Radiation effect on terminal velocity of a vertically falling non-spherical particle by diagonal Pade’ [3/3] Approximants

Introduction

The problem of acceleration motion of vertically falling spherical and non-spherical particle in Newtonian and Non-Newtonian fluids is relevant to many situations of practical interest. It is necessary to know the detailed trajectories of the accelerating particles for the purposes of design or improved operation. For example, the measurements of terminal velocity of raindrop in Newtonian fluids using the falling ball method. It is also necessary to know the time and distance required to reach the particle at terminal point to determine the reliable results for design models. In present, the non-spherical particle is considered in the presence of Radiation effect. It is clear from previous literature the motion of particle is affected by Radiation force. Considerable attentions have been devoted to the study of the acceleration motion of nonspherical particles in fluids and an excellent account of theoretical development in this area has been given by Clift et al. [17] for spherical bodies. Less information is available in the previous literature for the case of motion of non-spherical particle with Radiation force. Many correlation for the drag coefficient in terms of the Reynolds number for motion of nonspherical particles were given in the literature [1,5] . From all these, one of the well-known analytical correlation between Reynolds numbers and drag coefficient for non-spherical particle is presented by Chien [18] 67.289 .

Where Re= (1) Where drag coefficient with Sphericity and Re is is Reynolds numbers. These are based on the equal volume sphere diameter [13] . Eq. ( 1) was stated to be valid in the ranges of 0.2< <1 and 0.001<Re<10000 for the different shapes of partical [18]. The analysis derived by Pade' approximant and Varinational iteration method (VIM). The results of current methods are compared with the well-known R-K 4 th order method in order to verify the accuracy of the proposed methods.

Problem Formulation

Consider a rigid body, single non-spherical particle is falling in an infinite extent of incompressible Radiated Newtonian fluid of density ρ and viscosity µ, u represents the velocity of the non-spherical particle at any instant time t, g is the acceleration due to gravity and H is a Radiation parameter [2,3,7] . Thus, the equation of the particle motion is given by

Where the drag coefficient, equivalent volume diameter D, mass m and particle density ρ . In right hand side of the eq.( 2), the 1 st term represent the buoyancy effect, the 2 nd term corresponds to drag resistance, 3 rd term is associated with the added mass effect which is due to acc. of fluid around the particle and 4 th term is Radiation Parameter. The non-linear terms due to non-linearity nature of the drag coefficient is the main difficulty in solving eq. ( 2) could be re-written as follows:

Where a= (m πD 3 ρ) , b=3.75πDµ, c= [7,12,16] [ ] = , where L=M (for diagonal pade' approximants) (4) Where

) and (t) are the polynomials of the degree of at most L and M respectively. The formal power series is give u (t) =∑ (5) From eq.( 4), u (t)-=0(t L+M+1 ) (

Find the coefficients , , , , …with help of Taylor's expansion for a=b=c=d=1 Equ.(3) becomes, ′ (t)=H+1-u 2 -u ,u(0)=0 (8d)

(A).2 Diagonal pade' [3/3] for H=0

The given equation (8d) becomes ′ (t)=1-u 2 -u, with initial condition u(0)=0 (9) Taylor's series about zero of equ. ( 9) is given by

Now compare equ. (9b) with equ.( 8), So 0,

Solving the system of linear equations (8c) by Gauss elimination method and using eq. (9c) =0, =1, =0, =0.083 and =1, =0.5 , =0.5 , =0.0417

For H=2, The given equation is

For H=4, The given equation is In fig. 1, u (vertically) denotes the velocity of particle w.r.t. time t (horizontally).Solution for the velocity of a single nonspherical particle during the acceleration motion is obtained by diagonal pade' approximants of order [3/3]. Fig. 1.shows the Radiation effect on velocity of a single particle with Radiation parameter H=2, 4. Due to Radiation effect, the velocity of a particle is increasing with increment in the Radiation parameter

(11d)

(B)

. Varinational iteration method (VIM) J. He, was introduced Varinational iteration Method (VIM) to solve the several nonlinear ordinary and partial differential equations in 1997 [6] . He's Varinational iteration method (VIM) has been extensively applied as a power tool for solving various kinds of problems [8,9,10] . Besides the He' VIM, Saeed. U. used Modified Varinational Iteration Method to solve Sine-Gordon Equation and comparison with numerical results. They conclude that VIM provides stable and accurate results. VIM also gives the useful result for solving WBK equation. Using VIM, Liu and Gurram have solved the problems of free vibration involving an Euler-Bernoulli beam and obtained accurate results and compare the result with ADM [20] . Slota use the VIM to obtained the results for the Heat equation which were same as the exact solution [4,21] . He's VIM also used for solving a semi-linear inverse parabolic equation. A new application of Varinational iteration method to solve chaotic Rossler system [19] . From review of literature, it is confirmed that He's VIM is most accurate and useful method for solving the nonlinear problems.

Where L is a linear operator, N is a nonlinear operator and g(t) is a non-homogeneous term. By using the Varinational iteration method, a correction functional can be constructed as (t)= ʎ ũ (12a) Where ʎ is a general Lagrange multiplier, the subscript n means the nth approximation; is restricted variation and δũ =0. [11,15] According to VIM, firstly we will find Lagrange multiplier and then trial function to get the successive iterations , n 0 Which converge to the exact solution. The solution is u= lim → To solve eq. (3) using VIM [22] , the correction functional can be constructed as follows:

(t)= ʎ (13) For a=b=c=d=1, Eq. ( 13) becomes

The stationary conditions can be obtained as follows:

Subsequently, the Lagrangian multiplier is obtained as: ʎ= -

Substituting eq. (14e) in eq. ( 14) and assuming 0, solution will be gained for velocity variation w.r. In Table 2, Solution for velocity of a vertically falling single non-spherical particle during the acceleration motion with Radiation effect and without Radiation effect is obtain by Varinational Iteration Method (VIM) of 4 th iteration. IV. Runge-kutta 4 th order method (Numerical Method), It is clear that the current problem is initial value problem (IVP) of 1 st order. So far a solution, we can apply numerical methods like trapezoidal method, Euler's method (1 st order R-K method), and R-K 4 th order method. Trapezoidal method is generally used for typical problems. The R-K 4 th order method is the modification in Euler's method by adding midpoint in the step which increase the accuracy. Thus R-K 4 th order method is a suitable numerical technique in present problem [14] Equ. (3) , ′ (t)=H+1-u 2 -u , is a 1 st order differential equation with initial condition u(0)=0 (16)

For H=0, f (t,u)= 1-u 2 -u ,u(0)=0 (16a) For H=2, f (t,u)= 3-u 2 -u ,u(0)=0 (16b) For H=4, f (t,u)= 5-u 2 -u ,u(0)=0 (16c) In fig. 3, velocity variation of a vertically falling single nonspherical particle in an infinite incompressible Radiated Newtonian fluid is investigated by numerical method. With increase in Radiation parameter, the velocity is also increasing. In fig. 4. the Acceleration variation of a single particle without Radiation effect and with Radiation effect as a Radiation parameter H=2 &H=4 is investigated by numerical (R-K 4 th order) methods. It shows that the particle with Radiation effect H=4 reaches early at steeling velocity as compare to others like H=2,H=0. So it is clear that acceleration motion of a particle with high Radiation effect becomes zero (not accelerated) in short time interval as compared to and low Radiation effect and without Radiation effect 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0189 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0220 0.0206 0.0589 0.1320

Results and discussion

Conclusion

The achievement of this work is to apply the current methods i.e. diagonal pade' and VIM in order to study the nonlinear differential equation of 1 st order with initial condition that governed from the Radiation effect on accelerated motion of vertically falling single non-spherical particle in incompressible Radiated Newtonian fluid. The current methods are applied without using any linearization, discretization, restrictions or transformations. From above discussion, it is clear that due to increase in Radiation parameter velocity is also increasing and particle with high Radiation effect reaches early at terminal velocity. The diagonal pade' method [3/3] has a good agreement with numerical method and provides highly reliable results. In addition, this method does not require many iterations like VIM to reach accurate results. Both methods gives the accurate results in short time, but Diagonal pade' method is also suitable for long time. Also, the current method (diagonal pade') can be used to develop the valid solution of other nonlinear differential equation of order more than one.

Acknowledgment

The author would like to thank to the Department of Mathematics at IK Gujral Punjab Technical University, Kapurthala (Punjab) where the author worked as a visiting scholar.

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