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Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates
Advances in Difference Equations volume 2021, Article number: 486 (2021)
Abstract
Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions \((r,t,z)\) due to the presence of the term \(\frac{1}{r}\), which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.
1 Introduction
Suppose that the Euler equations have the form [1–4]
That describes the dynamics of incompressible, axisymmetric flow with swirl [3], where \(w ( r,t,z )\), \(u ( r,t,z )\), and \(v ( r,t,z )\) are the components of the velocity in the cylindrical coordinates \((r,\phi ,\text{ and }z)\), and \(p ( r,t,z )\) is the pressure. The flow is called axisymmetric flow if the velocity component and the pressure are independent of ϕ. Navier–Stokes and Euler equations in the cylindrical coordinates can describe any pipe fluid flow that has more applications, especially in the medical field. For example, blood flow in stenoses narrow artery [5–8]. System (1) had been solved using numerical methods in [1, 2, 9]. Manipulation of the results in most applications needs explicit solutions. The Lie symmetry analysis is one of the most important and powerful methods for obtaining closed-form solutions [10, 11]. The method proves its dependence in the fluid mechanics, turbulence field, and turbulent plane jet model [12–18]. Other researchers apply the method to other applications [19–25]. In (2007), Oberlack et al. [3] deduced five Lie point symmetries for Euler equations. Here, we use the commutative product to explore new Lie infinitesimals for system (1), then we use the investigated Lie vectors to reduce system (1) to the system of ODEs. By solving these ODEs, we explore new analytical solutions for Euler equations.
2 Investigation of Lie infinitesimals for Euler equations
System (1) possesses Lie infinitesimals as follows:
There are an infinite number of possibilities for these vectors as the presence of arbitrary functions \(f_{i} ( t )\), \(i=1\dots 8\). Using the commutative product between these infinitesimals listed in Table 1 authorizes us to specialize these vectors through the same procedure as in [10, 26]. Firstly, we generate the commutator table as follows in Table 1, where
The specialization process generates a nonlinear system of ODEs:
Through manual calculations this system has been solved, and the results are
Substituting from (5) into (2), we obtain
We use these vectors (6) to reproduce the commutator table (Table 2).
3 Reduction of the independent variables in Euler equations
3.1 Using Lie vector \(\boldsymbol{{X}_{{1}}}\)
To snaffle the similarity variables, we solve the associated Lagrange system
The similarity variables of system (1) are
Substituting from (8) into (1), we get the following system with two independent variables:
System (9) has five Lie vectors as follows:
3.1.1 Using vector \(\boldsymbol{V}_{\boldsymbol{3}}\)
This Lie vector will reduce system (9) to
where the new dependent variables have been obtained from solving the characteristic equation that the \(V_{3}\) was generated.
The solutions for system (11) are as follows:
Back substitution to the original variables using similarity variables in (8) and (12) leads to
where \(\boldsymbol{\delta}= \frac{(z- \ln ( t ) )}{r} \).
The solutions have been plotted for different values of time as depicted in Figs. 1–4.
3.1.2 Using \(\boldsymbol{V} = \boldsymbol{V}_{\boldsymbol{1}} + \boldsymbol{V}_{\boldsymbol{4}}\)
This vector produces a system of nonlinear ODEs as follows:
where the new dependent variables are
By solving system (15), new solutions for Euler equations have been produced:
Using the similarity variables in (8) and (16) leads to back substitution to the original variables:
3.1.3 Using Lie vector \(\boldsymbol{V} = \boldsymbol{V}_{\boldsymbol{1}} + \boldsymbol{V}_{\boldsymbol{5}}\)
Through the same previous procedure system (9) has been reduced to
where the similarity variables are
System (19) has closed form solutions as follows:
Back substitution using the similarity variables in (20) and (8) is as follows:
The solutions have been plotted in Figs. 5–8.
3.2 Using Lie vector \(\boldsymbol{X} = \boldsymbol{X}_{\boldsymbol{3}} + \boldsymbol{X}_{\boldsymbol{4}}\)
By solving the subsidiary equation, we explore the similarity variables
which reduce system (1) to
This system possesses three Lie vectors as follows:
-
Using \(\boldsymbol{V} = \boldsymbol{V}_{\boldsymbol{1}} + \boldsymbol{V}_{\boldsymbol{2}}\)
Following the same procedure system (24) will be reduced to
with new variables
By solving system (26), we have
Using the similarity variables in (23) and (27) authorizes us to back substitution to the original variables
4 Conclusions
We deduce an infinite number of Lie infinitesimals, and through commutative product properties, we minimize these vectors to four Lie vectors. Through some combinations between these vectors, we explore exact solutions for Euler equations. The results illustrate that the velocity components decrease with increasing the spatial or temporal coordinates. The pressure may be appearing as a negative value, and this is reasonable according to the human pressure in the case of the tapered artery [6].
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References
Verzicco, R., Orlandi, P.: A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123(2), 402–414 (1996)
Saiac, J.H.: Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid. Int. J. Numer. Methods Fluids, 10(2) 141–160 (1990)
Frewer, M., Oberlack, M., Guenther, S.: Symmetry investigations on the incompressible stationary axisymmetric Euler equations with swirl. Fluid Dyn. Res., 39(8), 647 (2007)
Leprovost, N., Dubrulle, B., Chavanis, P.-H.: Dynamics and thermodynamics of axisymmetric flows: theory. Phys. Rev. E, 73(4), 046308 (2006)
Chakravarty, S., Sen, S.: A mathematical model of blood flow in a catheterized artery with a stenosis. J. Mech. Med. Biol. 9(3), 377–410 (2009)
Chakravarty, S., Mandal, P.: Mathematical modelling of blood flow through an overlapping arterial stenosis. Math. Comput. Model. 19(1), 59–70 (1994)
Prasad, K.M., Thulluri, S., Phanikumari, M.: Investigation of blood flow through an artery in the presence of overlapping stenosis. J. Nav. Archit. Mar. Eng. 14(1), 39–46 (2017)
Akbar, N.S.: Blood flow analysis of Prandtl fluid model in tapered stenosed arteries. Ain Shams Eng. J. 5(4), 1267–1275 (2014)
Barbosa, E., Daube, O.: A finite difference method for 3D incompressible flows in cylindrical coordinates. Comput. Fluids 34(8), 950–971 (2005)
Ali, M.R., Sadat, R., Ma, W.X.: Investigation of new solutions for an extended \((2 + 1)\)-dimensional Calogero-Bogoyavlenskii-Schif equation. Front. Math. China 16, 925–936 (2021). https://doi.org/10.1007/s11464-021-0952-3
Sadat, R., et al.: Investigation of Lie symmetry and new solutions for highly dimensional non-elastic and elastic interactions between internal waves. Chaos Solitons Fractals 140, 110134 (2020)
Sadeghi, H., Oberlack, M., Gauding, M.: On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation. J. Fluid Mech. 854, 233–260 (2018)
Wacławczyk, M., Grebenev, V., Oberlack, M.: Lie symmetry analysis of the Lundgren–Monin–Novikov equations for multi-point probability density functions of turbulent flow. J. Phys. A, Math. Theor. 50(17), 175501 (2017)
Wacławczyk, M., Oberlack, M.: Symmetry analysis and invariant solutions of the multipoint infinite systems describing turbulence. In: Journal of Physics: Conference Series. IOP Publishing, Bristol (2016)
Sahoo, S., Saha Ray, S.: On the conservation laws and invariant analysis for time-fractional coupled Fitzhugh-Nagumo equations using the Lie symmetry analysis. Eur. Phys. J. Plus 134, 83 (2019)
Jyoti, D., Kumar, S., Gupta, R.K.: Exact solutions of Einstein field equations in perfect fluid distribution using Lie symmetry method. Eur. Phys. J. Plus 135, 604 (2020)
Zhao, Z., Zhang, Y., Han, Z.: Symmetry analysis and conservation laws of the Drinfeld-Sokolov-Wilson system. Eur. Phys. J. Plus 129, 143 (2014)
Kumar, D., Kumar, S.: Solitary wave solutions of pZK equation using Lie point symmetries. Eur. Phys. J. Plus 135, 162 (2020). https://doi.org/10.1140/epjp/s13360-020-00218-w
Ali, M.R., Sadat, R.: Lie symmetry analysis, new group invariant for the \((3 + 1)\)-dimensional and variable coefficients for liquids with gas bubbles models. Chin. J. Phys. 71, 539–547 (2021), ISSN 0577-9073
Jadaun, V., Kumar, S.: Symmetry analysis and invariant solutions of \((3+ 1)\)-dimensional Kadomtsev–Petviashvili equation. Int. J. Geom. Methods Mod. Phys. 15(8), 1850125 (2018)
Ali, M.R., Sadat, R.: Construction of Lump and optical solitons solutions for \((3+1)\) model for the propagation of nonlinear dispersive waves in inhomogeneous media. Opt. Quantum Electron. 53, 279 (2021). https://doi.org/10.1007/s11082-021-02916-w
Agarwal, P., Deniz, S., Jain, S., Alderremy, A.A., Aly, S.: A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Phys. A, Stat. Mech. Appl. 542, 122769 (2020) ISSN 0378-4371. https://doi.org/10.1016/j.physa.2019.122769
Salahshour, S., Ahmadian, A., Senu, N., Baleanu, D., Agarwal, P.: On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem. Entropy 17, 885–902 (2015). https://doi.org/10.3390/e17020885
Zhang, Y., Agarwal, P., Bhatnagar, V., Balochian, S., Yan, J.: Swarm intelligence and its applications. Sci. World J. 2013, Article ID 528069 (2013). https://doi.org/10.1155/2013/528069
Zhou, S.-S., Areshi, M., Agarwal, P., Shah, N.A., Chung, J.D., Nonlaopon, K.: Analytical analysis of fractional-order multi-dimensional dispersive partial differential equations. Symmetry 13, 939 (2021). https://doi.org/10.3390/sym13060939
Zhang, Y., Agarwal, P., Bhatnagar, V., Balochian, S., Zhang, X.: Swarm intelligence and its applications 2014. Sci. World J. 2014, Article ID 204294 (2014). https://doi.org/10.1155/2014/204294
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Sadat, R., Agarwal, P., Saleh, R. et al. Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates. Adv Differ Equ 2021, 486 (2021). https://doi.org/10.1186/s13662-021-03637-w
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DOI: https://doi.org/10.1186/s13662-021-03637-w
Keywords
- Euler equations
- Axisymmetric flow
- Lie point symmetries
- Analytical solutions