Invariance, Conservation Laws, and Exact Solutions of the Nonlinear Cylindrical Fin Equation

Invariance, Conservation Laws, and Exact Solutions of the Nonlinear Cylindrical Fin Equation

Saeed M. Ali^a, Ashfaque H. Bokhari^a, Fiazuddin D. Zaman^a, and Abdul H. Kara^b

aDepartment of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

bSchool of Mathematics, Centre for Differential Equations Continuum Mechanics and Applications University of the Witwatersrand Johannesburg, Wits 2050, South Africa Reprint requests to A. H. K.; E-mail:abdul.kara@wits.ac.za

Z. Naturforsch.69a, 195 – 198 (2014) / DOI: 10.5560/ZNA.2014-0008

Received August 6, 2013 / revised January 6, 2014 / published online April 2, 2014

Fins are heat exchange surfaces which are used widely in industry. The partial differential equation arising from heat transfer in a fin of cylindrical shape with temperature dependent thermal diffusivity are studied. The method of multipliers and invariance of the differential equations is employed to obtain conservation laws and perform double reduction.

Key words:Nonlinear Cylindrical Fin Equation; Exact Solutions; Conservation Laws.

1. Introduction

Fins are extended surfaces used, inter alia, to in- crease the heat exchange from a hot or cold surface to surrounding areas. Uses of fins include compres- sors, cooling of computer processors, air-cooled craft engines in air conditioning, etc. The heat transfer by fins of different shapes and profiles with a variety of boundary conditions is described by mathematical models [1]. There have been studies using a number of techniques to discuss the heat transfer through fins of different shapes. For example, [2] discussed the prob- lem

∂²θ

∂X²+∂²θ

∂Y²=0 (1)

using separation of variables and a Newton–Raphson method to compute the temperature profiles and heat transfer per fin length. More recently, Pakdemirli and Sahin [3,4] studied the problem

∂

∂X

k(θ)∂ θ

∂X

−N²f(X)θ=θ_t (2) using the Lie symmetries of the governing partial dif- ferential equation. Bokhari et al. [5] further studied the above nonlinear fin equation (2). They consid- ered group theoretic analysis that led to some new

exact solutions. In the same series of studies, Moit- sheki and Harley [6] considered a two-dimensional pin fin equation with length L and radius R having the form

1 R

∂

∂R

Rk(u)∂u

∂R

+ ∂

∂x

k(u)∂u

∂x

=s(u). (3) Using the Lie symmetry approach, they gave certain solutions of this equation for different cases ofs(u).

Extending this work in this area, we perform a dou- ble reduction of the nonlinear (2+1) fin equation by considering cylindrical fins with nonlinear thermal conductivity and variable heat transfer coefficient de- scribed by

div k(u)gradu

−N²f(X)u=u_t (4) for which the equivalent cylindrical coordinates of the (2+1)fin equation is given by

1 R

∂

∂R

Rk(u)∂u

∂R

+1 R

∂

∂ θ 1

Rk(u)∂u

∂ θ

+ ∂

∂z

k(u)∂u

∂z

−N²f(R)u=u_t.

(5)

If we assume no variation in axial direction (invari- ance in the z-coordinate), we would ignore the term

© 2014 Verlag der Zeitschrift für Naturforschung, Tübingen·http://znaturforsch.com

196 S. M. Ali et al.·Invariance, Conservation Laws, and Exact Solutions of Fin Equation

∂

∂z(k(u)^∂u

∂z). Then, relabelling the radial coordinateR byxand the angleθ byy, the cylindrical coordinates of the(2+1)fin equation become

1 x

∂

∂x xk(u)u_x +1

x

∂

∂y 1

xk(u)u_y

−N²f(x)u=u_t

(6)

so that

x²k(u)u_xx+x²k(u)_uu²_x+xk(u)u_x+k(u)_uu²_y +k(u)u_yy−N²x²f(x)u−x²u_t=0. (7) 2. Conservation Laws and Double Reduction

In this section, we derive the conservation laws of the(2+1)fin equation (6) using the method of multi- pliers and invariance of the differential equation and corresponding conservation under symmetries. The conserved vector(T^t,T^x,T^y)of (7) satisfies the diver- gence relation

D_tT^t+D_xT^x+D_yT^y=Q

x²k(u)u_xx+x²k(u)_uu²_x (8) +xk(u)u_x+k(u)_uu²_y+k(u)u_yy−N²x²f(x)u−x²u_t

.

Thus, E_u

Q

x²k(u)u_xx+x²k(u)_uu²_x+xk(u)u_x +k(u)_uu²_y+k(u)u_yy−N²x²f(x)u−x²u_t

=0, (9)

where E_u is the respective Euler–Lagrange opera- tor and Q is called a multiplier. If we suppose Q to be up to second order in derivatives, i. e., Q= Q(t,x,y,u,u_t,u_x,u_y,u_tt,u_xx,u_yy,u_tx,u_ty,u_xy), then ap- plication to (9) leads to cumbersome calculations for which the results cannot be presented here. Neverthe- less, solving the system, we obtain some nontrivial multipliers Q; each one leads to a conserved vector determined by, inter alia, a homotopy operator [7,8].

We consider the special case f(x) =cwherecis an arbitrary constant. In accordance with this choice, we obtain the following forms ofQ:

Q₁=−e^N2ct

x , Q₂=−e^N2ct

x ln(x). (10) The corresponding conserved vector ofQ₁is given by

T¹= (T^t,T^x,T^y)

=

e^N2txu,−e^N2txk(u)u_x,−e^N2tk(u)u_y x

,

(11)

andQ₂is given by T²=

e^N2txln(x)u, Z 1

0

e^N2t

−xλln(x)uk⁰(u)u_x+k(λu)

·(u−xln(x)u_x)

dλ,−e^N2txk(u)ln(x)u_y x

.

(12)

We perform the double reduction for the particular case k(u) =γ(αu)^α1 (as an example of a power law nonlin- earity),N²=1, andf(x) =1.

Equation (7), with the above choice ofk, f, andN², admits the symmetry generators

X₁=−sin(y)∂

∂x−cos(y) x

∂

∂y, X₂=cos(y)∂

∂x−sin(y) x

∂

∂y, (13)

X₃=x∂

∂x+2αu ∂

∂u, X₄=e^αt

α²∂

∂t−α²u ∂

∂u

, X₅= ∂

∂t, X₆= ∂

∂y.

Firstly, we show that X₄is associated with T¹ by using the result

X^[1]₄



 T^t T^x T^y



−





Dtτ Dxτ Dyτ D_tξ D_xξ D_yξ D_tη D_xη D_yη







 T^t T^x T^y





+ (D_tτ+D_xξ+D_yη)



 T^t T^x T^y



=0. (14) We have

X^[1]₄ =X₄−α²e^αtu_x ∂

∂u_x−α²e^αtu_y ∂

∂u_y (15)

−

αue^αt +

α²e^αt −αe^αtu_t ∂

∂u_t

.

Calculating the above quantities yield

X^[1]₄ T^t=0, X^[1]₄ T^x=αe^t−αt x(αu)^α1u_x, X^[1]₄ T^y=αe^t−αtx(αu)^α1u_y

x ,

(16)

S. M. Ali et al.·Invariance, Conservation Laws, and Exact Solutions of Fin Equation 197 and we haveξ=η=0,τ=α²e^αt , andφ=−α²ue^αt.

Therefore,





D_tτ D_xτ D_yτ D_tξ D_xξ D_yξ D_tη D_xη D_yη



=





αe^αt 0 0

0 0 0

0 0 0



 (17) and

D_tτ+D_xξ+D_yη



 T^t T^x T^y



=αe^αt



 T^t T^x T^y



. (18) Substituting (16), (17), and (18) into (14), we conclude thatX₄is associated withT¹= (T^t,T^x,T^y)with (Q₁=

−^e_x^t). Thus, we can get a reduced conserved vector by X₄whereX₄has a canonical formY₄= ^∂

∂qwhen e^−tα dt

α² = dx 0 = dy

0 = e^−tα du

−α²u

= dr 0 = ds

0 = dq 1 = dw

0 .

(19)

The invariants ofX₄from (19) are given by b1=x, b2=y, b3=e^tu, b4=r, b5=s, b6+q=−1

αe^−tα , b7=w, (20) whereb₄,b₅,b₆, andb₇are arbitrary functions all de- pendent onb₁,b₂, andb₃.

By choosingb₇=b₃,b₅=b₂,b₆=0, andb₄=b₁, we obtain the canonical coordinates

r=x, s=y, q=−1

αe^−tα , w=e^tu, (21) wherew=w(r,s), sinceY4= ^∂

∂q.

From (21), the inverse canonical coordinates are given by

x=r, y=s, t=−αln(−αq), u=e^−tw. (22) In the light of the above similarities, the partial deriva- tives ofuare given by

u_x=e^−tw_r, u_xx=e^−tw_rr, u_y=e^−tw_s, u_yy=e^−tw_ss, u_t=−e^−tw. (23) Consequently, (7) reduces to

r(αw)^α1w_r+r²(αw)^α1−1w²_r+r²(αw)^α1w_rr + (αw)^α1−1w²_s+ (αw)^α1wss=0.

(24)

The inverseA⁻¹is given by

A⁻¹=





D_tr D_ts D_tq D_xr D_xs D_xq Dyr Dys Dyq



=





0 0 ¹

α²e^−tα

1 0 0

0 1 0



. (25) In order to get the reduced conserved form, we use the formula



 T^r T^s T^q



=J(A⁻¹)^T



 T^t T^x T^y



. (26) We haveJ=det(A) =α²e^αt, therefore,



 T^r T^s T^q



=α²e^αt





0 1 0

0 0 1

1

α²e^−tα 0 0







 T^t T^x T^y



. (27) Thus, the reduced conserved form is

D_rT^r+D_sT^s=0, (28) where

T^r=−α²r(αw)^α1w_r, T^s=−α²(αw)^α1

r ws, T^q=rw.

(29)

The reduced conserved form admits the inherited symmetry

Xb₂=cos(s)∂

∂r−sin(s) r

∂

∂s. (30)

This symmetry is associated with the reduced con- served form. To show that this symmetry is associated, from

Xb^[1]₂ =cos(s)∂

∂r−sin(s) r

∂

∂s−sin(s) r² w_s ∂

∂w_r +

cos(s)

r ws+sin(s)w_r ∂

∂w_s,

(31)

we get Xb^[1]₂

T^r T^s

−

D_rξ^r D_sξ^r D_rξ^s D_sξ^s

T^r T^s

+ D_rξ^r+D_sξ^s T^r

T^s

=0.

(32)

198 S. M. Ali et al.·Invariance, Conservation Laws, and Exact Solutions of Fin Equation We transform Xb₂ to its canonical form Y = ^∂

∂m. Thus, the canonical coordinates are

n=rsin(s), m=r cos(s)+sin(s)

, v(n) =w. (33) Consequently, we have

A⁻¹=

D_rn D_rm D_sn D_sm

=

sin(s) cos(s) +sin(s) rcos(s) r(−sin(s) +cos(s))

.

(34)

In accordance with the similarities in (33), the partial derivatives ofware given by

w_r=sin(s)v_n, w_rr=sin²(s)v_nn, w_s=rcos(s)v_n, w_ss=−rsin(s)v_n+r²cos²(s)v_nn. (35) Thus, (24) reduces to

(αv)v_nn+v²_n=0. (36) To obtain the reduced conserved form, we apply the formula

Tⁿ T^m

=J(A⁻¹)^T T^r

T^s

. (37)

Thus, Tⁿ

T^m

=−r

sin(s) rcos(s)

cos(s) +sin(s) r cos(s)−sin(s)

· T^r

T^s

, (38)

whereJ=det(A) =−r. Using (35) from which we conclude

D_nTⁿ=0 so that

α²(αv)^α1v_n=C₁, (39) whereC₁is a constant,n=xsiny, andv=e^tu. Thus,

α³⁺

1 α

1+α

!

v^1+α1 =C1n+C₂, α6=−1,0, C₂is a constant. Then, we re-cast above in the origi- nal coordinates: the exact solution of (7) with k(u) = (au)^α1, f(x) =1, andN²=1 is then given by

u(x,y,t) =exp(−t)

C₁xsin(y) +C_21+α^α . (40)

3. Conclusion

The partial differential equation arising from heat transfer in a fin of cylindrical shape with temperature dependent thermal diffusivity is reduced, and exact so- lutions are obtained. This was achieved by construct- ing conservation laws and determining the associated invariances of the underlying differential equations.

[1] Q. D. Kern and D. A. Kraus, Extended Surface Heat Transfer, McGraw-Hill, New York 1972.

[2] S. W. Ma, A. I. Behbahani, and Y. G. Tsuei, Int. J. Heat Mass Trans.34, 79 (1991).

[3] M. Pakdemirli and A. Z. Sahin, Int. J. Eng. Sci. 42, 1875 (2004).

[4] M. Pakdemirli and A. Z. Sahin, Appl. Math. Lett.19, 378 (2006).

[5] A. H. Bokhari, A. H. Kara, and F. D. Zaman, Appl.

Math. Lett.19, 1356 (2006).

[6] R. J. Moitsheki and C. Harley, Math. Prob. Eng.,2011, 1563 (2011).

[7] W. Hereman, Int. J. Quant. Chem.106, 278 (2006).

[8] A. H. Kara, Math. Comput. Appl.15, 658 (2010).