J. Cent. South Univ. (2017) 24: 2353-2359

DOI: https://doi.org/10.1007/s11771-017-3647-x

Analysis of nano droplet dynamics with various sphericities using efficient computational techniques

Ali Zolfagharian¹, Milad Darzi², S.E. Ghasemi³

1. School of Engineering, Deakin University, Geelong, Victoria 3216, Australia;

2. Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, Missouri 65211, USA;

3. Young Researchers and Elite Club, Sari Branch, Islamic Azad University, Sari, Iran

Central South University Press and Springer-Verlag GmbH Germany 2017

Abstract: Motion of a vertically falling nano droplet in incompressible Newtonian media with initial velocity is investigated. The instantaneous velocity and acceleration are carried out by using the variational iteration method (VIM) and homotopy perturbation method (HPM), which are analytical solution techniques. The obtained results are compared with Runge–Kutta method in order to verify the accuracy of the proposed methods. The results show that, the analytical solutions are in good agreement with each other and with the numerical solution. Also, the effects of sphericity (f) on the velocity and acceleration profiles of the nano droplet are explained. Moreover, the results demonstrate that the VIM-Padé and HPM-Padé are very effective in generating analytical solutions for even highly nonlinear problems.

Key words: nano droplet; velocity; acceleration; homotopy perturbation method; variational iteration method

1 Introduction

Mathematical modelling is a vantage point to reach a solution in an engineering problem, so the accurate modelling of nonlinear engineering problems is an important step to obtain accuratre solutions [1–5]. Most technical problems in fluid mechanics and heat transfer problems are inherently nonlinear. These problems and phenomena can be modeled by ordinary or partial nonlinear differential equations to find their behavior in the environment. Therefore, some different methods have been introduced to solve these equations, such as the variational iteration method (VIM) [6], adomian decomposition method (ADM) [7], homotopy perturbation method (HPM) [8, 9], differential transformation method (DTM) [10, 11], Keller-box method (KBM) [12–14], modified homotopy perturbation method (MHPM) [15], differential quadrature method (DQM) [16, 17], least square method (LSM) [18–20], Galerkin method (GM) [21], collocation method (CM) [22, 23], parameterized perturbation method (PPM) [24], optimal homotopy asymptotic method (OHAM) [25, 26] and Exp-function method [27].

The problem of describing the acceleration motion of a falling sphere and non-sphere nano droplet in Newtonian fluids is relevant to many situations of practical interest. Typical examples include unit operations, such as classification, centrifugal and gravity collection and separation, where it is often important to know the detailed trajectories of the accelerating nano droplets for purposes of design or improved operation. The theoretical development in this area has been given by CLIFT et al [28] for spherical bodies. Several theoretical and empirical correlations for the drag coefficient on a spherical nano droplet are available in the technical literature [29, 30], and each non-spherical model may give the specific value for drag coefficient on a spherical nano droplet. The sphericity (f) of a nano droplet is the ratio of the surface area of a sphere (with the same volume as the given nano droplet) to the surface area of the nano droplet, so sphericity of a sphere nano droplet is unit. As mentioned previously, one of the well-known analytical correlations between Reynolds numbers and drag coefficient for non spherical bodies is presented by CHIEN [30]:

                      (1)

[31] commented on some analytical solutions to a class of Boussinesq-like equations derived recently by means of the homotopy perturbation method (HPM) and showed that one may obtain exactly the same result by means of the Taylor series in the time variable. Nanoparticles can be used to improve heat transfer and energy efficiency in a variety of engineering systems. In this field, recently physical properties of R600a with nanoparticles and without nanoparticles were studied experimentally [32, 33]. In another valuable study, SAZHIN et al [34] studied the effects of droplet evaporation, break-up and air entrainment on diesel fuel spray penetration theoretically at the initial stage of spray penetration when the influence of air entrainment is small. Also, SAZHIN et al [35] investigated the spherical nano droplets treatment in the oscillating flows analytically.

In this work, we aim to apply HPM and VIM as two efficient analytical techniques for nano droplet’s breakup modeling. It is assumed that during the process, breakup will not occur for the nano droplet. The analytical results are also compared with Runge–Kutta method in order to verify the accuracy of the proposed methods.

2 Description of problem

Consider a nano droplet with sphericity of f, diameter (D), mass (m) and density (ρ_s) falling in an infinite extent of an incompressible Newtonian fluid of density (ρ) and viscosity (μ). Let represent the velocity of the nano droplet at any instant time () and g the acceleration due to gravity. The unsteady motion of the nano droplet in a fluid can be described by the Basset–Boussinesq–Ossen (BBO) equation. For a dense nano droplet falling in light fluids and by assuming ρ<<ρ_s, Basset history force is negligible. Thus, the equation of nano droplet motion is given as

(2)

where C_D is the drag coefficient. In the right hand side of the Eq. (2), the first term represents the buoyancy effect, the second term corresponds to drag resistance, and the last term is due to the added mass and force effects which are due to acceleration of fluid around the nano droplet and the unsteady viscous shear on the non-sphere surface with equivalent diameter D, respectively. The second term which is non-linearity due to the nature of the drag coefficient C_D, is the main difficulty in solution of Eq. (2). By substituting Eq. (1) in Eq. (2), Eq. (2) could be rewritten as follows:

      (3)

                             (4)

                                (5)

                       (6)

                            (7)

Equation (3) is a nonlinear ordinary differential equation (ODE) which could be solved by numerical techniques such as the Runge–Kutta method. The VIM-Padé and HPM-Padé are employed to solve the motion equation of a vertically falling nano droplet with initial velocity in incompressible Newtonian media. Also, the validity of the results of the VIM-Padé and HPM-Padé solution are verified via comparison with numerical results obtained using the Runge-Kutta method.

3 Analytical solution

3.1 Homotopy perturbation method

In this section, the principle of the Homotopy Perturbation Method (HPM) is introduced. To illustrate the basic ideas of the HPM, we consider the following nonlinear differential equation:

                        (8)

With the boundary conditions of

                        (9)

where A, B, g(r) and Γ are respectively a general differential operator, a boundary operator, a known analytical function, and the boundary of domain Ω.

The operator A can be divided into a linear part L and a nonlinear part N(u). Equation (8) can therefore, be rewritten as

                       (10)

We construct a homotopy of Eq. (9) , which satisfies

                       (11)

Or

(12)

where is an embedding parameter, while u₀is an initial approximation which satisfies the boundary condition.

We consider u as follows:

                (13)

Setting p=1 yields in the approximate solution of equation to:

                   (14)

3.2 Application of HPM

After introducing the principle of HPM, we can construct a homotopy of our system as follows:

       (15)

Assuming v′=0 and substituting from Eq. (13) into Eq. (15) and rearranging based on powers of p-terms, we have:

     (16)

                 (17)

              (18)

          (19)

And so on, solving the above equations results in the following answers:

                     (20)

   (21)

(22)

According to Eq. (14) we can obtain u as follows:

        (23)

3.3 Padé approximant

Padé approximant is the ratio of two polynomials constructed from the coefficients of the Taylor series expansion of a function y(x). The [L/M] Padé approximants to a function y(x) are given by [36]:

                            (24)

where A_L(x) is polynomial of degree at most L and B_M(x) is a polynomial of degree at most M. The formal power series:

                                (25)

                   (26)

Determine the coefficients of A_L(x) and B_M(x) by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave [L/M] unchanged, we imposed the normalization condition:

                              (27)

Finally, we require that A_L(x) and B_M(x) have non- common factors. If we write the coefficient of A_L(x) and B_M(x) as

            (28a)

           (28b)

Then by Eq. (29), we may multiply Eq. (26) by B_M(x), which linearized the coefficient equations. We can write out Eq. (28) in more details as

            (29a)

                 (29b)

To solve these equations, we start with Eq. (29a), which is a set of linear equations for all the unknown b`s. Once the values of b are known, then Eq. (29b) gives and explicit formula for the unknown a, which complete the solution. If Eqs. (29a) and (29b) are non-singular, then we can solve them directly and obtain Eq. (30), where Eq. (30) holds, and if the lower index on a sum exceeds the upper, the sum is replaced by zero.

(30)

To obtain diagonal Padé approximants of different order such as [1/1], [4/4], [8/8] or [10/10] we can use the symbolic calculus software such as Maple or Mathematica.

Now back to HPM solution, with Eqs. (20)–(23) in mind the Padé approximation was obtained as follows:

                          (31)

Therefore, we are able to give an accurate approximate solution of the considered problem.

3.4 Variational iteration method

In order to illustrate the basic concepts of variational iteration method (VIM), the following nonlinear partial differential equation can be considered:

            (32)

where R is a linear operator which has partial derivatives with respect to x, L is the linear time derivative operator, Nu(x, t) is a nonlinear term and g(x, t) is an inhomogeneous term.

According to VIM, the following iteration formula can be constructed.

  (33)

where λ is the general Lagrange multiplier which can be identified optimally via the variational theory. We obtain the following stationary condition for Eq. (3):

                              (34)

3.5 Application of VIM

In this case, the Lagrange multiplier for VIM can be identified as follows:

                              (35)

Now according to the Eqs. (32) and (33), the correction functional of Eq. (3) can be settled as follows:

                       (36)

with the initial function:

                     (37)

Using the above variational formula (36), substituting the initial function (37) and applying the Padé approximation, the solution was obtained for

HPM-Padé and VIM-Padé had the same result thus for brevity, we do not list the equations herein.

4 Results and discussion

In Fig. 1, the results of the applied methods are shown and compared with the results of a numerical solution with high accuracy using the symbolic algebra package Maple and extensively tested Runge–Kutta algorithm. As an example, Eq. (3) was solved with the following constants:

Fig. 1 Comparison between HPM-Padé, VIM-Padé and numerical solutions for velocity field

Figure 1 clearly shows that the VIM-Padé and HPM-Padé are in good agreement with numerical solution. In the beginning, initial velocity and the gravity force make the nano droplet fall in the fluid but the drag force is more powerful than the buoyance force and eventually it will slow down the nano droplet’s velocity until the drop’s velocity reaches to its terminal value.

We consider an injected non-spherical nano droplet with sphericity of f, equivalent diameter D, mass and density ρ_s injected into a hot air storage tank with density ρ and viscosity μ. Let u represent the velocity of the nano droplet at any instant time, t, and g the acceleration due to gravity.

Considering the nano droplet life time, we use the VIM-Padé to obtain the velocity and acceleration fields of the nano droplet. Figure 2 shows the velocity and acceleration profiles of the nano droplet versus the time.

Fig. 2 Velocity (a) and acceleration (b) profiles

As soon as the nano droplet was sent into the tank due to high velocity of the drop, drag force is so great and when the time passes the velocity of the droplet will decrease and the buoyance force will be more sensible and thus the drag force will be reduced and so is the magnitude of the acceleration.

In Fig. 3, the falling velocity profiles of the particles versus time for different sphericity for the fuel are plotted.

It is apparent from this figure as the sphericity of the nano droplet increases the changes of velocity decreases during the time of process due to the surface area that is straightly in touch with the air and will influence the drag force.

Figure 4 shows acceleration profiles of nano droplet with various sphericities. As it is clearly demonstrated in this figure, there is a constant initial acceleration for each system and it is obvious from this figure as the sphericity of the nano droplet increases, the magnitude of acceleration decreases. In addition, we can see that as the sphericity increases, the time that the acceleration reaches to zero, i.e., terminal velocity, decreases and the nano droplet will reach its terminal velocity.

Fig. 3 Velocity profiles of nano droplet with different sphericities

Fig. 4 Acceleration profiles of nano droplet with different sphericities

5 Conclusions

VIM-Padé and HPM-Padé were presented to study the motion of a single nano droplet moving in a continuous fluid phase with a well-known drag coefficient. A very good agreement has been seen between numerical method and the current applied mathematical methods. Instantaneous velocity and acceleration were obtained as the results. The sphericity parameter (f) effects on dynamic motion of the nano droplet in this system were presented and discussed in details. We concluded that, as the sphericity of the nano droplet increases, the terminal velocity increases and the time taken by the system to attain zero acceleration decreases during the time of process. As the sphericity of the nano droplet increases, the constant initial acceleration of the droplet decreases.

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(Edited by HE Yun-bin)

Cite this article as: Ali Zolfagharian, Milad Darzi, S.E. Ghasemi. Analysis of nano droplet dynamics with various sphericities using efficient computational techniques [J]. Journal of Central South University, 2017, 24(10): 2353–2359. DOI:https://doi.org/10.1007/s11771-017-3647-x.

Received date: 2016-05-09; Accepted date: 2016-12-23

Corresponding author: S.E. Ghasemi, PhD; E-mail: s.ebrahim.ghasemi@gmail.com