Startseite A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
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A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform

  • Amit Prakash EMAIL logo , Doddabhadrappla Gowda Prakasha und Pundikala Veeresha
Veröffentlicht/Copyright: 12. Juli 2019
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 8 Heft 1

Abstract

In this paper, numerical solution of fractional order Navier-Stokes equations in unsteady viscous fluid flow is found using q-homotopy analysis transform scheme. Fractional derivative is considered in Caputo sense. The proposed technique is a blend of q-homotopy analysis scheme and transform of Laplace. It executes well in efficiency and provides h-curves that show convergence range of series solution.

Keywords: Navier-Stokes (N-S) equations; q-homotopy analysis transform method (q-HATM); Caputo fractional derivative

1 Introduction

Every phenomenon in fields of Science and Engineering may be alternately modelled using fractional order derivatives. It is due to their non-local property, intrinsic to several complex systems. They are used as modelling tools in nanotechnology, viscoelasticity, anomalous transport, control theory, financial & biological modelling etc. The most important among such models are those described by arbitrary order PDEs. Adomian decomposition [1], homotopy analysis, residual power series, fractional reduced differential transform, fractional variational iteration method [2, 3, 4], Laplace homotopy technique [5], Laplace variational iteration method [6], homotopy perturbation transform method [7], q-homotopy analysis transform method [8, 9, 10], modified trial equation method [11], new iterative Sumudu transform method [12] and Laplace perturbation method [13] etc. are some important methods which are applied to find numerical solution of these problems.

N-S equation define motion [14] of incompressible Newtonian fluid flow extending from enormous scale atmospheric motions to lubrication of bearings and express conservation of mass and momentum. Consider an unsteady, unidimensional viscous fluid motion in a long circular pipe. Fluid is initially at rest. Constant pressure gradient along pipe axis is abruptly imposed to set fluid in motion. Flow is taken as axially symmetric. It is given in [15] as,

ρut=−pz+μDr2u+1rDru,(1)

axis of pipe is z-axis and r is radial direction.

El-Shashed and Salem [16] generalized N-S Eq. by fractional N-S Eq. of order α, as

Dtαu=P+μDr2u+1rDru,0<α≤1,(2)

where P=−∂pρ∂z, α is order of fractional derivative. If α = 1, fractional N-S Eq. reduces to classical N-S equation. Here, efficiency of q-HATM is considered to numerically solve time-fractional N-S Eq. (2). It is an elegant union of q-HAM [17, 18] & transform of Laplace. The q-HAM is actually improvement of parameter q ∈ [0, 1] in HAM to q ∈ [0, 1n], n ≥ 1. Its authority is its potential of adjusting two strong computational methodologies for probing FDEs. Eq. (2) has been handled by Momani and Odibat by ADM [19], Khan et al. by HPM and variation iteration method [20], Ragab et al. by HAM [21], Kumar et al. by fractional modified Laplace decomposition method [22] and HPTM [23], Wang and Liu by modified reduced differential transform method [24] etc. But it has not been solved by q-HATM.

2 Preliminaries

Definition 2.1

A real valued function f(t), t > 0 is in space Cμ, μ ∈ R if ∃ a real number p(> μ) s.t. f(t) = tp f1(t);f1 ∈ C[0, ∞), and is in space Cμm if fm ∈ Cμ, M ∈ N ⋃ {0} [25, 26].

Definition 2.2

Caputo fractional derivative [9] of f, f ∈ C−1m, m ∈ N ⋃ {0} is

Dtαft=dnftdtn,α=n∈N,1Γ(n−α)∫0tt−τn−α−1fnτdτ,n−1<α<n,n∈N.

Definition 2.3

Laplace transform (LT) of Caputo fractional derivative is

LDtnαur,t=snαLur,t−∑k=0n−1snα−k−1ukr,0,n−1<nα≤n.

3 Implementation of q-HATM on Navier-Stokes equation

To illustrate its efficiency, we take a fractional nonlinear nonhomogeneous PDE,

Dtαur,t+Rur,t+Nur,t=fr,t,n−1<α≤n,(3)

where Dtαu is Caputo fractional derivative, R, N are linear, nonlinear differential operators, f(r, t) is source term.

Taking LT on Eq. (3) and applying its differentiation property, we get

Lur,t−1sα∑k=0n−1sα−k−1ukr,0+1sαLRur,t+LNur,t−Lfr,t=0.(4)

Nonlinear operator N is

Nφr,t;q=Lφr,t;q−1sα∑k=0n−1sα−k−1φkr,t;q0++1sαLRφr,t;q+LNφr,t;q−Lfr,t,(5)

here q ∈ [0, 1n] and ϕ is real function.

Build homotopy as

1−nqLφr,t;q−u0r,t=hqHr,tNφr,t;q,n≥1(6)

q is embedding parameter, H ≠ 0 is auxiliary function, h ≠0 is auxiliary parameter, u0 is initial guess.

For = 0, 1n; following results holds:

φr,t;0=u0r,t,φr,t;1n=ur,t.(7)

As q increases, φ varies from u0 to solution u(r, t).

Expanding φ about q by Taylor’s theorem, we get

φr,t;q=u0+∑m=1∞um(r,t)qm,(8)

where

ukr,t=1k!∂kφ(r,t;q)∂qk|q=0.(9)

With suitable choice of u0, n, ℏ, H, series (8) converges at q = 1n, we get

ur,t=u0+∑m=1∞um(r,t)1nm(10)

Define vectors as

u→m=u0r,t,u1r,t,…,um(r,t).(11)

Differentiating deformation Eq. (6)m-times w. r. tq, dividing by m! & taking q = 0, we get

L[umr,t−kmum−1r,t]=ℏH(r,t)Rmu→m−1.(12)

Using inverse transform, we get

umr,t=kmum−1r,t+ℏL−1H(r,t)Rmu→m−1,(13)

where

Rmu→m−1=1m−1!∂m−1N[φr,t;q]∂qm−1|q=0,(14)

and

ks=0,s≤1,n,s>1.(15)

4 Numerical Experiments

Now we implement this method on few test problems.

Example 4.1

Consider a time-fractional N-S equation

Dtαu=Dr2u+1rDru,0<α≤1,(16)

with initial condition

ur,0=r.(17)

Taking transform of Laplace on each side of Eq. (16) and simplifying, we get

Lu−rs−1sαLDr2u+1rDru=0.(18)

N is defined as,

Nφr,t;q=Lφr,t;q−1−kmnrs−1sαL∂2φ(r,t;q)∂r2+1r∂φ(r,t;q)∂r.(19)

The mth-order deformation eqn. for H = 1 is

Lumr,t−kmum−1r,t=ℏRmu→m−1,(20)

where

Rmu→m−1=Lum−1−1−kmnrs−1sαL∂2um−1∂r2+1r∂um−1∂r.(21)

Taking inverse transform on Eq. (20), we find

um=kmum−1+ℏL−1Rmu→m−1.(22)

Taking u0 and solving, we get

u1=−ℏtαrΓ(α+1),u2=−n+ℏℏtαrΓα+1+ℏ2t2αr3Γ(α+1),u3=−n+ℏ2ℏtαrΓα+1+2n+ℏℏ2t2αr3Γ2n+1−9ℏ3t3αr5Γ3α+1,…

Hence, next values may be got. The series solution of Eq. (16) is

ur,t=u0+∑m=1∞um(r,t)1nm.(23)

If n = 1, ℏ = −1 in Eq. (23), we arrive at

ur,t=r+∑n=1∞12.32….2n−32r2n−1tnαn!.(24)

For α = 1 in Eq. (24), we gain solution of standard N-S equation as

ur,t=r+∑n=1∞12.32….2n−32r2n−1tnn!,

Example 4.2

Now, consider time-fractional N-S equation

Dtαu=P+Dr2u+1rDru,0<α≤1,(25)

with initial condition

ur,0=1−r2.(26)

Taking LT on Eq. (25) and simplifying, we get

Lu−1−r2s−Psα+1−1sαLDr2u+1rDru=0.(27)

Also,

Nφr,t;q=Lφr,t;q−1−kmn1−r2s−Psα+1−1sαL∂2φ(r,t;q)∂r2+1r∂φ(r,t;q)∂r.(28)

For H = 1, deformation equation is

Lumr,t−kmum−1r,t=ℏRmu→m−1,(29)

Here

Rmu→m−1=Lum−1−1−kmn1−r2s−Psα+1−1sαL∂2um−1∂r2+1r∂um−1∂r.(30)

By inverse transform,

um=kmum−1+ℏL−1Rmu→m−1.(31)

Taking u0 and solving, we get

u1=ℏ(4−P)tαΓ(α+1),u2=n+ℏ4−P−PℏtαΓ(α+1),u3=n+ℏ24−P−n+ℏP−PℏtαΓ(α+1),

and so on ….

Series solution of Eq. (25) is

ur,t=u0+∑m=1∞um(r,t)1nm.

5 Numerical simulations and discussion

Figs. 1 and 7 show plots of numerical results of Eqs. (16) and (25) respectively when α = 0.5 and 1 for Ex. 4.1 and 4.2. Fig. 2 displays behaviour of solution for arbitrary order α and standard case α = 1 at r = 0.5, ℏ = −1, n = 1 for Ex. 4.1. Fig. 8 shows behaviour of solution for arbitrary order α and standard case α = 1 at r = 1.5, ℏ = −1, n = 1 = P for Ex. 4.2. It is clear from Fig. 2 and 8 that as α tends to 1, the q-HATM solution converges. Fig. 3 and 9 represent convergence control parameter curves for Ex. 4.1 and 4.2. The value of ℏ should be negative. From Fig. 3 and 9, it is clear that, as ℏ decreases, rate of covergence increases. Figs. 4–6 and 10–12, show ℏ-curves at distinct order of fractional derivative at n = 1, 2, 3 for Ex. 4.1 and 4.2. In h-curves, horizontal line exhibits convergence range of solution. We observe that as arbitrary order of derivative increases, range of convergence increases. Also, from Figs. 4–6, 10–12, it is clear that range of convergence depends positively on n. Table 1 shows comparison of results by HAM, FMLDM, HPTM, q-HATM at different values of rand twhen α = 1 = n, ℏ = −1 for Ex. 4.1. It can be observed from Table 1 that our results are accurate and agree with existing methods.

Fig. 1 Plots of q-HATM solution u(r, t) Vs. r, t when ℏ = −1, n = 1, for (a) α = 0.5 (b) α = 1, for Ex. 4.1
Fig. 1

Plots of q-HATM solution u(r, t) Vs. r, t when ℏ = −1, n = 1, for (a) α = 0.5 (b) α = 1, for Ex. 4.1

Fig. 2 α-curve of solution u(r, t) Vs. t when ℏ = −1, n = 1, r = 1.5 for Ex. 4.1
Fig. 2

α-curve of solution u(r, t) Vs. t when ℏ = −1, n = 1, r = 1.5 for Ex. 4.1

Fig. 3 Plot of solution u(r, t) Vs. t when n = 1 = α = r for Ex. 4.1
Fig. 3

Plot of solution u(r, t) Vs. t when n = 1 = α = r for Ex. 4.1

Fig. 4 ℏ-curve of q-HATM solution u(r, t) Vs. h when t = 0.01, r = 1, n = 1 for Ex. 4.1
Fig. 4

ℏ-curve of q-HATM solution u(r, t) Vs. h when t = 0.01, r = 1, n = 1 for Ex. 4.1

Fig. 5 ℏ-curve of q-HATM solution u(r, t) Vs. h when t = 0.01, r = 1, n = 2 for Ex. 4.1
Fig. 5

ℏ-curve of q-HATM solution u(r, t) Vs. h when t = 0.01, r = 1, n = 2 for Ex. 4.1

Fig. 6 ℏ-curve of q-HATM solution u(r, t) Vs. h when t = 0.01, r = 1, n = 3 for Ex. 4.1
Fig. 6

ℏ-curve of q-HATM solution u(r, t) Vs. h when t = 0.01, r = 1, n = 3 for Ex. 4.1

Fig. 7 Plots of q-HATM solution u(r, t) Vs. r, t when P = 1, ℏ = −1, n = 1, (a) α = 0.5, (b) α = 1, for Ex. 4.2.
Fig. 7

Plots of q-HATM solution u(r, t) Vs. r, t when P = 1, ℏ = −1, n = 1, (a) α = 0.5, (b) α = 1, for Ex. 4.2.

Fig. 8 α-curve of solution u(r, t) Vs. t when P = 1, ℏ = −1, n = 1, r = 1.5 for Ex. 4.2.
Fig. 8

α-curve of solution u(r, t) Vs. t when P = 1, ℏ = −1, n = 1, r = 1.5 for Ex. 4.2.

Fig. 9 Plot of solution u(r, t) Vs. t when P = 1, n = 1, α = 1, r = 1, for Ex. 4.2.
Fig. 9

Plot of solution u(r, t) Vs. t when P = 1, n = 1, α = 1, r = 1, for Ex. 4.2.

Fig. 10 h-curve of solution u(r, t) Vs. ℏ obtained when t = 0.01, r = 1, n = 1, for Ex. 4.2
Fig. 10

h-curve of solution u(r, t) Vs. ℏ obtained when t = 0.01, r = 1, n = 1, for Ex. 4.2

Fig. 11 h-curve of solution u(r, t) Vs. ℏ obtained when t = 0.01, r = 1, n = 2 for Ex. 4.2
Fig. 11

h-curve of solution u(r, t) Vs. ℏ obtained when t = 0.01, r = 1, n = 2 for Ex. 4.2

Fig. 12 h-curve of solution u(r, t) Vs. ℏ when t = 0.01, r = 1, n = 3 for Ex. 4.2
Fig. 12

h-curve of solution u(r, t) Vs. ℏ when t = 0.01, r = 1, n = 3 for Ex. 4.2

Table 1

Comparison of results by HAM, FMLDM, HPTM, q-HATM at different values of r and t when α = 1, ℏ = −1, n = 1 for Ex. 4.1.

rtHAM[21]FMLDM [22]HPTM [23]q-HATM
1.250.251.47368000001.47368000001.47368000001.4736800000
1.501.67901234571.67901234571.67901234571.6790123457
1.751.90011602311.90011602311.90011602311.9001160231
2.002.12963867192.12963867192.12963867192.1296386719
1.250.501.77544000001.77544000001.77544000001.7754400000
1.501.89506172841.89506172841.89506172841.8950617284
1.752.07046171242.07046171242.07046171242.0704617124
2.002.27148437502.27148437502.27148437502.2714843750
1.250.752.20136000002.20136000002.20136000002.2013600000
1.502.16666666672.16666666672.16666666672.1666666667
1.752.26960492652.26960492652.26960492652.2696049265
2.002.42993164062.42993164062.42993164062.4299316406
1.2512.79752000002.79752000002.79752000002.7975200000
1.502.51234567902.51234567902.51234567902.5123456790
1.752.50611352412.50611352412.50611352412.5061135241
2.002.60937500002.60937500002.60937500002.6093750000

6 Conclusion

In this paper, approximate analytic solution of time-fractional N-S equation is gained by the q-HATM. It is worth mentioning that q-HATM is capable of reducing time and work of computation in comparison to existing numerical methods keeping higher accuracy of results intact. The q-HATM contains parameters ℏ and n, that can be adopted to manage convergence of solution. It makes this scheme more powerful and exciting.

Acknowledgement

The authors are extremely thankful to the reviewer’s for carefully reading the paper and useful comments and suggestions which have helped to improve the paper.

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Received: 2018-05-23
Revised: 2018-08-20
Accepted: 2018-11-06
Published Online: 2019-07-12
Published in Print: 2019-01-28

© 2019 Amit Prakash et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
https://doi.org/10.1515/nleng-2018-0080
Schlagwörter für diesen Artikel
Navier-Stokes (N-S) equations; q-homotopy analysis transform method (q-HATM); Caputo fractional derivative
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