MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution

1 Introduction

Engineering nanoparticles and the underlying technology are known to man from medieval period. The church paintings of medieval ages are dyed with long lasting metal colloids, and now we know that they are particle size varied noble metal nanoparticles which retain their color unusually over a large period. The notion of faraday reports his optical studies on gold colloids which exhibited a red color than the bulk glittering yellow. The physical and mechanical properties of nanostructures are found to be superior over the bulk materials due to the spatial confinement (finite size), increased surface area and surface energy of these nanoparticles. The increased surface area is advantageous for catalysts and high sensitive chemical sensors, and the functionalization of these nanoparticles presents many applications in colloidal chemistry and biomedical fields. The electronic band structure of the surface and inner atoms differ considerably and this gives rise to novel properties. The mechanical properties such as elasticity, hardness, fatigue, toughness etc., are found to be modified due to high ordered nature of crystallization. The imperfections can diffuse to surface on annealing in turn resulting in improved mechanical properties. Carbon fibers are reported to have high tensile strength and young’s modulus. Composites of nanoparticles and polymers combine the dual properties of both ensuing improved mechanical, thermal and functional properties due to better mixing of the two.

Nanofluids are fine suspensions of nanoparticles in base liquid (Glycol, water, engine-oil etc.), originally proposed to improve the heat transfer properties in heat engineering by increasing the thermal conductivity. The large surface area provides more heat transfer interface between the solid particles and liquid. High stability and dispersibility, reduced clogging in channels, adjustable wettability make these nanofluids superior over conventional solid liquid composites. Choi [1] was the first to play a lead role to introduce that the nanoparticles can be prepared by the metallic or, metallic oxides particles (Cu, Au, CuO, Al₂O₃), carbon nanotubes (Single walled or Multi walled), Ceramics and many other such materials. Eastman et al. [7] conducted an experimental work on the enhancement of thermal conductivity with the suspensions of Copper nanoparticles and ethylene glycol type base fluid. Eastman et al. [7] have conducted an experimental study on thermal conductivity enhancement of ethylene glycol based nanofluids containing Copper nanoparticles. This results shows that the effective thermal conductivity of ethylene glycol increased up to 40% for a nanofluid consisting of ethylene glycol and approximately 0.3% volume of copper nanoparticles of 10 nm diameter. Das et al. [27] have observed 10-25% increase n thermal conductivity in water with 1-4 vol.% of Al₂O₃ nanoparticles. Pak and Cho [3] have reported abnormal increase in viscosity and single phase convective heat transfer coefficient in nanofluid relative to convectional fluids. Thereafter Eastman et al. [10] Kim et al. [9], Wang and Majumdar [36], Kakac and Pramuanjaroenkij [23] reviewed the theoretical and experimental thermal conductivity and viscosity of nanofluids. A benchmark study of thermal conductivity of nanofluids has been published by Buongiorno et al. [8] and Das et al. [22]. Thermophoresis and Brownian motion effects on the convective transport of nanofluids with various models and methods have been proposed by Khanafer and Vafai [11], Tiwari and Das [18], Oztop and Abu-Nada [6], Wang and Wei [33] etc. Kuznetsov and Nield [1] examined the influence of nanoparticles on natural convective boundary layer flow past a vertical plate using the Buongiorno model Nield and Kuznetsov [4] studied the Cheng-Minkowycz problem on natural convection past a vertical plate in a porous medium flooded by a nanofluid. Kuznetsov and Nield [2] discussed the double diffusive natural convective boundary layer flow and heat transfer over a vertical plate. Khan and Pop [34] analyzed the boundary layer flow of nanofluid past a stretching sheet. Khan and Aziz [31] worked on the investigation of physical parameters associated with the nanofluids towards a vertical surface with a constant heat flux. Some pioneer work on the recent development of nanofluids can be found in [38, 39, 40, 41, 42, 43, 44, 45, 46] and the references therein

The Oldroyd-B fluid model was employed to describe the rheological behavior of viscoelastic nanofluid. The Oldroyd-B fluid model important because of its numerous applications such as production of plastic sheet and extrusion of polymers through a slit die in polymer industry, biological solution pant tars glues, etc. The detail study of Oldroyd-B fluid model suspended with nanoparticles can be cited in [5, 17, 18, 24, 26, 28, 30, 35, 37].

Motivated and inspired by the ongoing research in this area the purpose of the current study is to analyze the three dimensional MHD boundary layer flow and heat transfer of Oldroyd-B nanofluid along stretching sheet. The governing equations of momentum, energy and concentration profiles are first reduced to self-similar form and then solved by Differential Transform Method (DTM). The concept of DTM was first introduced by Zhou [12], who solved linear and nonlinear problems in electrical circuits. This method was successfully applied to solve various problems [13, 14, 15, 16, 19, 20, 21] especially in fluid dynamics. All of these successful applications verified the validity, efficiency and flexibility of the DTM.

In the next section we have discussed the implementation of DTM-Padé approximation to the transformed coupled ordinary differential equations and a comparative study is made between 20^th order DTM solution to other semi-analytical techniques through tables and graphs.

2 Mathematical Formulations

Let us consider steady three dimensional (3D) flow of a free convective incompressible Oldroyd-B nanofluid over a stretching surface. Flow is induced by a bidirectional stretching surface at z = 0. Effect of inclined magnetic field, nonlinear chemical reaction, viscous dissipation, heat generation/absorption and variable viscosity are not considered in this study. Presence of electric and induced magnetic field is not accounted. The fluid is considered electrically conducting in the presence of a uniform magnetic field of magnitude B₀ is applied in the z-direction. Let the sheet is stretched in the direction of x- and y- axis and z-axis is taken normal to it. Let U_w(x) = ax and V_w(x) = by be the velocities of the stretching surface along x- and y- directions as shown in Fig. 1. Let the stretching sheet kept at a constant temperature T_w and concentration C_w. the ambient fluid temperature and concentration at free stream form the sheet are T_∞ and C_∞respectively. The governing boundary layer equations for the steady three dimensional flow of an Oldroyd-B nanofluid are [7, 8, 9, 10].

Fig. 1

Geometry of the problem

The boundary conditions for flow situation are:

The similarity variables are introduced as:

Using the following similarity transformations in Eq. (1)–(5), we get the following nonlinear ordinary differential equations as:

subject to the boundary conditions are as follows

Where M is the dimensionless magnetic parameter, β₁ and β₂ are the Deborah numbers in terms of relaxation times and retardation times respectively, N_b is the Brownian motion parameter, N_t is the thermophoresis parameter, Le is the Lewis number, Pr is the Prandtl number and λ is the ratio of stretching rates parameters which are defined as:

The local Nusselt number and the local Sherwood number are defined as

By using the following similarity transformation we have

Where Rex=xuw(x)ν is the local Reynolds’s number

3 Differential Transform Method (DTM)

Let us consider a function f(η) which is analytic in a domain Γ and let η = η₀ represent any point in the domain Γ. The function f(η) is then expressed by a power series as follows:

Here F(k) is called transformed function of f(η). The inverse differential transform of F(k) is defined as

By combining Eqs. (17) and (18), we can obtained f(η) as

Eq. 19 is well known Differential Transform Method derive from the Taylor’s series expansion.

In real world situation, the function f(η) in equation (19) is expressed by a finite series and can be written as

Which means that f(η)=∑k=N+1∞F(k)(η−η0)k is negligibly small, where N is series size.

4 The Padé Approximants

Let us consider a function f(x) representing the following power series

The Padé approximation of any function is a rational fraction and it is denoted by the following notation [61].

Where P_L(x) is a. polynomial of degree at most L and Q_M(x) is a polynomial of degree at most M. We have:

It has been. perceived that in Eq. (22) there are L+1 numerator coefficients and M+1 denominator coefficients. Since we can simply multiply the numerator and denominator by a constant and leave [L, M] unchanged, we impose the following condition

So there are L+1 independent numerator coefficients and M+1 denominator coefficients, making L+M+ 1 unknown coefficient in all. By using the following results given in [62, 63] the series can be written as:

By cross-multiplying Eq. (27), we find that

From Eq. (28) the following set of equations are obtained.

and

Where a_n = 0 for n < 0 and q_ij = 0 for j > M.

For nonsingular equations (29) and (30) the solution can be put in the following form

The construction of [L, M] approximants involves any algebraic operations. The basic criteria is to find the approximants of [L, M] is such that L = M. In this paper here we implement the approximants by using Maple software. Since the diagonal approximant is the most accurate approximant for finding the unknown parameter at infinity. So, here we construct only diagonal approximants which found many applications in heat transfer and pertinent.

5 Numerical Method

It is obvious that the type of the current problem is boundary value problem (BVP) and the appropriate numerical method needs to be selected. The numerical solution is performed using Runge Kutta Fourth order method (RK4).

The fourth order formula is

6 Analytical Approximations by using DTM-Padè

Taking one dimensional differential transform method from Table 1, each term of the governing equations (9)–(11) can be transformed as

and so on

Now putting the above transformed functions into eqns. (8–11) with the transformed boundary conditions,

where F(k), G(k),Θ (k) and Φ (k) are the transformed functions of f(η), g(η),θ (η) and ϕ (η) respectively and are given by

The analytic. approximate solution obtained by DTM cannot satisfy boundary conditions at infinity. Therefore it is essential to combine the series solution with the Padè approximation to provide an efficient tool to handle boundary value problems in infinite domains. By using Maple 2016 built-in function command “Padè” we find the values of A, B, C and D at η → ∞.

The approximate analytic solution by DTM-Padè is given as follows for β₁ = 0.1, β₂ = 0.1,

So here we selected A   =   − 2.32178,B = 1.65456, C = 0.912736, D = −1.51015. we get the 20^th order DTM solution of f(η), f(η), θ (η) and ϕ (η).

7 Results and Discussions

The aim of this section is to examine the effects of various physical parameters on the velocity, temperature and nanoparticles profiles respectively. Table 1 gives the transformation of original function by DTM. Table 2 elucidates the values of governing parameters by Padè approximants. Table 3 and 4 proves the validity of 20^th order DTM-Padè solution by comparing with the existing results. Figs 2–4 give the existence of DTM-Padè solution with Runge Kutta Fourth order method (RK4). Fig. 5 illustrate that the velocity is reduced with an increasing value of magnetic parameter M. As M increases, a resistive force like a drag force is produced, which is called a Lorentz force. The nature of Lorentz force retards the force on the velocity which slows down its motion. The similar effects are measured for normal velocity component in Fig. 6. Fig. 7 shows the effect of Deborah number β₁ on the temperature profile θ (η). An increases of Deborah number both the fluid temperature and thermal boundary layer thickness increases. This is due to the fact that the Deborah number involves relaxation time λ₁ leads to enhancement in thermal boundary layer thickness and temperature. While β₂ depends on the retardation time λ₂. Larger retardation time leads to a weaker temperature profile shows in Fig. 8. Fig. 9 depicts the effects of magnetic parameter M on the temperature profile, which shows that the temperature and boundary layer thickness rises when the value of M are enhanced. in such case, ore heat is generated that corresponds to higher temperature and thicker boundary layer. In Fig. 10 effects of nanoparticles concentration with thermophoresis parameter Nt is plotted. It has been visualized that the concentration profile increases with increase of Nt. This is due to fact that the larger value of Nt increases the thermophoresis force in the boundary layer region which leads to the higher nanoparticles concentration. Fig. 11 showed the impact of Deborah number β₁ on the nanoparticles concentration profile ϕ (η). Increasing values of Deborah number β₁ repots enrichment in concentration profile. Fig. 12 plotted the influence of thermophoresis parameter N_t with nanoparticles concentration profile ϕ (η). Thermophoresis parameter has a proportional relationship with thermos-diffusion constant. Larger value of N_t corresponds to a higher thermo- diffusivity to reunited the nanoparticles within the boundary layer region, leads to an enhancement in concentration profile. Fig. 13 represents the effect of ϕ (η) for various value of Nb. It has been noticed that an increases of N_b, the thermophoresis force is decreases in the thermal boundary layer which results ground, concentration of boundary layer thickness decreases. Therefore the nanoparticles concentration profile is decreases with increases of Nb. In Fig. 14 the effects of Lewis number with nanoparticles concentration profile has been plotted. It shows that the larger values of Lewis number Le causes a reduction in the nanoparticles concentration. Therefore the nanoparticles concentration profile is decreases with increases of Le.

Fig. 2

Comparison of velocity profile with RK4 and 20^th order DTM-Padè solution

Fig. 3

Comparison of temperature profile with RK4 and 20^th order DTM-Padè solution

Fig. 4

Comparison of nanoparticles concentration profile with RK4 and 20^th order DTM-Padè solution

Fig. 5

Influence of f′(η) for various values of M

Fig. 6

Influence of g′(η) for various values of M

Fig. 7

Influence of Deborah number in terms of relaxation time β₁ on θ(η)

Fig. 8

Influence of Deborah number in terms of relaxation time β₂ on θ(η)

Fig. 9

Influence of magnetic parameter M on θ(η)

Fig. 10

. Influence of Thermophoresis parameter N_t on θ(η)

Fig. 11

Influence of Deborah number in terms of relaxation time β₁ on ϕ(η)

Fig. 12

Influence of Thermophoresis parameter N_t on ϕ(η)

Fig. 13

Influence of Brownian motion parameter N_b on ϕ(η)

Fig. 14

Influence of Lewis number Le on ϕ(η)