. https://doi.org/10.29121/granthaalayah.v8.i6.2020.549
Angular Momentum Conservation Laws Non-Symmetrical Stress
Tensor Conjugate Problem the Navie-Stokes Separate Problem Using the Ostrogradsky-Gauss theorem to construct the laws of conservation and replacement of the integral over the surface by the integral over the volume, we neglect the integral term outside, i.e. neglect the circulation on the sides of the elementary volume (in the two-dimensional case, this is clearly visible). Circulation means the presence of rotation, which in turn means the presence of a moment of force (angular momentum). As a result, we have a symmetric stress tensor, a symmetric velocity tensor, etc. Static pressure, as follows from kinetic theory, there is a zero-order quantity; the terms associated with dissipative effects are first-order quantities. It does not follow from the Boltzmann equation and from the phenomenological theory that the pressure in the Euler equation is equal to one third of the sum of the pressures on the corresponding coordinate axes. The inaccuracy of determining the velocities in the stress tensor in the stress tensor does not strongly affect the results at low speeds. All these issues are discussed in the work. As example in this paper suggests task of flowing liquid at little distance of two parallel plates.
## 1. INTRODUCTIONThe aim of the work
is to study the consequences of using the Ostrogradsky-Gauss
theorem in continuum mechanics in deriving conservation laws and numerically
solving received equations. Conservation laws were obtained experimentally and
therefore were originally written in integral form. Differential laws are
obtained in two ways: using the finite volume method for an elementary volume
and using the Ostrogradsky Gauss theorem by replacing the surface integral to the volume integral, that is, taking the
integral by parts with further use of the theorems on the conditions Integral
turning in zero. Usually the derivation of conservation laws is analyzed using
the Ostrogradsky-Gauss theorem for a fixed volume
without moving. The theorem is a consequence of the application of the
integration in parts at the spatial case. In reality, in mechanics and physics
gas and liquid move and not only progressively, but also rotate. Discarding the
term means ignoring the velocity circulation over the surface of the selected
volume. When studying vortices, the
whole theory is based on the action of the law of conservation of moments [1], [2] partially the moment is used when
considering stresses in beams. When
taking into account the motion of a gas, the extra-integral term is difficult
to introduce into the differential equation. Therefore, to account for all
components of the motion, it is proposed to use an integral formulation. The classical
Boltzmann equation does not fulfill the law of conservation of momentum. From
the definition of pressure, both from the classical Boltzmann equation and from
the modified one, it does not follow that hydrostatic pressure is one third of
the sum of the pressures at the coordinate surfaces. The equations of motion
obtained from the Boltzmann equation correspond to the zero the order for the
Euler equations and the first order for the Navier-Stokes
equations. Hydrostatic pressure is a zero-order value, but the theory remains
the same when determining different pressures at each surface, i.e. The use of one pressure is
possible under equilibrium conditions (Pascal's law), but for nonequilibrium
conditions the fact is not obvious. This is highlighted in the textbook [1]. Back in 1970, in the
textbook it was said, “First of all, we
note that along with the distribution of
volume and surface forces, for the sake of generality, we should also consider
the volume and surface distributions of pairs of forces (moments), on the possibility of which in continuous
media is currently indicated ”. In the theory of
elasticity, when considering the relationship between the components of the
strain tensor and the stress tensor, the experimental fact is used that the
components (stresses) normal to the
side of
elementary element are
proportional to the sum of the stresses of the other components and they all
differ. The introduced components are the result of the action of moments of
force. The prevailing theory is connected with the fact that the derivation of
conservation laws in the theory of elasticity excludes the contribution of the
distributed moment to the equilibrium of forces. As a result, the law of
equilibrium of forces and moments of forces are considered separately. In solving specific problems, the contribution
of the distributed moment is often studied, but the symmetry condition of the
stress tensor remains. The
existing classical theory is constructed so that the main role is played by
pressure forces. This is probably why, at low speeds, results coincidences with
experiment are obtained. However, calculations even for a potential flow lead
to the formation of a vortex sheet. For Euler's equations, this means that
Pascal's law in the non-equilibrium case does not work and it is necessary to
consider separately not pressure but An example of the importance of separating
the individual components is the study of the processes of wave propagation in
a rotating fluid [3] with the introduction of a item related to
rotation. in pressure gradient. Sometimes, when solving problems in the theory of
elasticity, the first invariants are used, but this can be done if the tensor
is symmetric. When considering the vortex motion, the tensor is asymmetric [4]. The equilibrium condition is fulfilled if there are no internal and
external forces, however, any surface forces can be converted by changing
variables into internal forces and deformations occur. When studying and
writing down conservation laws, it is important to distinguish between
equilibrium and no equilibrium cases. At equilibrium in the mechanics of a
continuous medium, equilibrium means the homogeneity of the distribution of all
macro parameters. However, any additional external influence leads to the
emergence of certain disturbances and creates a distributed moment, which
creates additional force and gives
asymmetric pressure values. Writing separately the law of equilibrium for forces and separately
for the moments of forces without taking into account the mutual influence, and
the moment creates additional force, we come to the conclusion about the
symmetry of the stress tensor. In previous works, taking into
account the angular momentum law nonsymmetrical stress tensor is received. The
method for calculation of nonsymmetrical part was suggested. The work discusses
the listed issues and, if possible, gives answers to some of them. ## 2. EQUATIONS One of the options for deriving symmetry
conditions for pressure tensor: We have angular momentum
However, moment creates additional force and
the symmetry of the stress tensor is broken. Taking into account the equilibrium equation,
we obtain
In
classical case:
In our case
## 3.
THE
INFLUENCE OF THE MOMENT IN THE PROBLEM OF FLUID MOTION BETWEEN TWO CLOSELY
SPACED PARALLEL PLATES
Consider the flow of a very viscous fluid
between two parallel plates, the distance h between which we will consider very
small. Reynolds number is small, external forces are absent. Initial statement
of the problem +) ,
+) , +) . The axes lie in one of the boundary planes, the axis
Oz is directed perpendicular to these planes and the equations of the boundary
planes z = 0, z = h. It is assumed that the velocity is directed parallel to
the boundary planes, so that Due to the small distance between the plates, the order of
the derivative is
large compared to the derivatives and .
The order of the derivatives is large compared to the orders of the
derivatives and .Then the equations take the form. + = 0. We show the influence of the moment in this problem. We take into account that the Euler equations provide the
main order. In classical theory the pressure is selected in the center of the
cell and is the same for all coordinate axes and is equal to p. Suppose that the moment acts and creates a
small additional force. = , = , The result of the additional force will be
new velocity = + ∊ , = + ∊. = 0. By the boundary conditions, we have , , Under the assumption of a small influence of the moment . We
get two equations that pressure must satisfy and = 0. Therefore, the following equation must be satisfied = 0. The results allow us to hope that the pressure distribution
along the corresponding axes is responsible for large eddies. Dispersion create small waves. Viscosity is
responsible for dissipation. ## 4. THE PROPOSED VERSION OF THE EQUATIONSWe will use the
expansion that is in the textbooks [],
but we will make a decomposition with respect
near the center of inertia of the elementary volume = We will not divide the
speed into divergent and vortex parts. We leave Newton's law for the effect of
viscosity. In modern computational mechanics, no difficulties will arise. Then the viscous stress
tensor will have the form For an incompressible
fluid, the equations remain the same in the case for the non-compressible
liquid without the angular momentum,
the pressure gradient will change. Really,
.
+ for the remaining
components is similar. It is significant that in this case the velocity remains
equal to the initial one and the equations coincide in speed with the Lamb
equation. In general, the influence of the moment is added. It follows that all
the conclusions of the classical theory will be preserved. ## APPENDICES
A new model of a
continuous medium is proposed, based on taking into account the angular
momentum through calculation outside the integral term using the Ostrogradsky-Gauss theorem. The model does not use the
hypothesis of equal pressure of one third of the sum of pressures on the sides
of the elementary volume parallel to the coordinate axes. The proof is the
occurrence of a vortex sheet when solving flow problems according to the Euler
model. The contribution of the moment becomes decisive in calculating the flow
separation near the wings. The symmetry of the stress tensor is possible only
if we ignore the additional term,which
takes into account the rotation of the elementary volume, that creates additional force in the
equation for the momentum. ## SOURCES OF FUNDINGNone. ## CONFLICT OF INTERESTNone. ## ACKNOWLEDGMENTNone. ## REFERENCES
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