea [17] and from paper by Cattaneo [16], an extensive amount of literature [18–20] has contributed to elimination of the paradox of instantaneous propagation of thermal disturbances. The approach is known as extend irreversible thermodynamics, which introduces time derivative of the heat flux vector, Cauchy stress tensor and its trace into the classical Fourier law by preserving the entropy principle. Joseph and Preziosi [20] give a detailed history of heat conduction theory. In addition to discussing various other models of heat conduction, these authors state that Cattaneo’s equation is the most obvious and simple generalized form of Fourier’s law that gives rise to a finite speed of propagation. Puri and Kythe [21] investigated the effects of using the (Maxwell–Cattaneo) model in Stoke’s second problem for a viscous fluid. They also studied the effects of discontinuous boundary data on the velocity gradients temperature fields occurring in Stoke’s first problem for a viscous fluid. They also noted that in the theory of generalized thermofluid, the nondimensional thermal relaxation time t0 defined as t0 ¼C Pr , where C and Pr are the Cattaneo and Prandtl numbers, respectively, is of the order of 10  2. The Fourier law, modified in this way, established an impact equation relating heat flux vector, velocity and temperature. The energy equation in terms of the heat conduction vector q is

rCp

D T ¼ rUq þ F Dt

ð3Þ

4189

where r is the density of the fluid and Cp the specific heat at constant pressure and "    2  2   @u 2 @v @w @w @v 2 þ F¼m 2 þ2 þ2 þ @x @y @z @y @z  2  2 # @u @w @v @u þ þ þ þ ð4Þ @z @x @x @y is the internal heat due to viscous stresses and the operator D/Dt defined as D @ ¼ þ ðVUrÞ Dt @t

ð5Þ

is the rate of dissipation of energy per unit time per unit volume. The generalized Fourier’s and Ohm’s laws in MHD for a thermoelectric medium are proposed by Shercliff [22] q ¼ krT þ PJ

ð6Þ

J ¼ s0 ðE þ V  BSrTÞ

ð7Þ

where J is the conduction current density vector, E and B are, respectively, the electric density and the magnetic flux density vectors and V¼(u,v,w) is the vector velocity of the fluid. Substituting Eq. (6) in Eq. (3) we shall obtain the well-known energy equation D 2 T ¼ kr TrUPJ þ F ð8Þ Dt Theoretically, Fourier’s heat-conduction equation leads to the solutions exhibiting infinite propagation speed of thermal signals. It was shown by Cattaneo [16] and Lebon and Rubi [18] that it is more reasonable physically to replace Eq. (6) by the following generalized Fourier’s law of heat conduction including the current density effect to obtain [23]

rCp

@q ¼ krT þ PJ ð9Þ @t Using the new fractional Taylor’s series of fractional order [24], the generalized Fourier law of heat conduction including the current density effect, can be considered as

q þ t0

qþ

ta0 @a q ¼ krT þ PJ, 0 o a r 1 a! @ta

ð10Þ

Now taking the partial time derivative of fraction order a of Eq. (3), we get    a  D @a T @ q @a F rCp r U ð11Þ ¼  þ a a a Dt @t @t @t Multiplying Eq. (11) by ta0 =a! and adding to Eq. (3) we obtain       ta @a T ta @a q ta @a F D T þ 0 a ¼ rU q þ 0 a þ F þ 0 a rCp ð12Þ Dt a! @t a! @t a! @t Substituting from Eq. (10), we get     ta @a T ta @a F D 2 T þ 0 a ¼ kr TrUPJ þ F þ 0 a rCp Dt a! @t a! @t

ð13Þ

Taking into account the definition of D/Dt from Eq. (5), we arrive at    ta @a T @ þ ðVUrÞ T þ 0 a @t a! @t   ta @a F k 1 1 2 r T rUPJ þ Fþ 0 a ð14Þ ¼ rCp rCp rCp a! @t Eq. (14) is the generalized energy equation with fraction time derivative of order a (0 o a r1), taking into account the relaxation time t0 . This generalization eliminates the paradox of infinite speed of propagation of heat in thermoelectric fluid.

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M.A. Ezzat / Physica B 405 (2010) 4188–4194

3. The magnetic induction has one non-vanishing component

3. Limiting cases

Bz ¼ m0 H0 ¼ B0 ðconstantÞ:

Taking into consideration 8 f ðy,tÞf ðy,0Þ a-0 > > > > > @f ðy,tÞ < a @ 0oao1 Ia1 , f ðy,tÞ ¼ @t a > @t > > @f ðy,tÞ > > a¼1 : @t

4. The Lorentz force F ¼ J4B, has one component in the x-direction is @T Fx ¼ s0 B20 us0 k0 B0 : ð19Þ @y

Ia

where the notion is the Riemann–Liouville fractional integral is introduced as a natural generalization of the well-known n-fold repeated integral In f ðtÞ written in a convolution-type form as in Ref. [25] Ia f ðtÞ ¼

Zt 0

5. The continuity equation @u ¼ 0: @y

ð20Þ

6. The equation of motion with modified Ohm’s law @u @2 u @3 u s0 B20 s0 k0 B0 @T ¼ u 2 l0 :  u 2 @t r r @y @y @t@y

) ðtBÞa1 f ðBÞ dB GðaÞ a 40

ð21Þ

7. The generalized energy equation

I0 f ðtÞ ¼ f ðtÞ the heat Eq. (14) in the limiting case a ¼1 transforms to      @ @T k 1 1 @F þðVUrÞ T þ t0 ¼ r2 T rUPJ þ F þ t0 @t @t rCp rCp rCp @t

rC p

@ @t

þ



Tþ



ta0 @a T @2 T @u þQ ¼ ðk þ s0 p0 k0 Þ 2 þ p0 s0 B0 a @y a! @t @y

ta0 @a Q : a! @ta

ð22Þ

ð15Þ which is the same equation obtained by the generalized theory [23]. In the limiting case and using homogenous initial condition a ¼0, Eq. (14) reduces to   @ k 1 1 þðVUrÞ T ¼ r2 T rUPJ þ F ð16Þ @t rCp rCp rCp

Let us introduce the following non-dimensional variables: y ¼

U y, u

Pr ¼

Cp m

t ¼ ,

U2 t, u

M¼

t0 ¼

us0 B20 , rU 2

U2 t0 , u q ¼

u ¼ u

u , U

l0 ¼

U2 l, u2

q,

k kT0 U k0 s0 B0 T0 p0 us0 B0 TT1 u2 Q K0 ¼ , P0 ¼ , y¼ , Q ¼ kT0 T0 rU 2 kU 2 T0

which is the same equation obtained by the coupled theory of thermoelectric MHD [22].

ð23Þ Eqs. (21) and (22) are reduced to the non-dimensional equations

4. Application: Stokes’ first problem with heat source distribution

  @u @ @2 u @y ¼ 1l0 , MuK 0 @t @t @y2 @y

ð24Þ

Consider the laminar flow of an incompressible conducting viscoelastic fluid above the non-conducting half-space y 40. Taking the positive y-axis of the Cartesian coordinate system in the upward direction and the fluid flows through half space y40 above and in contact with the plane surface occupying xz-plane. A constant magnetic field of strength H0 acts in the z direction. The induced electric current due to the motion of the fluid that is caused by the buoyancy forces does not distort the applied magnetic field. The previous assumption is reasonably true if the magnetic Reynolds number of the flow (Rm ¼ UL0 s0 me ) is assumed to be small, which is the case in many aerodynamic applications where rather low velocities and electrical conductivities are involved. Under these conditions, no flow occurs in the y and z directions and all the considered functions at a given point in the half space depend only on its y-coordinate and time t and V ¼ ðu,0,0Þ are the vector velocity of the fluid. Given the above assumptions, we have

From now on, we shall consider a heat source of the form Q ¼ Q 0 dðyÞHðtÞ. To simplify the algebra, only problems with zero initial conditions are considered. Appling the Laplace transform defined by the formulas [26] R1 LfgðtÞg ¼ gðsÞ ¼ 0 est gðtÞ dt

1. The figure-of-merit ZT at some reference temperature, T0 ¼ Tw T1 , is defined as

8 9 80 > > > y ðy,sÞ > > > > > > > > > > <0 < = uðy,sÞ d ¼ C dy > > > yuðy,sÞ > > > > > > > > : uuðy,sÞ > ; > :0

s0 k20 T ZT0 ¼ k 0

ð17Þ

where k0 is the Seebeck coefficient at T0, Tw the temperature of the plate and T1 the temperature of the fluid away from the plate. 2. The first Thomson relation at room temperature is

p0 ¼ k0 T0 , where p0 is the Peltier coefficient at T0.

ð18Þ

Pr

! ta @a þ 1 ta @a Q @ @2 y @u þ 0 a þ 1 y ¼ ð1 þZT0 Þ 2 þ P0 þQ þ 0 : @t a! @t @y a! @ta @y

LfDn gðtÞg ¼ sn LfgðtÞg

n 40

ð25Þ

ð26Þ

on both sides of Eqs. (24) and (25) and writing the resulting equations in matrix form results in 0 0

1 0

0 sþ M 1l0 s

0

98 9 8 9 > > yðy,sÞ > > 0> > > > > > > >> > > > > > <0> = =< uðy,sÞ = , Q 0 bdðyÞ D >> > > > 1 y uðy,sÞ > > > >> > > > > > : > ; > > : > ; 0 ; uuðy,sÞ 0 0 1

K0 1l0 s

ð27Þ where

C¼

  ta s Pr 1 þ 0 sa , 1 þZT0 a!

D¼

P0 1 þ ZT0

and b ¼

  ta 1 1þ 0 sa : 1 þ ZT0 a!

M.A. Ezzat / Physica B 405 (2010) 4188–4194

where

Eq. (27) can be written in constricted form as Guðy,sÞ ¼ AðsÞGðy,sÞ þ Bðy,sÞ,

ð28Þ

where Gðy,sÞ denotes the state vector in the transform domain whose components consist of the transformed temperature, velocity and induced magnetic field as well as their gradients. In order to solve the system (28), we need first to find the form of the matrix exp(A(s)y). The characteristic equation of the matrix A(s) has the form   s þM K0 D ðs þ MÞC k4   þ C k2 þ ¼ 0, ð29Þ 1l0 s 1l0 s 1l0 s where k is a characteristic root. The Cayley–Hamilton theorem states that the matrix A satisfies its own characteristic equation in the matrix sense. Therefore, it follows that   sþM K0 D ðs þMÞC A4   þ C A2 þ I ¼ 0: ð30Þ 1l0 s 1l0 s 1l0 s Eq. (30) shows that A4 and all higher powers of A can be expressed in terms of A3, A2, A and I, the unit matrix of order 4. The matrix exponential can now be written in the form exp½Ay ¼ a0 ðy,sÞI þ a1 ðy,sÞAðsÞ þ a2 ðy,sÞA2 ðsÞ þ a3 ðy,sÞA3 ðsÞ:

ð31Þ

The scalar coefficients of Eq. (30) are now evaluated by replacing the matrix A by its characteristic roots 7 k1 and 7 k2 , which are the roots of the biquadratic Eq. (29), satisfying the relations   sþM K0 D  þC , ð32aÞ k21 þ k22 ¼ 1l0 s 1l0 s k21 k22 ¼

ðs þ MÞC : 1l0 s

ð32bÞ

expð 7k1 yÞ ¼ a0 7 a1 k1 þ a2 k21 7a3 k31 ,

ð33aÞ

expð 7k2 yÞ ¼ a0 7 a1 k2 þ a2 k22 7a3 k32 :

ð33bÞ

By solving the system of linear Eq. (11), we can determine a0  a3 (see Appendix A). Substituting for the parameters a0  a3 into Eq. (31), computingA2 and A3 and using Eqs. (32a) and (32b), one can obtain after some lengthy algebraic manipulations i,j ¼ 1,2,3,4

ð34Þ

where the elements ‘ij (y,s) are given in Appendix B. In the actual physical problem the space is divided into two regions, accordingly as y Z 0 or yo 0, inside the region 0 r y o1 the positive exponential terms, not bounded at infinity, must be suppressed. Thus, for y Z 0 we should replace each sinh(ky) by (1/2)exp(  ky) and each cosh(ky) by (1/2)exp( ky). In the region yr 0 the negative exponentials are suppressed instead. We shall now proceed to obtain the solution of the problem for the regiony Z0. The solution for the other region is obtained by replacing each y by –y. The formal solution of Eq. (28) can be expressed as   Z y Gðy,sÞ ¼ exp½Aðy,sÞy Gð0,sÞ þ exp½AðsÞzBðz,sÞ dz ð35Þ

3 k1 k2 þs þ M 6 ð1l0 sÞðk1 þ k2 Þ 7 6 7 7 Q0 b 6 60 7 nðsÞ ¼  6 7 7 4s 6 1 6 7 4 5 K0 ð1l0 sÞðk1 þ k2 Þ Eq. (36) expresses the solution of the problem in the Laplace transform domain in terms of the vector nðsÞ representing the applied heat source and the vector Gð0,sÞ representing the conditions at the plane source of heat. In order to evaluate the components of this vector, we note first that due to the symmetry of the problem, the temperature is a symmetric of y while the velocity is anti-symmetric. It thus follows that uð0,tÞ ¼ 0 or uð0,sÞ ¼ 0:

ð37Þ

Gauss’s divergence theorem will now be used to obtain the thermal condition at the plane source. We consider a short cylinder of unit base whose axis is perpendicular to the plane source of heat and whose bases lie on opposite sides of it. Taking limits as the height of the cylinder tends to zero and noting that there is no heat flux through the lateral surface, upon using the symmetry of the temperature field we get qð0,tÞ ¼

Q0 Q0 HðtÞ or qð0,sÞ ¼ : 2 2s

ð38Þ

Using Fourier’s law of heat conduction in the non-dimensional form, namely q¼

ð½1 þZT0 yu þ P0 uÞ 1 þ ðta0 =a!Þsa

ð39Þ

we obtain the condition

bQ0 2s

:

ð40Þ

Eqs. (37) and (40) give two components of the vector Gð0,sÞ. In order to obtain the remaining two components, we substitute y¼0 on both sides of Eq. (36) obtaining a system of linear equations whose solution gives   bQ0 k1 k2 þðs þ MÞ=ð1l0 sÞ , ð41Þ yð0,sÞ ¼ 2sk1 k2 ðk1 þ k2 Þ uuð0,sÞ ¼

bK0 Q0 2sð1l0 sÞðk1 þk2 Þ

ð42Þ

Inserting the values from Eqs. (37), (41) and (42) into the righthand side of Eq. (36) and performing the necessary matrix operations, we obtain  2  k1 sM 7 k1 y k22 sM 7 k2 y b Q0 yðy,sÞ ¼ e  e , ð43Þ k1 k2 2sðk21 k22 Þ uðy,sÞ ¼

h i 7 bK0 Q0 e 7 k1 y e 7 k2 y : 2 2 2sð1l0 sÞðk1 k2 Þ

ð44Þ

In the above equations the upper (plus) sign indicates the solution in the region yo0, while the lower (minus) sign indicates the region yZ0.

5. Numerical inversion of the Laplace transforms

0

Evaluating the integral in Eq. (35) using the integral properties of the Dirac delta function, we obtain Gðy,sÞ ¼ Lðy,sÞ½Gð0,sÞ þ nðsÞ,

2

yuð0,sÞ ¼ 

This leads to the system of equations

exp½AðsÞy ¼ Lðy,sÞ ¼ ½‘ij ðy,sÞ,

4191

ð36Þ

In order to invert the Laplace transform in the above equations, we adopt a numerical inversion method based on a Fourier series expansion [27]. In this method, the inverse g(t) of the Laplace

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M.A. Ezzat / Physica B 405 (2010) 4188–4194

transform gðsÞ is approximated by the relation "     !#  N X ec t 1 ikpt ikp , gðtÞ ¼ gðc Þ þ Re exp g c þ t1 2 t1 t1 k¼1

generalized case (a 40). In the first and older theory, the waves propagate with infinite speeds, so the value of any of the functions is not identically zero (though it may be very small)

0 rt r2t 1 ,

ð45Þ

0.003

n

where c is an arbitrary constant greater than all the real parts of the singularities of g(t) and N is sufficiently large integer chosen such that      iN pt iNp  ec t Re exp g c þ ð46Þ r e, t1 t1

 = 0.5 G–F Theory

6. Numerical results and discussion

0.0015

0.001

0

2

4 6 8 Figure-of-merit, ZTo

10

12

Fig. 2. A plot of the temperature as a function of the thermoelectric figure of merit for different values of fractional order a.

0.15  = 0.0 C–Theory  = 0.5 G–F Theory

0.1

 = 1.0 G– Theory

0.05

-6

-4

-2

0

0

2

4

6

Velocity,u

significant effect on all the fields. It is noticed that all the waves reach the steady state depending on the value of the fractional order a. The important phenomenon observed in all computations is that the solution to any of the functions considered vanishes identically outside a bounded region of space surrounding the heat source at a distance from it equal to yn(t), say yn(t) is a particular value of y depending only on the choice of t and is the location of the wavefront. This demonstrates clearly the difference between the solution corresponding to the classical use of the Fourier heat equation (a ¼ 0.0) and to the use of

0

Velocity,u

 In all figures, it is noticed that the fractional order a has a



 = 1.0 G– Theory

0.0005

The investigation of the effect of the fractional order a on the flow of a fluid over the boundaries with heat sources distribution in the presence of a magnetic field has been carried out in the preceding sections. The computations were performed for a value of time, namely, t¼1. The numerical technique outlined above was used to obtain the temperature y, velocity component u and heat flux q. The results are shown in Figs. 1–4. The graphs show the three curves predicted by the different theories of thermoelectric MHD, classical Fourier’s law for coupled theory (C-Theory, a ¼0.0), generalized fractional order theory (G-F Theory, a ¼0.5) and generalized theory with one relaxation time (G-Theory, a ¼1.0). Hence, we conclude with the following points:



 = 0.0 C–Theory

0.002 Temperatue, θ

where e is a prescribed small positive number that corresponds to the degree of accuracy required. The cited numerical procedure is used to invert the expressions of temperature and velocity fields in Laplace transform domain.

0.0025

-0.05

-0.1

-0.15 Distance, y Fig. 3. A plot of the velocity against distance for different values of fractional order a.

0.6 0.4  = 0.0 C–Theory

0.5

 = 1.0 G– Theory

 = 0.0 C–Theory

Temperatue, θ

Temperatue, θ

 = 0.5 G–F Theory  = 1.0 G– Theory

0.2

Heat flux, q

0.4 Heat flux, q

0.3

 = 0.5 G–F Theory

0.3

0.2

0.1 0.1

-6

-4

-2

0

0 Distance, y

2

4

6

Fig. 1. A plot of the temperature against distance y for different values of fractional order a.

-6

-4

-2

0

0

2

4

6

Distance, y Fig. 4. A plot of the heat flux against distance y for different values of fractional order a.

M.A. Ezzat / Physica B 405 (2010) 4188–4194









for any large value of y. In the non-Fourier theory, the response to the thermal and mechanical effects does not reach infinity instantaneously, but remains in a bounded region of space given by 0oyoyn(t) for the semi-space problem and by Min (0, yn(t)  c) oyoy+yn(t) for the whole space problem [28]. In Fig. 1 we notice that the temperature increment in the generalized fractional order theory is a continuous function, which means that the particles transport the heat to the other particles easily and this makes the decrease in rate of the temperature greater than the other ones. In the coupled theory the temperature exhibits a jump discontinuity at y¼0.3, while in the generalized theory the thermal waves exhibit two jumps discontinuities at (y¼0.3 and y¼2.85), but cut the x-axis rapidly than the other ones. Fig. 2 presents some data on the figure-of-merit ZT0 as a function of temperature of various thermoelectric fluids [23]. We notice that the efficiency of a thermoelectric material figure-of-merit is proportional to the temperature of the fluid particles [29]. The velocity distributions for a problem for the infinite space in the presence of heat sources are represented graphically in Fig. 3 and for different values of fractional order a. From this figure we learn that the velocity of the fluid particles increase with decrease in the fractional order a. This means that the boundary layer region around the surface is dependent on the fractional order a. Investigation of heat transfer involves measuring a heat flux on a surface along which a liquid moves [30]. It is observed from Fig. 4 that heat flux increases with increase in the fractional order a.

7. Conclusion The main goal of this work is to introduce a new mathematical model for Fourier law of heat conduction with time-fractional order and includes the thermoelectric figure-of-merit. This model enables us to improve the efficiency of a thermoelectric material figure-of-merit according to the choice of suitable values of fractional derivative order a. The result provides a motivation to investigate conducting thermoelectric materials as a new class of applicable thermoelectric non-Newtonian fluids. Experimental studies confirmed that the one-dimensional model could be used for heat calculation in fluids over the boundaries.

Appendix A a0 ¼

k21 cosh k2 yk22 cosh k1 y , k21 k22

a1 ¼

k31 sinhk2 yk32 sinh k1 y , k1 k2 ðk21 k22 Þ

a2 ¼ a3 ¼

cosh k1 ycosh k2 y , k21 k22

k2 sinh k1 yk1 sinhk2 y : k1 k2 ðk21 k22 Þ

Appendix B. The elements of the matrix L(s,y)

ðk21 CÞcosh k2 yðk22 CÞcosh k1 y , k21 k22 " # Dðs þ MÞ k2 sinhk1 yk1 sinh k2 y  2 2 ‘12 ¼  , 1l0 s k1 k2 k1 k2 ‘11 ¼

4193

‘13 ¼ k2 ðk21 ðs þMÞ=ð1l0 sÞÞsinhk1 yk1 ðk22 ðs þ MÞ=ð1l0 sÞÞsinhk2 y   , k1 k2 k21 k22 " ‘14 ¼ D

# cosh k1 ycosh k2 y , k21 k22

" # CK0 k2 sinhk1 yk1 sinh k2 y ‘21 ¼ , 1l0 s k1 k2 ðk21 k22 Þ ‘22 ¼ ðk21 ðs þ MÞ=ð1l0 sÞÞcosh k2 yðk22 ðs þ MÞ=ð1l0 sÞÞcosh k1 y , k21 k22 " # K0 cosh k1 ycosh k2 y ‘23 ¼ , 1l0 s k21 k22

‘24 ¼

k2 ðk21 CÞsinhk1 yk1 ðk22 CÞsinh k2 y , k1 k2 ðk21 k22 Þ

‘31 ¼ " # k2 ðk21 ðs þ MÞ=ð1l0 sÞÞsinh k1 yk1 ðk22 ðs þ MÞ=ð1l0 sÞÞsinhk2 y C , k1 k2 ðk21 k22 Þ " # Dðs þ MÞ cosh k1 ycosh k2 y ‘32 ¼  , 1l0 s k21 k22 ‘33 ¼ ðk21 ðs þ MÞ=ð1l0 sÞÞcosh k1 yðk22 ðs þ MÞ=ð1l0 sÞÞcosh k2 y , k21 k22 " ‘34 ¼ D

# k1 sinh k1 yk2 sinhk2 y , k21 k22

" # CK0 cosh k1 ycosh k2 y , ‘41 ¼ 1l0 s k21 k22 " # s þ M k2 ðk21 CÞsinhk1 yk1 ðk22 CÞsinh k2 y ‘42 ¼ , 1l0 s k1 k2 ðk21 k22 Þ " # K0 k1 sinh k1 yk2 sinhk2 y , ‘43 ¼ 1l0 s k21 k22

‘44 ¼

ðk21 CÞcosh k1 yðk22 CÞcosh k2 y : k21 k22

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