Granular material is the assembly of large amount of discrete solid particles. The particles of this matter can be homogeneous or heterogeneous in material, size and shape, and it can be in an elastic or plastic state and have varying degrees of strength. Typically, the spaces between particles are filled with air or water, so granular flow represents a multiphase process. However, if particles are close-packed or the density of particles significantly exceeds the intergranular fluid or gas, then only the particles will be essential in momentum transfer in the matter, and not the fluid (gas) or interaction between the fluid (gas) and particles. Thus the intergranular fluid (gas) may not be taken into account in this case. Granular material mostly gets into this category and therefore can be considered as a dispersed single-phase rather than a multiphase mixture.

Granular materials can behave like elastic-plastic solids, viscous fluids, and even "granular gases" under various external influences. This diversity of granular media modes of behavior explains the existence of different approaches for its description [1,2]:

Works [2,3,4,5,6,7,8] present studies on the granular media mechanics. Applied problems of granular media mechanics, associated with mining engineering are considered in [2]. The connection between granular mechanics and plasticity theory, geological mechanics, synergetics and other branches of science is shown. The paper indicates the existence of two ways in the development of mathematical models of granular material: the first - continuum modeling in the context of continuum mechanics and the second - discrete modeling in the context of discrete element method (DEM), for example. Granular media are standing at intermediate position between viscous fluids and solids (elastic or plastic). One of the problems of granular material mechanics is formulation of closed equations systems, simulating the deformation processes of such materials. [3] gives the experiments of sand deformation of different sizes, which shown that the main axes of stress tensor are deviate from the strain tensor axes. Unlike viscous fluids, granular materials are not sensitive to strain rate, so the internal stresses of packing derived to a stationary condition remain unchanged. Some technological processes in mining and the chemical industry are associated with deformation of granular materials under gas filtration conditions. Paper [4] presents an experimental set for study of the structural behavior of granular medium in shear with an air filtration flow influence without the occurrence of fluidization. The air flow during deformation of the stationary packing of particles causes softening of the material, associated with the decrease of normal and shearing stresses and a porosity increase. Work [5] investigates the stress state of a granular conical embankment. The results of experiments show that the embankment foundation stresses after formation have underestimated values in the center of its hollow above the apex. The reason of its weak growth is the multi-tier "arch structure" which can collapse by weak vibrations. The DEM-based numerical calculations confirm the experimental data. In [6] the uniform shear of a granular sample based on the DEM is considered. Usage opportunity of the numerical results for continuous models formulation is presented. Work [7] presents a DEM based numerical research of the whirling motion of a granular material in a bounded domain. In [8] the 2D problem of unidirectional mass transfer in a granular medium caused by a combined loading with continuous rotation of the strain tensor main axes. To describe this phenomenon a hypoplastic model of a granular material and a resemblance model of an incompressible fluid are used. For both models the deformation kinematics calculations are performed and the results obtained were compared with experimental data. It is shown that the models qualitatively describe the deformation process and the unidirectional mass transfer effect which are observed in experiments.

In many physical processes the granular materials are behave like liquid, thus the transition to hydrodynamic description of these materials is an important question for the flowability study [9,10,11,12,13,14,15,16,17,18,19]. At the moment, a lot of studies of flowability dedicated to the description of its behavior using continuum theories were performed [17,20,21,22,23]. A research of the granular medium motion in a vertically shaken opened container shown that experimental and numerical results can be explained using the hydrodynamics theory [24]. The hydrodynamic model is structurally close to the viscous gas equations system [25] with the dependence of viscosity on density. The solvability issues for related models are considered in [27,28,29,30,31]. The most common numerical methods for similar problems are given in [32,33,34].

This paper proves the solvability of an initial boundary-value problem for the equations of one-dimensional non-stationary isothermal and non-isothermal motion of granular material. The viscosity depends on density and temperature, and the pressure is the interpolation form of VanderWaals state equation. A numerical study of a model of one-dimensional isothermal motion of granular medium is presented. A difference scheme based on finite-difference approximation is proposed to obtain a numerical solution.

Let’s introduce the necessary notations. Let ${\left|\right|f\left|\right|}_{q,\Omega }=(\underset{\Omega }{\int }{\left|f\left(x\right)\right|}^q{dx)}^{1/q}$, ${\left|\right|u\left|\right|}_{q,r,Q_T}=$

$=(\underset{0}{\overset{T}{\int }}(\underset{\Omega }{\int }{|u(x,t)}^q{dx\left|\right)}^{r/q}{)}^{1/r}$ are the norm functions in Lebesgue’s spaces $L_q(\Omega )$, $L_{q,r}\left(Q_T\right)$, where $r,q\in [1,\infty ]$. We assume $\left|\right|·{\left|\right|}_q=\left|\right|·{\left|\right|}_{q,\Omega },\left|\right|·\left|\right|=\left|\right|·{\left|\right|}_{2,\Omega }$ for brevity. Also we use Sobolev’s spaces $W_p^l(\Omega )$, where l is natural, $1\le p\le \infty$, with the norm ${\left|\right|u\left|\right|}_{W_p^l(\Omega )}={\left|\right|u\left|\right|}_{p,\Omega }^{\left(l\right)}=$

$={\displaystyle \sum _{\left|\alpha \right|=0}^l}{\displaystyle \sum _{\left(\alpha \right)}}\left|\right|D_x^{\alpha }u{\left|\right|}_{L_p(\Omega )}$, where ${\displaystyle \sum _{\left(\alpha \right)}}$ means summing $\left|\alpha \right|={\alpha }_1+...+{\alpha }_n$ [25].

We consider a motion of a shallow, vertically shaken granular bed. The differential equations system, simulating the one-dimensional non-isothermal motion of a granular matter consists of mass, momentum and energy equations [24]: Here $(x,t)$ - Euler’s coordinates; n - number density; m - mass of a particle; u - velocity; p - pressure, $\mu$ - shear viscosity; $\lambda$ - second viscosity; g - gravity acceleration; T - granular temperature, determined by vibrations of particles’ mean velocity, i.e. $1/2k_{B}T=1/2m(<u^2>-<u{>}^2)$, where $k_B$ is the Boltzmann’s constant; I is the dissipative term due to the inelastic particle collisions; $\kappa$ is the heat conductivity coefficient. . In the applications we use the following relations [35,36,37]: Here $n_c$ is the number density of a hexagonal close-packed crystal; d - diameter of the particles; $l={\displaystyle \frac{n_c-n}{\sqrt{8}nd(n_c-an)}}$ - is the mean free path, where $a=1-\sqrt{3/8}$ [35]; $Pr$ is the Prandtl number; $\epsilon =(1-e^2)$ - inelasticity parameter [24], $e\approx 0,9$ is the normal restitution coefficient; $\alpha =0,6$ and ${\gamma }_c=2,26$ are constants from [35,37] respectively. Assuming that the flow density $\rho (x,t)$ equals $mn$ [11], we obtain the Navier-Stokes equations which can be solved in the $Q_T=\Omega \times (0,T)$, $\Omega =(0,L)$ domain with the initial-boundary conditions For the proof of problem (1) – (4) in a short period it is convenient to use another independent variables, namely Lagrange coordinates. Let $y=y(\zeta ,x,t)$ – is the solution of Cauchy problem: ${\displaystyle \frac{\partial y}{\partial \zeta }}{=u(y,\zeta ),y|}_{\zeta =t}=x$. Assume ${\xi =y(\zeta ,x,t)|}_{\zeta =0}$ and consider $\xi$ and t variables as new. Then, $\rho (\xi ,t)={\rho }^0\left(\xi \right)J(\xi ,t)$, where $J(\xi ,t)={\displaystyle \frac{\partial \xi }{\partial x}}$ - is the transition Jacobian [25]. Turning into mass Lagrangian variables $(\overline{x},t)$ from $(\xi ,t)$ with the rule ${\rho }^0\left(\xi \right)d\xi =d\overline{x},\overline{x}\left(\xi \right)=\underset{0}{\overset{\xi }{\int }}{\rho }^0\left(\eta \right)d\eta \in [0,L]$ and designate $\overline{x}$ as x, we obtain The initial-boundary conditions are In (5)-(7) we use the following dimensionless relations (see Appendix A): where $x_1=\underset{0}{\overset{L}{\int }}{\rho }^0\left(\xi \right)d\xi =m{n}_{c}d$, $l_1=d$,$t_1={\displaystyle \frac{d\sqrt{m}}{{\sqrt{T}}_0}}$, $u_1={\displaystyle \frac{{\sqrt{T}}_0}{\sqrt{m}}}$, $p({\rho }^{\prime },T^{\prime })=n_c{T}_0{p}^{\prime },$$\kappa ({\rho }^{\prime },T^{\prime })=n_{c}d\sqrt{{\displaystyle \frac{T_0}{m}}}{\kappa }^{\prime },$$\mu ({\rho }^{\prime },T^{\prime })=m\kappa Pr=m{n}_{c}d\sqrt{{\displaystyle \frac{T_0}{m}}}{\mu }^{\prime },$$T_0\propto m{\left(af\right)}^2$ is the given granular temperature at the bottom of container, amplitude a and frequency f are generated by a shaking table, where the container partially filled with beads is located. Therefore, the range of $x^{\prime }$ is a unit segment [0,1]and the system (5)–(7) becomes (primes omitted): where $G={\displaystyle \frac{mdg}{T_0}}$ is the dimensionless term, which contains the dimensionless shaking strength $S={\displaystyle \frac{T_0}{mgd}}={\displaystyle \frac{a^2{\omega }^2}{gd}}(\omega =2\pi f)$ - it is the ratio of kinetic energy, inserted into the system by the vibrating bottom and the potential energy associated with the particle diameter

The initial and boundary conditions are as follows: We formulate the following results of this work.

Definition 1. A generalized solution of the problem (9) – (13) is the set of functions $\left(u\right(x,t)$, $\rho (x,t)$, $T(x,t))$ from the following spaces: which satisfy equations (9) – (11) and inequations $0<\rho (x,t)<1,0<T(x,T)<\infty$ almost everywhere in $Q_T$ and take the given boundary and initial values in the matter of traces of functions from specified classes.

We will find a local generalized solution as a limit of approximate solutions $({\rho }^n,u^n)$, where $u^n$ are expressed as a finite sum: with unknown coefficients $u_i^n\left(t\right)$, $i=1,2,...,n$. To define these coefficients we require (10) be satisfied approximately: The function ${\rho }^n(x,t)$ will be determined from a solution of: From (17) we obtain the following relation for ${\rho }^n(x,t)$: Unknown coefficients $u_i^n\left(t\right)$ are found from the Cauchy problem solution for the ordinary differential system: and $u^n(x,t)={\displaystyle {\sum }_{i=1}^n}{u}_i^n\left(t\right)sin\left(\pi ix\right)$, ${\rho }^n(x,t)$ is determined by (18). Initial conditions for (19) will find from a sine expansion in the Fourier series of the initial functions $u^0\left(x\right)$: Thus, the generalized solution $(u^n,{\rho }^n)$ satisfies the Cauchy problem (19), (20), the local solvability of this problem follows from Cauchy - Picard theorem for each fixed n for an ordinary differential system [38].

We indicate a value for $t_0$, for which the problem given is solvable for every n on $[0,t_0]$. Another condition of which an interval quantity of $t_0$, is chosen associated with the requirement of positivity for ${\rho }^n(x,t)$. Because $0<b_0\le {\rho }^0\left(x\right)<B_0<1$, we demand that the following relations be satisfied for ${\rho }^n(x,t)$ for every n and with $x\in [0,1],t\in [0,t_0]$. Besides, from (18) and (21) we obtain