Effects of Heat Transfer and Nonlinear Slip on the Steady Flow of Couette Fluid by Means of Chebyshev Spectral Method

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Effects of Heat Transfer and Nonlinear Slip on the Steady Flow of Couette Fluid by Means of Chebyshev Spectral Method

Rahmat Ellahi^a,b, Xinil Wang^c, and Muhammad Hameed^c

a Department of Mechanical Engineering Bourns Hall, University of California Riverside, 92521 USA

b Department of Mathematics and Statistics, IIIUI, H-10, 44000, Islamabad, Pakistan

c Department of Mathematics & Computer Science, University of South Carolina Upstate 800 University Way, Spartanburg, 29303, USA

Reprint requests to R. E.; E-mail:rellahi@engr.ucr.edu,rahmatellahi@yahoo.com Z. Naturforsch.69a,1 – 8 (2014) / DOI: 10.5560/ZNA.2013-0060

Received April 16, 2013 / revised August 7, 2013 / published online November 21, 2013

This article is concerned with the study of heat transfer and nonlinear slip effects on the Couette flow of a third-grade fluid. Numerical solutions are obtained by solving nonlinear differential equations using the higher-order Chebyshev spectral method. The results for no slip and no thermal slip become special cases of this study. Moreover, the results for Poiseuille flow can be obtained as a special case from the generalized Couette flow analysis by setting the plate velocity to zero. Graphical results for involved pertinent parameters are sketched and examined.

Key words:Third-Grade Fluid; Couette Flow; Nonlinear Slip Boundary Conditions; Heat Transfer;

Chebyshev Spectral Method.

1. Introduction

The study of non-Newtonian fluids [1–9] is gain- ing attention of researchers because of its practical importance in science and engineering. The classical Navier–Stokes equations have been proved inadequate to describe and capture the characteristics of complex rheological fluids as well as polymer solutions [10, 11]. These kinds of fluids are generally known as non- Newtonian fluids. Most of the biological and indus- trial fluids are non-Newtonian in nature. Few examples of such fluids are blood, tomato ketchup, honey, mud, plastics, and polymer solutions. The inadequacy of the classical theories to describe these complex fluids has led to increased interest in the constitutive modelling of nonlinear fluids and its applications to various engineering and biological problems. There are several models which have been proposed to describe such liquids, however their full potential has not been ex- ploited yet and several questions remain un-resolved.

These models have relevance in rheology of suspen- sions, in the transport of slurries etc. Among these, the fluid of differential third grade has received consider- able attention which is a subclass of differential type fluids [12]. A third-grade fluid has been studied suc-

cessfully in various types of flow situations [13–17]

and is known to capture the non-Newtonian affects such as shear thinning or shear thinking as well as normal stresses. The constitutive equations for the third- grade fluid, as in the case of most non-Newtonian fluids, give rise to a complicated and highly nonlinear set of equations. This intrinsic nonlinearity due to the presence of normal stresses makes the problem diffi- cult to solve analytically and numerically even in simple geometries.

In the present work, we numerically investigate the flow and heat transfer with nonlinear slip effects of a third-grade fluid. We use the Chebyshev spectral method to study the nonlinear slip effect on the Couette and generalized Couette flow. Our numerical method is seen to provide a direct scheme for solving these nonlinear problems without any need for linearization or any restrictive assumptions. Moreover, the method is found to greatly reduce the size of computational work while maintaining high accuracy. Spectral methods have been successfully used in finding the numerical solution of various problems and in studying computational fluid dynamics problems. Spectral methods are proved to offer a superior intrinsic accuracy for derivative calculations [18].

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2. Formulation

In this section, we derive the governing equations for the hydrodynamic flow of an incompressible and thermodynamically compatible third-grade fluid between two rigid parallel plates located at y^∗=±h.

A constant pressure gradient is applied to induce the flow. Both plates are heated such that the upper and lower plates have temperaturesT₁andT₀, respectively.

The velocity field V and temperature field T for the problem are considered to be uni-directional and are given by

V= [u^∗(y),0,0], T =T(y), (1) whereu^∗represents the x-component of the velocity.

The equations which govern the flow are continuity, momentum, and energy equations

divV =0, (2)

ρdV

dt =divS, (3)

ρdE

dt =S·L−divq. (4)

Body forces are neglected in (3). Hereρ denotes the fluid density andV the velocity vector. The operator d/dtdenotes the material time derivative,Eis the in- ternal energy, andqis the heat. The Cauchy stress tensor S for a thermodynamic third-grade fluid is given by

S=−pI+µA₁+α1A₂+α₂A²₁+β trA²₁

A₁, (5) whereIis the unit tensor,µis the coefficient of shear viscosity,prepresents the pressure,α₁,α₂are second- grade parameters, andβ is a material constant for the third-grade fluid. Whenβ=0, the results of a second- grade fluid are recovered as a special case of this problem. For detailed thermodynamical analysis the reader is referred to [19]. The Rivlin–Ericksen tensorsA₁,A₂ are given by

A₁= (gradV) + (gradV)^T, A₂= d

dtA₁+A₁(gradV) + (gradV)^TA₁. (6) Here T denotes transpose. The continuity equation (2) is identically satisfied with the choice of (1). Further-

more, using (1) and (5), (3) and (4) give us a coupled system of a nonlinear differential equations governing the flow and heat transfer of a third-grade fluid between two heated plates:

µd²u^∗ dy^∗2+6β

du^∗ dy^∗

2 d²u^∗ dy^∗² = dp

dx^∗, (7) Kd²T

dy^∗² +µ du^∗

dy^∗ 2

+2β du^∗

dy^∗ 4

=0, (8) where p is the modified pressure and ‘^∗’ represents a dimensional quantity.

The velocity and thermal slip boundary conditions are

u^∗(−h)−γ^∗

"

du^∗ dy^∗+2β

µ du^∗

dy^∗ 3#

y^∗=−h

=U₀, (9)

u^∗(h) +γ^∗

"

du^∗ dy^∗+2β

µ du^∗

dy^∗ 3#

y^∗=h

=0, (10) T(−h)−γ₁^∗

dT dy^∗

y^∗=−h

=T0, (11)

T(h) +γ₁^∗ dT

dy^∗

y^∗=h

=T₁. (12)

Note thatγ^∗represents the dimensional slip parameter, γ₁^∗ is the thermal slip parameter, andU₀ is the velocity of lower plate. The system of differential equations and the corresponding boundary conditions are non- dimensional using suitable parameters, and the non- dimensional set of equations and boundary conditions are as following:

d²u dy²+6β

du dy

2d²u

dy²=−B, (13)

d²θ dy² +λ

du dy

2

+2β λ du

dy 4

=0, (14) u(−1)−γ

"

du dy+2β

du dy

3#

y^∗=−1

=1, (15)

u(1) +γ

"

du dy+2β

du dy

3#

y^∗=1

=0, (16)

θ(−1)−γ1

dθ dy

y=−1

=0, (17)

θ(1) +γ1

dθ dy

y=1

=1, (18)

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where u= u^∗

U₀, y=y^∗

h, β =β^∗U₀² µh² , λ= µU₀²

K(T₁−T₀)=P_rE_c, B=− h² µU₀

dp dx, θ= T−T₀

T₁−T₀, γ=γ^∗

h, γ1=γ₁^∗ h.

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The non-dimensional parameters given in (19) are used to non-dimensionalise the equations and boundary conditions. In the above, λ represents the Brinkman number. In the next two sections, we will investigate two problems, namely the Couette flow and generalized Couette flow between heated parallel plates. The situation becomes more complicated when considering generalized Couette flow and computer algebra sys- tems produce complicated expressions. Here, we will present numerical solutions based on the Chebyshev spectral methods for Couette and generalized Couette flow as well.

3. Numerical Method

In this section, we present the numerical method used to find the numerical solutions of the problems of the Couette and generalized Couette flow for the third- grade non-Newtonian fluid. Our goal is to solve the non-dimensional set of coupled equations with nonlinear boundary conditions for these problems. We find numerical solutions using the Chebyshev spectral collocation method. Chebyshev polynomials have been successfully employed in finding numerical solutions of various boundary value problems and in the study of computational fluid dynamics [20–22]. In solving ordinary differential equations, linear or nonlinear, to high accuracy on a simple domain, and if the data defining the problem are smooth, then spectral methods are usually the best tool. They can often achieve a high order of accuracy (up to ten digits) as compared with the finite difference or finite element methods [23].

We first consider the Couette flow of a non-New- tonian third-grade fluid. The governing equations to be solved are given in (13) to (18). To solve the equations, we employ the spectral collocation method of Cheby- shev type. We look for an approximate solution ϕ^N, which is a global Chebyshev polynomial of degreeN defined on the interval[−1,1]by

T_N(y) =cosNθ, θ=cos⁻¹y. (20)

We discretize the interval by using collocation points to define the Chebyshev nodes in[−1,1], namely

y_j=cos jπ

N

, j=0,1, ...N.

The functionϕ(y)is approximated by an interpolating polynomial which is constructed in terms of the values of ϕ at each of the collocation points by employing a truncated Chebyshev series of the form

ϕ(y) =ϕN(y_j) =

N k=0

∑

ϕ˜_kT_k(y_j), j=0,1, . . .,N, (21) where ˜ϕkrepresents the series coefficients. The derivatives of the functions at the collocation points are given by

∂ ϕ_N dy (y_j) =

N

∑

k=0

D_jkϕ˜_k, (22) whereDrepresents the derivative matrix, given by

D_jk=c_j(−1)^j+k

c_ky_j−y_k , j6=k; k=0,1, . . .,N. (23) The higher-order derivatives are computed as simply multiple powers ofD, i.e.,

∂ⁱϕN

dyⁱ (y_j) =

N

∑

k=0

Dⁱ_jkϕ˜_k, k=0,1, . . .,N, (24) whereiis the order of the derivative.

As described above, the Chebyshev polynomials are defined on the finite interval[−1,1]which is our computational domain in these problems. We sample the unknown function u at the Chebyshev points to obtain the data vectoru= [u(y₀),u(y₁), . . .,u(y_N)]^T. The next step is to find a Chebyshev polynomial ϕ of degreeN that interpolates the data, i.e., ϕ(y_j) =u_j, j=0,1, . . .,N, and obtain the spectral derivative vectoru⁰⁰ by differentiating ϕ and evaluating at the grid points, i.e.,u⁰⁰_j=ϕ⁰⁰(t_j), for j=0,1, . . .,N. This transforms the nonlinear differential equation into nonlinear algebraic equations which can be solved by Newton’s iterative method.

4. Couette Flow

For the case of steady, incompressible, and thermodynamically compatible, laminar flow of a non- Newtonian incompressible fluid, there is no applied

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pressure gradientB=0. The upper plate remains stationary and the lower plate moves with velocity U₀ in positive x-direction. The flow is driven by the motion of the upper plate and in this case the flow is not pressure induced. The governing equations and boundary conditions in absence of pressure gradient are given as

d²u dy²+6β

du dy

2

d²u

dy² =0, (25)

d²θ dy²+λ

du dy

2

+2β λ du

dy 4

=0, (26)

u(−1)−γ

"

du dy+2β

du dy

3#

y^∗=−1

=1, (27)

u(1) +γ

"

du dy+2β

du dy

3#

y^∗=1

=0, (28)

θ(−1)−γ₁ dθ

dy

y=−1

=0, (29)

θ(1) +γ₁ dθ

dy

y=1

=1. (30)

After applying the Chebyshev spectral method, the equation and boundary conditions are transformed into a coupled system of following nonlinear algebraic equations:

Fig. 1. Velocity fieldufor flow of third-grade fluid. (a) Velocity field for different values ofβwith fixed slip parameterγ=1.

(b) Velocity field for different increasing values ofγwith fixed third-grade material constantβ=1.

N

∑

j=0

d_{k j}²u(x_j) +6β

N j=0

∑

d_{k j}¹u(x_j)

!2 N

∑

j=0

d_{k j}²u(x_j) =0,

N

∑

j=0

d_{k j}²Θ(x_j) +λ

N

∑

j=0

d_{k j}¹u(x_j)

!2

+2β λ

N j=0

∑

d_{k j}¹u(x_j)

!4

=0.

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The Chebyshev approximations for the boundary conditions take the form

u(x_N)−γ





N

∑

j=0

d¹_{N j}u(x_j)

+2β

N

∑

j=0

d¹_{N j}u(x_j)

!3



y^∗=−1

=1,

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u(x₀) +γ





N

∑

j=0

d_0,j¹ u(x_j)

+2β

N

∑

j=0

d¹_0ju(x_j)

!3



y^∗=1

=0,

(33)

T(x_N)−γ₁

"_N

∑

j=0

d_{N j}¹ Θ(x_j)

#

y=−1

=0, (34)

T(x₀) +γ1

"

N

∑

j=0

d¹_0jΘ(x_j)

#

y=1

=1. (35)

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Fig. 2. Temperature fieldθfor flow of third-grade fluid. (a) Temperature field for different values of Brinkman numberλwith fixed slip parameterγ=1 and fixedβ =1. (b) Temperature field for different increasing values ofγwith fixed third-grade material constantβ=1.

This transforms the nonlinear ordinary differential equations into a system of nonlinear algebraic equations which can be solved by Newton’s method.

5. Result for Couette Flow

Using the numerical method described in the previous section, we first consider the flow of a third-grade fluid between heated parallel plates. Numerical results are obtained and are given by Figures 1 and2. The results are presented for different values of the non- Newtonian parameter, and numerical results are found to be in good agreement with the existing analytical results.

6. Generalized Couette Flow

For the case of generalized Couette flow of steady, incompressible, and thermodynamically compatible, laminar flow of a non-Newtonian incompressible fluid, there is non-zero applied pressure gradientB6=0. The upper plate remains stationary and the lower plate moves with velocity U₀ in positive x-direction. The flow is driven by the motion of the upper plate as well as due to the applied pressure gradient. The governing equations for non-zero pressure induced flow, our system of equations, is given by

d²u dy²+6β

du dy

2 d²u

dy² =−B, (36)

d²θ dy² +λ

du dy

2

+2β λ du

dy 4

=0, (37) with same set of boundary conditions given by (27–30). After applying the Chebyshev spectral method, the equations and boundary conditions are transformed into a coupled system of following nonlinear algebraic equations:

N

∑

j=0

d_{k j}²u(x_j) +6β

N

∑

j=0

d_{k j}¹u(x_j)

!2 N

∑

j=0

d²_{k j}u(x_j) +B=0,

N

∑

j=0

d_{k j}²Θ(x_j) +λ

N

∑

j=0

d_{k j}¹u(x_j)

!2

+2β λ

N

∑

j=0

d_{k j}¹u(x_j)

!4

=0.

(38)

The Chebyshev approximations for the boundary conditions take the form

u(x_N)−γ





N

∑

j=0

d¹_{N j}u(x_j)

+2β

N

∑

j=0

d¹_{N j}u(x_j)

!3



y^∗=−1

=1,

(39)

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u(x₀) +γ





N

∑

j=0

d¹_0ju(x_j)

+2β

N

∑

j=0

d_0j¹u(x_j)

!3



y^∗=1

=0,

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T(x_N)−γ₁

"

N

∑

j=0

d¹_{N j}Θ(x_j)

#

y=−1

=0, (41)

T(x₀) +γ1

"

N

∑

j=0

d_0j¹Θ(x_j)

#

y=1

=1. (42)

Fig. 3. Velocity fieldufor generalized Couette flow of third-grade fluid for different values of the applied pressure gradientB.

Fig. 4. (a) Temperature fieldθfor generalized Couette flow of third-grade fluid for different values of the Brinkman number λ and fixed applied pressure gradientBand slip parameters.

7. Results for Generalized Couette Flow

Using the Chebyshev collocation method, numerical results are obtained and are given by Figures3and4.

The results are presented for different values of material constant of the fluid, slip parameter, and pressure gradients.

8. Discussion and Conclusions

The parallel-plate flow and heat transfer of a third- grade fluid subject to nonlinear partial slip is considered. The partial slip is controlled by a dimen-

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sionless slip factor, which can vary from zero (to- tal adhesion) to infinity (full slip). The constitutive equation is modelled for a third-grade fluid. The heat transfer analysis is also carried out. The im- portant finding in this communication is the com- bined effects of the nonlinear slip, heat transfer, and the third-grade fluid parameter on the velocity, skin- friction, and the temperature field. Graphs are presented in order to study the influence of impor- tant flow parameters such as third-grade parameter β, pressure gradient B, Brinkman number λ, and slip parameters γ,γ1 on the velocity and temperature profiles.

Accurate numerical solutions are obtained using highly accurate Chebyshev spectral methods. To the best of our knowledge, this is the first attempt to apply this robust and highly effective analytical technique as well as the highly accurate Chebyshev spectral method to study the flow of a third-grade fluid in this geome- try. The proposed numerical method offers a superior intrinsic accuracy for the derivative calculations. The numerical results indicate the usefulness of the spectral methods in obtaining accurate solutions to the nonlinear problems arising in the flow of third fluid. As compared to other numerical techniques, such as finite dif- ferences, the nonlinearity is not a major complication for spectral methods.

The first set of results displays the solutions for the case of Couette flow. Figure 1a and b shows velocity fields for different values of third-grade parameter β and slip parameterγ. The behaviour of the temperature under the influence of third-grade parameterβand slip parameterγis shown in Figure1. It is evident that by keeping all the other parameters fixed, an increase in the third-grade parameter results in the decreased velocity; similar behaviour is found by increasing the slip parameter. The influence of Brinkman numberλ and slip parameter is shown in Figure2. It is obvious from the results that an increase in the third-grade parameter increases the temperature. A similar effect on the

temperature is seen in the case when we the increase Brinkman number by keeping third-grade parameter and slip fixed.

In a second set of results, we extend the Couette flow results to generalized Couette flow by including the effect of pressure gradient(B6=0). The effects of different parameters in the case of generalized Couette flow are shown in Figures3 and4. It shows that velocity increases by increasing the magnitude of the pressure gradient and decreases by increasing the third-grade parameter. The effect of the third-grade parameter on the temperature is opposite to that of velocity. More- over, both velocity and temperature increase by increasing the pressure gradient. The effect of Brinkman number on the temperature is depicted in Figure4. It shows that the temperature increases by an increase in the Brinkman number as was observed in the previous case.

The results for the case of a Poiseuille flow can be obtained as a special case from the generalized Couette flow analysis by setting the plate velocity to zero, and the flow is driven by pressure gradient only. Further- more, the results for the zero fluid slip and no thermal slip also become special cases of this work, and the results can be recovered by settingγandγ₁to zero. When β =0, the results of second-grade fluid are recovered as a special case of this problem

To the best of our knowledge, no such analysis is available in the literature which can describe the heat transfer and nonlinear slip effects simultaneously on Couette and generalized Couette flow. The results presented in this paper will now be available for exper- imental verification to give confidence for the well- posedness of this nonlinear boundary value problem.

Acknowledgements

R. Ellahi thanks to M. Hameed for providing kind hospitality during his visit of University of South Car- olina Upstate, Spartanburg.

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