Finite Element Solution for Unsteady Flow of a Third-Grade ...Finite Element Solution for Unsteady Flow of a Third-Grade Fluid... 89 Figure 1: Geometry of the problem. 2. Problem Formulation

International Journal of Advanced Mechanical Engineering (IJAME).ISSN 2250-3234 Volume 8, Number 1 (2018), pp. 87–99© Research India Publicationshttp://www.ripublication.com/ijame.htm

Finite Element Solution for Unsteady Flow of aThird-Grade Fluid between two Plates

I. Nayak

Department of Mathematics,Veer Surendra Sai University of Technology,

Burla, Sambalpur, 768018, India.

A. K. Nayak1

Department of CS & IT, Faculty of Engineering and Technology,SOA (Deemed to be University), Bhubaneswar, India.

S. Padhy

Department of CSE, Silicon Institute of Technology,Bhubaneswar, India.

Abstract

This work studies the unsteady flow of a third grade fluid passed between two hor-izontal plates, where the lower plate suddenly starts moving on its own plane witha time varying velocity F (t) and the top plate is fixed. The governing non-linearpartial differential equations are then solved using finite element method. An ap-proximate solution is obtained for the flow velocity and the effect of the variousphysical parameters on velocity are studied for two cases, (i) variable acceleration,(ii) constant acceleration. One of the important observation obtained is that thevelocity increases when both the elastic parameter values are increasing for bothconstant and variable acceleration cases.

AMS subject classification: 65L12, 65M06, 76S05.Keywords: Third grade fluid, finite element method, unsteady flow, and time vary-ing velocity.

1Corresponding author. Email: [email protected]

88 I. Nayak, A. K. Nayak, and S. Padhy

1. Introduction

Several investigators are engaged in studying the flows of non-Newtonian fluids duringthe past few decades in view of their potential applications in industry and technology.Many fluid models are proposed to characterize the non-Newtonian properties of suchfluids. The class of third order fluid is one such model, which exibit the visco-elasticcharacteristic such as shear thinning thickening phenomenon observed in many non-Newtonian fluids [3], [5]. This motivated many researchers like Erdogan [8], Fosdickand Rajagopal [12], Hayat et al. [15], Kaloni, and Siddique [10], Ariel [9], Ellahi et al.[11], Nayak et al. [4]. Recently Sahoo and Do [2] have studied the entrained flow andheat transfer of an electrically conducting non-Newtonian fluid due to a stretching surfacesubject to partial slip. By suitable similarity transformation they reduce the resultinghighly non-linear partial differential equations into ordinary differential equations andadopting a second order numerical scheme they solve the differential equation. Okoya[14] studied the thermal transition of a reactive flow of a third grade fluid with viscousheating and chemical reaction between two horizontal flat plates where the top plate ismoving with a uniform speed and the bottom plate is fixed in the presence of imposedpressure gradient. He obtains an approximate explicit solution for the flow velocity usinghomotopy-perturbation technique. In the present work we employ finite element methodto study an unsteady flow of a third grade fluid between two porous plates when the lowerplate suddenly starts moving with a time varying velocity F (t). The governing non-linearpartial differential equations are reduced to a system of non-linear algebraic equationsusing finite element method and then the system is solved using damped-Newton method.

Whereas earlier traditional perturbation techniques which is strongly based on thepresence of small elastic parameters, the homotopy perturbation method (HPM) solutionis also valid only for weak non-linearity. It is proved by Sajid et al. [6], [7] that HPMdoes not overcome the disadvantages of traditional perturabation methods. They alsoproved by considering examples of the thin film flows that the HPM results are divergentfor strong non-linearity.

Finite element method is an efficient and widely used method which over-comesaforementioned limitations. This method is widely used to solve problems arising influid dynamics, civil engineering, computational finance, aerospace engineering etc.The present work provides an application of this method to a strong non-linear partialdifferential equation.

Rest of the paper is organized as follows. In section 2 the problem is formulated asnon-linear partial differential equations with appropriate boundary conditions. Section 3employs the finite element method to solve the non-linear partial differential equations.Section 4 discusses the effect of various parameters on the velocity field. Finally thepaper is concluded in section 5.

Finite Element Solution for Unsteady Flow of a Third-Grade Fluid... 89

Figure 1: Geometry of the problem.

2. Problem Formulation

Let x′- axis be chosen along the lower wall, y

′-axis be normal to it as shown in Fig. 1,

and the upper plane be specified by the equation y′ = L where the number L is defined

later. It is also supposed that the walls extends to infinite in both the sides of the x′-axis.

The lower plate suddenly starts moving on its own plane with a time varying velocityF (t). Then the velocity component u

′at any point in the flow field compatible with the

equation of continuity can be obtained as

u′ = u

′(y

′, t

′) (1)

The constitutive equation of an incompressible third grade fluid considered is, as givenby Coleman and Noll [1]

P = −pI +n∑

i=1

Si (2)

where,S1 = µA1, S2 = α1A2 + α2A

21, and S3 = β1A3 + β2(A2A1 + A1A2) + β3(trA

21)A1.

I the identity tensor and An represents the kinematical tensors defined by

A1 = ∇u + (∇u)T , An+1 = (∂

∂t+ u.∇)An + ∇u.An + (∇u.An)T , n = 1, 2

P is the hydrostatic pressure, µ is coefficient of viscosity.αi (i = 1, 2) and βi (i = 1, 2, 3) are material constants.

A detailed thermodynamic analysis of the model, represented by eq. (2) is given byFosdick and Rajagopala [12]. It was shown that if all motion of the fluid are to be com-patible with thermodynamics in the sense that these motions meet the Clausious-Duheminequality and if it is assumed that the Helmholtz free energy is a minimum when thefluid is locally at rest, thenµ ≥ 0, α1 ≥ 0, | α1 + α2 |≤ √

24µβ3, β1 = β2 = 0, β3 ≥ 0.


In present analysis, we assume that the fluid is thermodynamically compatible and there-fore eq. 2 reduces to

P = −pI + µA1 + α1A2 + α2A21 + β3(trA

21)A1 (3)

The non-zero components for stress for the problem under consideration are

Px′x′ = −p + α2

(∂u′

∂y′

)2

Py′y′ = −p + (2α1 + α2)

(∂u′

∂y′

)Pz′z′ = −p

Px′y′ = µ′ ∂u′

∂y′ + α1∂2u′

∂y′∂t ′+ 2β3

(∂u′

∂y′

)3

Px′z′ = 0

Pz′y′ = 0

andPx′y′ = Py′x′ , Py′z′ = Pz′y′ , Pz′x′ = Px′z′

Using these stress components, the governing equation of motion is obtained as

ρ

(∂u

′

∂t′

)= µ

∂2u′

∂y′2 + α1

∂3u′

∂y′2∂t

′ + 6β3

(∂u

′

∂y′

)2∂2u

′

∂y′2 (4)

The initial and boundary conditions to which equation (4) is subjected are assumed as

t′ = 0 :u

′ = 0, ∀y′

t′> 0 :u

′ = F (t) f or y′ = 0

u′ = 0, f or y

′ = L

where F (t) =Atn

(5)

We introduce the following dimensionless variables and parameters

u = u′

A, y = y

′√

νT, t = t

′

T, α = α1

ρνT, γ = 6β3A

2

ρν2T

where, ν = µ

ρis the kinematic coefficient of viscosity, ρ is the density, T is the tem-

perature, t ′ is time. Using the above notations, the equation of motion (4) reduces to thefollowing non-dimensional form

∂u

∂t= ∂2u

∂y2+ α

∂3u

∂y2∂t+ γ

(∂u

∂y

)2 (∂2u

∂y2

)(6)


and the initial and boundary conditions become

t = 0 :u = 0, ∀y

t > 0 :u = tn, for y = 0

u = 0, for y = 1

(7)

where α denotes second grade elastic parameter and γ denotes third grade elastic pa-rameter.

3. Method of Solution

We use Galerkin Finite Element method to obtain solution of the problem (6) and (7).The weak form of equation is obtained by multiplying both sides of the equation by eachof the global shape functions φi , f or i = 1, 2, . . . , N as

∫ 1

0φi(y)

(∂u

∂t− ∂2u

∂y2− α

∂3u

∂y2∂t− γ

(∂u

∂y

)2∂2u

∂y2

)dy = 0 (8)

Integrating the above product using integration by parts and then using the initial andboundary conditions we obtain the above equations as

∫ (φi

∂u

∂t+ φ

′i

∂u

∂y+ φ

′i

∂2u

∂y∂t+ φ

′i

(∂u

∂y

)3)

dy = 0, i = 1, 2, . . . , N (9)

The global shape functions φi(i = 1, 2, . . . , N ) are chosen as linear, piecewise functionsin [i − 1, i + 1] defined as follows

φi = 0, x ≤ xi−1

= x − xi−1

xi − xi−1, xi−1 ≤ x ≤ xi

= xi+1 − x

xi+1 − xi

, xi ≤ x ≤ xi+1

= 0, otherwise

(10)

where xi − xi−1 = h, for i = 1, 2, . . . , N .Taking the solution of the equation (9) in the form u =

∑j

uj (t)φj (y) we obtain

∫ 1

0

⎡⎢⎣∑

j

duj (t)

dt

(φiφj + φ

′iφ

′j

)+

∑j

uj (t)φ′iφj +

⎛⎝∑

j

uj (t)φ′j

⎞⎠

3

φ′i

⎤⎥⎦ dy = 0

(11)


Now replacingduj

dtby

unj − un−1

j

�tand uj by un

j ≈ uj (n�t), for n = 1, 2, . . . , K ,

We obtain the time discretise form as

∫ 1

0

⎡⎢⎣∑

j

unj − un−1

j

�t

(φiφj + φ

′iφ

′j

)+

∑j

unjφ

′iφj +

⎛⎝∑

j

unjφ

′j

⎞⎠

3

φ′i

⎤⎥⎦ dy = 0 (12)

for i = 1, 2, . . . , N ; n = 1, 2, . . . , Kusing (10) and (7), the above set of equations for each n = 1, 2, . . . , K reduces to thefollowing system of non-linear algebraic equations.

Ri ≡ − γ(un

i−1

)3

h3+

(h

6t+ 1

2− α

th

)un

i−1 + 2γ(un

i

)3

h3

+(

4h

6t+ 1 + 2α

th

)un

i − γ(un

i+1

)3

h3+

(h

6t+ 1

2− α

th

)un

i+1

+ 3γ(un

i−1

)2un

i

h3− 3γ un

i−1

(un

i

)2

h3+ 3γ un

i

(un

i+1

)2

h3− 3γ

(un

i

)2un

i+1

h3

+(

α

th− h

6t

)un−1

i−1 +(−2α

th− h

6t

)un−1

i

+(

α

th− h

6t

)un−1

i+1 = 0,

(13)

for i = 1, 2, . . . , N ; n = 1, 2, . . . , K .For the solution of this set of equations we used Damped Newton iterative method

as described in [13]. For the implementation of this method the residuals Ri , i =1, . . . , N and non-zero elements of the Jacobian matrix

(∂Ri

∂uj

), i = 1, 2, . . . , N , and

j = 1, 2, . . . , M of the system are given in Appendix.The Algorithm 1 given below is designed to solve the problem according to the

schemes discussed in this section.

4. Results and Discussion

The study exhibits the flow field between two parallel plates when lower plate suddenlystarts moving with a time varying velocity tn on its own plane. The effect of elasticparameters on the velocity field are investigated through simulations and results areproduced as graphs for two major cases, n = 0.5 i.e. when the plate starts moving witha variable acceleration and n = 1, i.e. plate starts moving with a constant acceleration.

The effect of second order elastic parameter α on velocity field is depicted in Fig. 2 andFig. 3. It is observed that for large value of the elastic parameter α the velocity increaseswith increase of the parameter values for both variable and constant acceleration. From


Algorithm 1 Velocity Computation1: Input the parameters �t , h, α, γ.2: Initialize velocity by solving the tri-diagonal system,

taking α = γ = 0.3: Compute velocity using damped-Newton method[13].4: Update the velocity for next time-step.5: Repeat step-3 and step-4 for each time-step up to M

time-steps.6: Output the result of velocity distribution.

Figure 2: Effect of α on Velocity field. (γ = 0.5 and n = 0.5)

Figure 3: Effect of α on Velocity field. (γ = 0.5 and n = 1)


Figure 4: Effect of α on Velocity field. (γ = 0.5 and n = 0.5)

Figure 5: Effect of α on Velocity field. (γ = 0.5 and n = 1)

Fig. 4 and Fig 5 a similar effect is observed on velocity field for small values of elasticparameters.

From Fig. 6 and Fig. 7 it is observed that velocity increases with increase of thirdorder elastic parameter γ both for variable and constant acceleration. Whereas fromFig. 8 and Fig. 9 it is observed that velocity increases with increase of third order elasticparameter γ , the variation of velocity in case of small values of elastic parameter is notvery significant as compared to larger values of elastic parameters.

Fig. 10 shows the effect of time t on the velocity distribution for the cases n = 1 andn = 0.5. It is seen that the velocity increases when t increases. It is also observed forsmall values of t the maximum velocity is attained at the lower plate while for relativelyhigher values of t the maximum velocity attained little away from the plate.


Figure 6: Effect of γ on Velocity field.(α = 1 and n = 0.5)

Figure 7: Effect of γ on Velocity field. (α = 1 and n = 1)

Figure 8: Effect of γ Velocity field. (α = 1 and n = 0.5)


Figure 9: Effect of γ on velocity field. (α = 1 and n = 1)

Figure 10: Effect of T ime on velocity field. (α = 1, γ = 0.5, and n = 0.5)

5. Conclusion

An unsteady flow of a third grade fluid between two plates is solved using finite elementmethod, where the weak form is obtained by taking piecewise, linear, global shapefunctions. The discussed method is an efficient method to find better result in case ofstrong non-linear problem and it is valid for all values of elastic parameters. The majorobservation obtained in the present work is that the velocity increases when both theelastic parameter values are increasing both for constant and variable acceleration cases.


AppendixA. Residuals

R1 = −γ(un

0

)3

h3+

(h

6t+ 1

2− α

th

)un

0 + 2γ(un

1

)3

h3

+(

4h

6t+ 1 + 2α

th

)un

1 − γ(un

2

)3

h3+

(h

6t+ 1

2− α

th

)un

2

+ 3γ(un

0

)2un

1

h3− 3γ un

0

(un

1

)2

h3+ 3γ un

1

(un

2

)2

h3− 3γ

(un

1

)2un

2

h3

+(

α

th− h

6t

)un−1

0 +(−2α

th− 4h

6t

)un−1

1

+(

α

th− h

6t

)un−1

2 = 0

(14)

Ri = −γ(un

i−1

)3 +(

h

6t+ 1

2− α

+ h

)un

i−1 + 2γ(un

i

)3

h3

+(

4h

6t+ 1 + 2α

th

)un

i − γ(un

i+1

)3

h3+

(h

6t+ 1

2− α

th

)un

i+1

+ 3γ(un

i−1

)2un

i

h3− 3γ un

i−1

(un

i

)2

h3+ 3γ un

i

(un

i+1

)2

h3− 3γ

(un

i

)2un

i+1

h3

+(

α

th− h

6t

)un−1

i−1 +(−2α

th− 4h

6t

)un

i +(

α

th− h

6t

)un−1

i+1 = 0

(15)

RN = −γ(un

N−1

)3

h3+

(h

6t+ 1

2− α

+ h

)un

N−1 + 2γ(un

N

)3

h3

+(

4h

6t+ 1 + 2α

th

)un

N + 3γ(un

N−1

)2un

N

h3− 3γ un

N−1

(un

N

)2

h3

+(

α

th− h

6t

)un−1

N−1 +(

2α

th− 4h

6t

)un−1

N = 0

(16)

B. Jacobians

∂R1

∂un1

≡ J (1, 1) = 6γ(un

1

)2

h3+

(4h

6t+ 1 + 2α

th

)+ 3γ

(un

0

)2

h3− 6γ un

0un1

h3

+ 3γ(un

2

)2

h3− 6γ un

1un2

h3

(17)


∂R1

∂un2

≡ J (1, 2) = −3γ(un

2

)2

h3+

(h

6t+ 1

2− α

th

)+ 6γ un

1un2

h3− 3γ

(un

1

)2

h3(18)

∂Ri

∂uni−1

≡ J (i, i − 1) = −3γ(un

i−1

)2

h3+

(h

6t+ 1

2− α

th

)+ 6γ un

i−1uni

h3

−3γ(un

i

)2

h3

(19)

∂Ri

∂uni

≡ J (i, i) = 6γ(un

i

)2

h3+

(4h

6t+ 1 + 2α

th

)+ 3γ

(un

i−1

)2

h3− 5γ un

i−1uni

h3

+ 3γ(un

i+1

)2

h3− 6γ un

i uni+1

h3

(20)

∂Ri

∂uni+1

≡ J (i, i + 1) = −γ(un

i+1

)2

h3+

(h

6t+ 1

2− α

th

)

+ 6γ uni un

i+1

h3− 3γ

(un

i

)2

h3

(21)

∂RN

∂unN−1

≡ J (N , N − 1) = −γ(un

N−1

)2

h3+

(h

6t+ 1

2− α

th

)

+ 6γ unN−1un

N

h3− 3γ

(un

N

)2

h3

(22)

∂RN

∂unN

≡ J (N , N ) = 6γ(un

N

)2

h3+

(4h

6t+ 1 + 2α

th

)+ 3γ

(un

N−1

)2

h3

− 6γ(un

N−1

)2un

N

h3

(23)

References

[1] B. D. Coleman, W. Noll, An approximation theorem for functional with applicationin continuum mechanics Arch. for Rat. Mech. and Anal. 6, 355–370 (1965).

[2] B. Sahoo, D. Younghae, Effects of slip on sheet-driven flow and heat transfer of athird grade fluid past a stretching sheet Int. Comm. in Heat and Mass Transfer 37,1064–1071 (2010).


[3] G. S. Beavers, D. D. Joseph, The rotating rod viscometer J. of Fluid Mech. 69,475–511 (1975).

[4] I. Nayak, A. K. Nayak, S. Padhy, Numerical solution for the flow and heat transferof a third-grade fluid past a porous vertical plate Adv. Stud. in Theor. Phy. 6,615–624 (2012).

[5] D. D. Joseph, R. L. Fosdick, The free surface on a liquid between cylinders rotatingat different speeds Arch. for Rat. Mech. and Anal. 49, 321–401 (1973).

[6] M. Sajid, T. Hayat, S. Asghar, Comparison of the ham and hpm solutions of thinflim flows of non-newtonian fluids on a moving belt Nonlinear Dynamics 50, 27–35(2007).

[7] M. Sajid, T. Hayat, The application of homotopy analysis method to thin film flowsof a third order fluid Chaos, Solutions & Fractals 38, 506–515 (2008).

[8] M. E. Erdogan, Plane surface suddenly set in motion in a non-newtonian fluid ActaMechanica 108, 179–187 (1995).

[9] P. D. Ariel, Flow of a third grade fluid through a porous flat channel Int. J. of Engg.Sc. 41, 1267–1285 (2003).

[10] P. K. Kaloni, A. M. Siddique, A note on the flow of a visco elastic fluid betweeneccentric disc J. of Non-Newtonian Fluid Mech. 26, 125–133 (1987).

[11] R. Ellahi, T. Hayat, F. M. Mahomed, S. Asghar, Effects of slip on the non-linearflows of a third grade fluid Nonlinear Analysis: Real World Applications 11, 139–146 (2010).

[12] R. L. Fosdick, K. R. Rajagopal, Thermodynamics and stability of fluids of thirdgrade Proc. of Royal Society London-A, 351–377 (1980).

[13] S. D. Conte, C. D. Boor, Elementary Numerical Analysis an Algorithmic Approach,3rd ed. McGraw-Hill Inc, New York 1980.

[14] S. S. Okoya, On the transition for a generalized coquette flow of a reactive thirdgrade fluid with viscous dissipation Int. Comm. in Heat and Mass Transfer, 188–19635 (2008).

[15] T. Hayat, S. Nadeem, S. Asghar, A. M. Siddiqui, Fluctuating flow of a third orderfluid on a porous plate in a rotating medium Int. J. of Non-Linear Mech. 36, 901–916(2001).

Finite Element Solution for Unsteady Flow of a Third-Grade ...Finite Element Solution for Unsteady Flow of a Third-Grade Fluid... 89 Figure 1: Geometry of the problem. 2. Problem Formulation

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