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Dispersion and Group Analysis of Dusty Burgers Equations

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Abstract

We investigate the system of non-stationary one-dimensional equations consisting of a parabolic Burgers equation for the velocity of a viscous gas and a hyperbolic Hopf equation for the velocity of solid particles. The Burgers and Hopf equations are connected into a system due to relaxation terms simulating the momentum transfer between the carrier phase (gas) and the dispersed phase (particles). The momentum transfer intensity is inversely proportional to the relaxation time of the particle velocity to the gas velocity (stopping time).

A dispersion relation is constructed for this system. A particular solution corresponding to the damping of a low-amplitude sound wave is found. For an infinitely short velocity relaxation time, the effective viscosity of the gas-dust medium is derived, which is determined by the viscosity of the gas and the mass fraction of particles in the mixture.

The Lie algebra of symmetries of Burgers–Hopf system is found. Invariant submodels with respect to the basis operators of the symmetry algebra are derived. These submodels are explicitly integrated, except for one that defines stationary motion. For this submodel, a code has been developed for the numerical generation of particular solutions of the system. It is shown that the invariant solution determined by this submodel, in the asymptotic case of infinitely short velocity relaxation time also makes it possible to obtain the effective viscosity of the gas-dust mixture. Moreover, this effective viscosity coincides with the viscosity value determined from the dispersion relation.

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Fig. 1

Notes

  1. A brief description of approaches of different levels for modeling two-phase dispersed media can be found, for example, in the works [1] Section 1.2, [2], Section 1.2.

  2. Further in paragraphs 2–4 the functions \(u(t,x)\) and \(v(t,x)\) will be assumed to be real-valued, sufficiently smooth and defined in some two-dimensional domain of real variables \((t,x)\). The solution of the system (1) will be understood as a classical solution.

  3. The one-dimensional Burgers equation under consideration is a special case of the equation of motion from the Navier–Stokes system, corresponding to zero pressure in the medium. This is a model of a continuous medium, which is affected only by viscosity—internal friction, that reduces the velocity difference between two layers of the medium moving with different velocities. The Hopf equation is the Burgers equation with zero viscosity.

  4. Note that our goal is to find real-valued functions of a real variable that are classical solutions of the system (1). Dispersion analysis involves searching for a solution in the form of a complex-valued function of a real variable and subsequent contraction the values of the function to the space of real numbers. This means that we will specify the perturbation of a constant solution in the form (3), then we will work with complex-valued functions, and then we will take the real or imaginary part of the found complex-valued functions as real solutions.

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ACKNOWLEDGMENTS

The authors are grateful to A.V. Panov for a fruitful discussion of the results and careful reading of the manuscript.

Funding

The dispersion analysis (paragraph 2) was done by O.P. Stoyanovskaya (funded by the RSF grant 23-11-00142), the group analysis (paragraph 3) was done by N.M. Yudina as a part of her bachelor thesis at Novosibirsk State University, invariant solutions (paragraph 4) were derived by G.D. Turova (funded by the RSF grant 22-21-20063).

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Authors and Affiliations

  1. Lavrentiev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, 630090, Novosibirsk, Russia

    O. P. Stoyanovskaya

  2. Chelyabinsk State University, 454001, Chelyabinsk, Russia

    G. D. Turova

  3. Novosibirsk State University, 630090, Novosibirsk, Russia

    N. M. Yudina

Authors
  1. O. P. Stoyanovskaya
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  2. G. D. Turova
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  3. N. M. Yudina
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Correspondence to O. P. Stoyanovskaya, G. D. Turova or N. M. Yudina.

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Appendices

Appendix A

NUMERICAL FINDING OF STATIONARY SOLUTIONS OF THE BURGERS—HOPF SYSTEM

To find stationary solutions (1) with an arbitrary value of the velocity relaxation time \(\tau\) we will solve the system (25) numerically. To do this, let’s move from a second-order system of two equations to a first-order system of three equations by introducing a new variable \(p=\varphi_{5}^{\prime}\). We approximate the spatial derivative by the first-order operator

$$y^{\prime}=\frac{y^{n+1}-y^{n}}{h},$$

where \(h\) is the grid step of the finite-difference scheme. As a result, we get the formulas

$$\begin{cases}\varphi_{5}^{n+1}=\varphi_{5}^{n}+hp^{n},\\ \varphi_{6}^{n+1}=\varphi_{6}^{n}+\displaystyle\frac{h}{\tau\varphi^{n}_{6}}(\varphi^{n}_{5}-\varphi^{n}_{5}),\\ p^{n+1}=(1+\displaystyle\frac{h}{\nu}\varphi^{n}_{5})p^{n}+\displaystyle\frac{h\varepsilon}{\nu\tau}(\varphi^{n}_{5}-\varphi^{n}_{6}),\end{cases}$$
(A1)

allowing to find the values of velocities in the computational domain by the given values of the velocity and acceleration of the carrier phase, the velocity of the dispersed phase at a given point in space

$$\varphi_{5}^{0}=u^{0},\quad\varphi_{6}^{0}=v^{0},\quad p^{0}=\displaystyle\frac{\partial u}{\partial x}^{0}.$$

Note that the scheme (A1) is not an asymptotic preserving (AP) scheme. That is, if \(\nu\) or \(\tau\) are small parameters of the problem, for stability it is required to adjust the value of \(h\) from the values of small parameters.

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Stoyanovskaya, O.P., Turova, G.D. & Yudina, N.M. Dispersion and Group Analysis of Dusty Burgers Equations. Lobachevskii J Math 45, 108–118 (2024). https://doi.org/10.1134/S1995080224010505

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