The exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids

Peer review under responsibility of University of Bahrain.

Abstract

Usually, the temperature distribution of nanofluids and the nanoparticles’ concentration are finally governed by second-order ordinary differential equations with polynomial coefficients. In this work, a class of second-order boundary value problems with applications on nanofluids has been theoretically solved in terms of the Kummer function. Several lemmas have been presented to relate the Kummer function with the generalized incomplete gamma function. Accordingly, the current solutions reduce to those in the literature at certain values of the coefficients as special cases. Furthermore, the present results are very useful in obtaining the solutions for any future similar problems without any need to perform further calculations.

Keywords:

• Ordinary differential equation
• Hypergeometric series
• Boundary value problem
• Exact solution
• Laplace transform
• Nanofluid

1 Introduction

Recently, it has been found that the temperature distribution and the nanoparticles’ concentration of nanofluids (Aly et al., 2013a,b, 2016; Ebaid et al., 2014; Ebaid and Al Sharif, 2015; Hamad, 2011; Kameswaran et al., 2012; Khan et al., 2013") are basically governed by partial differential equations. These basic equations are then reduced to a set of ordinary differential equations by means of a similarity variable. The final transformed ODEs are usually of second order with polynomial coefficients. In this paper, we consider a generalized second-order ordinary differential equation describing the temperature of nanofluids and/or the nanoparticles concentration in the form:(1) subject to the following set of boundary conditions(2) where , Q, and R are physical parameters which are related to the densities, the thermal conductivities, and the heat capacitances of the base-fluids and the nanofluids. The parameter is often used to describe the convective heat condition and it takes some particular values according to the physical problem. In this filed of research, many useful results have been recently reported by Bhatti et al. (2016, 2017)", Ellahi et al. (2016), Shehzad et al. (2016), Sheikholeslami and Zeeshan (2017a,b)", andZeeshan et al. (2016). The objective of this paper is to derive a general exact solution for the system (1)–(2). The paper is organized as follows. In Section 2, the solution of Eqs. (1)–(2) will be obtained in terms of hypergeometric function. In Section 3, some special cases are discussed. Section 4 is devoted to obtain some useful properties through some lemmas. The solution in terms of the generalized incomplete gamma function is introduced in Section 5. Moreover, it will be proved in Section 6 that the current general solution reduces to the those in the literature as special cases.

2 Solution in terms of hypergeometric function

Following Ebaid et al. (2017), the analytical solution of Eqs. (1)–(2) can be obtained as(3) where c is an integration constant and is Kummer’s function. In addition, the assumption in (3) leads to the satisfaction of the first boundary condition , while the second boundary condition determines c as(4)

Therefore is finally given by(5)

It will be shown in a later section that the general solution (5) reduces to the same results in the literature as special cases. However, the limitation of the current method can be found if Eq. (1) is non-homogenous, i.e., the R.H.S. of this equation contains some terms in t.

3 Special cases

3.1

When , Eq. (1) becomes(6) which arises in Hoda et al. (2017). Hence, the exact solution in this case can be directly obtained from (5) as(7)

3.2

When vanishes, the solution directly comes from (7) as(8)

3.3

When both of R and vanish, the solution is obtained from (7) or (8) as(9)

It is important here to refer to that the solutions (7) and (9) can be expressed in terms of the generalized incomplete gamma function. In order to do that, some relations between the Kummer function and the generalized incomplete gamma function shall be proved in the following section.

4 Analysis

4.1 Theorem 1

For , we have(10)

Proof

Using the definition of Kummer’s function, we have(11) where(12) is the generalized incomplete gamma function. □

4.2 Lemma 1

For , we have(13)

Proof

On using the definition of the generalized incomplete gamma function, we have

4.3 Lemma 2

For , we have(14)

Proof

The proof follows immediately from Theorem 1 and Lemma 1. □

5 Solution in terms of the generalized incomplete gamma function

5.1

On inserting (13) and (14) into (7), it then follows(15) which is more simpler form than the corresponding hypergeometric solution (7).

5.2

In this case we have from Eq. (15) at that(16) which is also another simpler form than corresponding hypergeometric solution (9).

6 Applications

6.1 Heat transfer of carbon-nanotubes with convective condition

In Hoda et al. (2017), the heat transfer of carbon-nanotubes suspended nanofluids in the presence of convective condition has been studied and given by the following ODE:(17) where m and n are physical parameters. In Hoda et al. (2017), the Laplace transform was applied to solve Eq. (17) with the BCs (2). However, we can directly use the formula (15) to derive the same exact solution. Before doing that, we compare Eq. (1) with Eq. (17) to assign , Q, and R as(18)

On inserting (18) into (15), yields(19) which is the same exact solution obtained in Hoda et al. (2017, Eq. (24)) via Laplace transform.

6.2 Heat transfer of carbon-nanotubes with no convective condition

In Ebaid and Al Sharif (2015), the heat transfer of carbon-nanotubes suspended nanofluids in the absence of convective condition, i.e., at , is given by the following ODE:(20)

In this case, we have(21)

Therefore, the solution (16) becomes(22) which is also the same exact solution obtained in Ebaid and Al Sharif (2015, Eq. (19)).

7 Conclusion

In this paper, a class of second-order ordinary differential equations arises in nanofluids has been solved exactly. The generalized solution was firstly obtained in terms of the Kummer function. Some properties have been proved for the relation between Kummer’s function and the generalized incomplete gamma function through some lemmas. These properties were then used to express the solutions in simpler forms in terms of the generalized incomplete gamma function. In addition, the later solutions have been applied on some problems in the literature. It was found that the current solutions reduce to those in the literature for certain values of the coefficients. Finally, the current general solution is very useful in obtaining the analytic solution for similar problems which may arise in future without any need to perform any additional calculations.

Conflict of interest

The authors have no conflict of interest.

Acknowledgment

The authors would like to acknowledge financial support for this work from the Deanship of Scientific Research (DSR), University of Tabuk, Tabuk, Saudi Arabia, under Grant No. S-028-1438.

Notes

Peer review under responsibility of University of Bahrain.