Influence of the Angular Momentum in Problems Continuum Mechanics

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2021.16.1 Evelina Prozorova

 Influence of the Angular Momentum in Problems
 Continuum Mechanics
 EVELINA PROZOROVA
 Mathematical-Mechanical Department
 St. Petersburg State University
 Av. 28 , Peterhof, 198504
 RUSSIA

 Abstract: - For continuum mechanics a model is proposed, that is built with consideration outside
 the integral term when deriving conservation laws using the Ostrogradsky-Gauss theorem.
 Performed analysis shows discrepancy between accepted classical conservation laws and
 classical theoretical mechanics and mathematics. As a result, the theory developed for potential
 flows was extended to flows with significant gradients of physical parameters. We have
 proposed a model that takes into account the joint implementation of the laws for balance of
 forces and angular momentums. It does not follow from the Boltzmann equation that the pressure
 in the Euler and Navier-Stokes equations is equal to one third of the sum the pressures on the
 corresponding coordinate axes. The vector definition of pressure is substantiated. It is shown that
 the symmetry condition for the stress tensor is one of the possible conditions for closing the
 problem. An example of solving the problem of the theory of elasticity is given.

 Keywords: angular moment, potential flow, circulation, Euler equation, Pascal’s law

 Received: September 26, 2020. Revised: February 13, 2021. Accepted: February 26,
 2021. Published: March 4, 2021.

 1. Introduction calculation of stresses in the rods [1,2] and
 In aeromechanics, shipbuilding, the physics in the calculations of viscoelastic problems
 of earth and atmosphere, there are no [3]. The probable reason for ignoring the
 understanding of some processes associated effect in the general theory is the proof of its
 with the interaction of flows with body absence, built on the basis of the theory,
 which are moving at high speeds; there are taking into account the interaction of
 no adequate models for the formation of neighboring elementary volumes only along
 vortex structures, etc. The objects of study the normal to the surface [4-6]. In this case,
 in continuum mechanics are solids, liquid, one can only prove the consistency of the
 gas, and plasma. In a mathematical whole theory. The influence of the rotational
 description, the basic laws are conservation part of the stress tensor can be traced by
 laws. This unites them. Therefore, the work analyzing the Hamel solution and its
 discusses general questions arising when generalizations [7]. Dividing the speed into
 writing conservation laws. The law of divergent and rotational components about
 equilibrium of forces was put as basis of the an axis passing through an arbitrary point
 classical theory, the law of conservation does not right, as according to theory, only
 angular momentum was considered as movement around the axis of inertia need to
 consequence of the fulfillment the law of consider. A model that is ignoring the
 balance forces. Each of laws was considered rotation of the elementary volume leads to
 independently. However, the gradient of the the symmetry of the stress tensor, which
 distributed moment leads to the formation of violates the "continuity" of the medium [8].
 an additional force, that we need to take into When analyzing the derivation of
 account in the equations of motion. This is differential conservation laws in continuum
 evidenced by the works devoted to the mechanics, it turned out that the transition

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DOI: 10.37394/232011.2021.16.1 Evelina Prozorova

 from the integral formulation to the version, use only the equilibrium part of the
 differential formulation (integration by Chapman-Enskog function, the rest of the
 parts) was performed without taking into terms included in the function do not
 account outside integral component. In contribute. Consequently, the full account of
 classical version, the Ostrogradsky-Gauss the viscous contribution is ensured by the
 theorem is formulated for a motionless body full velocity, in fact, by its corresponding
 without rotation [9]. Thus, the generalization first derivatives. However, the
 of the theorem to body that is rotating [10] macroparameters included in the equilibrium
 leads to a stronger influence of the no distribution function must be calculated not
 symmetry of the stress tensor in the entire according to zero approximation from the
 mechanics of a continuous medium. After Euler equations, but from the equations of
 integration by parts, the transition from the the first approximation (the Navier-Stokes
 integral formulation to the differential equations, if they are used). The equations
 formulation is difficult. However, one can of motion obtained from the Boltzmann
 take advantage of physical and geometric equation correspond to the zero order for the
 considerations and derive equations from Euler equations and the first order for the
 them. At present, the need to divide the Navier-Stokes equations. Using only
 stress tensor into two parts is not clear. The divergent part of velocity without rotation it
 speeds of various processes at the time of we obtain speed different from the initial
 writing the equations were relatively low one. The situation is saved by the fact that
 compared to modern one. Subsequently, the this circumstance affects the viscous
 theory that was developed for potential component of stresses, which is a first-order
 flows was used to flows with significant quantity. Hydrostatic pressure is a zero-
 gradients of physical parameters was order quantity. Using Pascal's law for
 expanded. The lack of symmetry leads to an equilibrium, the pressure is chosen equal to
 increase in the order of the derivatives the one third of the pressure on the coordinate
 Navier-Stokes equations. The role of areas. It does not follow from the Boltzmann
 boundary conditions increases accordingly. equation that the pressure in the Euler
 Their specific form is determined by the equation is equal to one third of the sum of
 problem under consideration. Since the the pressures on the corresponding
 original formulation obtained in experiments coordinate axes. There is no experimental
 is integral, then numerical methods using confirmation of this fact. However, the
 conservation laws in integral form are theory remains the same when determining
 preferable. These methods include the finite the different pressure on each of the sites,
 volume method. The implementation of the i.e. the use of one pressure is possible under
 method is achieved in various ways. equilibrium conditions (Pascal's law), but for
 Equations of an incompressible fluid are no equilibrium conditions the fact is not
 best approximated. For a compressible fluid, obvious. Attention is drawn to this in the
 the computational process becomes more textbook [11] and articles [12,13]. In the
 complicated. It is important to use the theory of elasticity, when considering the
 integral conservation laws obtained from the relationship between the components of the
 experiment. The transition from the deformation tensor and the stress tensor, the
 currently used equations to integral ones due experimental fact of the change in the
 to the rejection to investigate of the rotation components (stresses) normal to the
 effects does not allow the conservation laws elementary site is used in proportion to the
 to be fulfilled. The experimentally sum of the stresses from others components,
 confirmed Newton's law on the relationship and they all differ. If problems of the theory
 between the stress and strain tensor of elasticity was solving with a symmetric
 concerned the total velocity with the no tensor, it is necessary to satisfy the
 symmetric tensor. In the kinetic theory, the compatibility conditions [5,6]. To get
 coefficients of viscosity and thermal around these conditions when solving the
 conductivity, even in no equilibrium problem numerically, the function values in

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 the cell are “averaged”. The result of the contribution to friction and pressure.
 calculation is then the values in the center. Measurements performed on the axes of
 This is not discussed in the articles, but in symmetry do not allow speaking about the
 private conversations the fact is admitted. degree of asymmetry of the stress tensor.
 Experimental fact of the no symmetry of the The existence of invariants higher than the
 stress tensor can be found in the works first and, accordingly, the principal axes of
 [14,15], theory in [16]. The existing theory the elementary volume in the proposed
 is connected with the fact that the derivation theory of elasticity becomes problematic.
 of conservation laws in the theory of The question of equilibrium in mechanics of
 elasticity excludes the contribution of the a continuous medium and the extension of
 distributed moment to the conditions of Pascal's law to the case of small
 equilibrium of forces. An interesting feature deformations, the need to replace the
 of the equations of hydrodynamics is the averaged pressure with its theoretical values
 absence of nesting of models equations into , , are discussed. The rejection
 each other. For example, the potential flow of pressure averaging makes it possible to
 cannot be obtained from the Euler equations. explain the formation of vortex bundles
 Static pressure, as follows from the kinetic when flowing around bodies in the vicinity
 theory, is a zero-order quantity, and the of the separation point. It is very difficult to
 terms associated with dissipative effects are judge the influence of the moment, limiting
 first-order quantities. As already noted, it ourselves to the results of calculating two-
 does not follow from the Boltzmann dimensional problems. If the tensor is
 equation that the pressure in the Euler asymmetric, then we have three equations
 equation is equal to one third of the sum of for four unknowns. The system is not
 the pressures on the corresponding closed. Therefore, additional assumptions
 coordinate axes. Inaccuracy in determining have to be made. One of these assumptions
 the velocities in the stress tensor does not is the assumption about the symmetry of the
 greatly affect the results. For an stress tensor. These issues are discussed in
 incompressible fluid, the speeds are the this work.
 same. The most important characteristic of
 an elementary volume is its position of the
 center of inertia and the correct projection of
 2. Origin of the angular
 the boundaries of an arbitrary volume on the momentum
 coordinate axes. Due to the arbitrariness of Consider the interaction of three particles
 the directions of motion, the viscous interacting with each other. Since the
 components of the velocity must be taken angular momentum is equal to the force
 into account even in one-dimensional acting on the center of inertia, that is vector
 motion. With any restructuring of the flow,
 multiplied by the radius vector, we can
 the position of the center of inertia changes,
 a moment of force is created, the gradient of calculate the changes in the moment after a
 which is the force itself. Additional force is certain period of time
 added with stresses and change the direction
 of movement and the speed of the
 molecules. The result is a change in the
 equation of state even under equilibrium
 conditions and the formation of fluctuation
 phenomena. The equation of state (Newton's
 law) in classical mechanics ensures the
 symmetry of the stress tensor. Note again
 that the classical equations do not include all
 Fig. 1. Interaction of three particles
 the speeds that need to be considered. The
 vortex part is discarded. The experiment + + 
 does not separate speeds, measuring the total = ,
 + + 

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 ( , 1 , 2 , . . , 1 , 2 , … ) 1 … 1 … 
 1 ( 1 + 1̇ ∆ )+ 2 ( 2 + 2̇ ∆ )+ 3( 3 + 3̇ ∆ ) . To construct an equation directly for the
 +∆ =
 1 + 2 + 3 velocity projection, we multiply the equation
 . by the velocity projection ξ on the
 At the next moment in time, the center of coordinate axis, for example, . Let us have
 inertia will shift. Therefore, an additional different pressure values 1 , 2 , 3 :
 force arises, since the gradient of the = 3 − ,
 moment is a force. Thus, the moment creates = − 3 + 1 ,
 a collective force for any movement of = 2 − 1 .
 particles with different speeds. The effect
 Really,
 works in the formation of fluctuations and
 must be taken into account when calculating
 
 the equation of state. In continuum
 [ ]
 mechanics, the integral conservation laws 1 2 3
 obtained experimentally are written in
 differential form. Consequently, the impossibility of the same
 When passing to differential equations, the values of pressure p. Pressure is a vector like
 Ostrogradsky-Gauss theorem is used for the force. In this case, the Lamb equation [7]
 selected fixed volume, ignoring the possible has the form
 rotation of the elementary volume. The 2
 discarded term represents the velocity + ( ) + × = −
 2
 1
 circulation, its rotor part. Circulation is the
 
 ⃗
 action of the moment. Consequently, the and there is a dependence of density as an
 rotational terms (moment) discarded in the average value over an elementary volume,
 construction of the Navier-Stokes equations then the Bernoulli equation makes sense.
 are formed not only in the case of large The finite volume method is often used to
 gradients, but also due to the additional numerically solve the Euler and Navier-
 circulation of the velocity around the Stokes equations. However, its formulation
 elementary volume [10]. is based on differential equations with a
 The influence of the moment is also symmetric stress tensor. In the mechanics of
 manifested through pressure. There are no liquid and gas, it is not customary to average
 experiments confirming Pascale's law in the calculation results over the calculated
 non-equilibrium cases. The kinetic theory cells. Consequently, since the mesh has
 (modified and classical) also does not follow finite dimensions, in old works, the no
 Pascal's law. symmetry of the tensor is partially taken into
 Consider the classical Boltzmann equation account in numerical calculations. In solid
 (the law of momentum does not hold) mechanics, averaging is offer performed
 1
 when solving problems by the finite element
 ( , ) = ∫ ( , , ) , method. With a large number of steps, this
 = ∫ ( , , ) , gives significant errors. The main argument
 ; = ∫ 2 ( , , ) , for choosing a symmetric tensor is the use of
 = − . the equilibrium equation for the force
 without the influence of the moment.
 ( + , + , + )drdξj =
 
 However, the momentum gradient is force.
 ( , , t ) + ( ) . Consequently, the requirement of
 
 distribution function, - radius vector; - simultaneous fulfillment of the laws of
 point coordinate; is the velocity of a point, balance of forces and moments of forces
 is the molecular weight, and, according to changes the equations.
 the definition of the distribution function , In general case
 the probability of finding the system at
 points ( , ) in the intervals is

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 Here all designations are
 ( + + + ) = 1
 standard, , , forces created by
 
 + +
 + + the moment, , , are external
 
 moments.
 
 ( + + + )= 3. Specific tasks
 
 + In the three-dimensional case, we have six
 2 + + + unknowns and six equations for the no
 
 symmetric tensor, but used only three. In
 the flat version, there is one equation for the
 ( + + + )= moments and two for movement, unknown
 
 + four. Closing can be by the symmetry
 3 + + + condition of the stress tensor. But you can
 
 do it in another way. Some problems with
 symmetric tensor cannot be solved. Let us
 ( + + + 3 ) − consider the problem when the stress
 
 distributions are given, with and without
 ( + + + 2 ) + taking into account the influence of the
 
 − + + = 0 angular momentum. The voltages can be the
 same, they can be different. Considered case
 of elongated plate. General view of
 
 ( + + + 2 ) − equations
 
 ( + + + 1 ) + 
 + 
 =0, 
 + 
 =0,
 
 − + + = 0
 
 ( + )
 
 ( + + + 1 ) − 
 − ( + )
 (
 
 +
 
 +
 
 + 2 ) + 
 + − = 0.
 − + + = 0

 Fig. 2. Elongated plate
 We will assume that there is a point 1 = 1 , = 0 , = 01 is a
 where the stresses coincide. We will begin solution to the system of equations
 with the system of equations represents the and = .
 first two equations, instead of the third
 equation we use the symmetry condition of Thus, an additional assumption that closes
 the stress the systems of equations on the plane in
 classical mechanics is the condition of the
 
 ( + )− ( + )+ stress tensor symmetry. Another option is
 used in work [11], the assumption about the
 − = 0. Then distribution of stress. For a symmetric
 tensor, consider the solution: let =

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 ( − ) + + ( ) or = + Later the theory was developed on the basis
 ( ). Then of Langevin and Fokker-Planck equations
 
 = − ∫0 ̇ = − ̇ + ( ) , 0 = [17-23]. The evolution of a Brownian
 2 particle (fluctuation) is determined by its
 − ∫ (− ̈ + ̇ ) = ̈ 2 - ̇ + ( ). interaction with the environment, which is
 1
 Let us satisfy the boundary condition always collective. In the kinetic
 0 ( = ) = 0. representation, the evolution of a system of
 2
 0 = ̈ ( − ) /2 - ̇ ( − ). Brownian particles is described by a
 = − ̇ ( − ) + ( ) nonlocal equation for the n-particle
 Equation for moment distribution function. The Langevin and
 (− ̇ + ̇ ) − ( ̈ ( − ) - ̇ − ̈ Fokker-Planck equations are obtained from
 ( − ) + ̇ ) = − = ( , ) = 0. the Liouville equation for specially selected
 The solution for the no symmetric tensor can models of integral kernels using
 be found as follows phenomenological conservation laws. In
 ( + ) recent years, the molecular modeling
 
 + 
 =0, 
 + =0,
 method has been widely used.
 The classical Langevin equation for one
 ( + )− (
 particle
 ( + ) 1
 + ) + = 0. = − ѷ V+ ( ) , where ( ) is a
 
 random force.
 At the first iteration, we consider equations A Markov Gaussian process is considered
 only for φ, leaving the coefficients from the with the condition that the average for an
 zero iteration. The first parenthesis vanishes. ensemble of particles < > = 0 . In our
 We represent the second parenthesis as version, this condition is fulfilled by virtue
 ( + ) + of the fulfillment of the theorem on the
 + = + conservation of the moment in a closed
 
 = 0 + volume. For equilibrium condition, this is
 . Then
 true. Langevin equation taking into account
 1
 1 − ∫ the influence of the angular momentum is
 + = 0. = 1 = ѷ 1 
 = − V + , where ѷ -
 1 
 − | 1 = , = coefficient of friction of the selected
 
 = ,
 2
 = 2 + ( ). particle, is the mass of the particle, is
 the moment of force acting on the particle, V
 Now you can find the corrected value of . – particle velocity.
 In this version, the tensor will be symmetric.
 The classical Langevin equation for one
 The tensor will not be symmetric if
 particle
 , = 0, ( ) = ( − ). In this 1
 case, = 0, = 0. We get the results 
 = − ѷ V+ ( ) , where ( ) is a
 of work random force.
 E. A. Bulanova [16]. If there are A more interesting situation arises in
 disturbances on both sides, then stitching is Coulomb plasma. The movement of the
 required at some point 1 , dependence on center of inertia entails a new distribution of
 can be arbitrary. The influence of the charges. The result is a change in the
 moment creates a collective effect. Let us strength of the electric field. In this case, the
 consider its effect in a rarefied gas. main force will be strength associated with
 Fluctuations are considered as random the movement of charges.
 deviations from the mean. This effect was
 first explained by A.Einstein. It consisted in 4. Conclusion
 the fact that the diffusion force should be Modern models of continuum mechanics do
 equal to the viscous Stokes drag force not give answers to many questions when
 np F s ∂np describing
 = DE
 cs ∂x

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 the processes of turbulent flows, flow [7]. N.E. Kochin, I.A. Kibel, N.V. Rose,
 separation, in matters of destruction of Theoretical Hydromechanics, Moscow: Fiz-
 solids, etc. It is proposed to extend the Mat Literature, 1963, 583
 suggested model to the case of large [8].A.A. Ilyushin, Asymmetry of strain and
 gradients and bring the model in line with stress tensors in the mechanics of a
 the main provisions of classical mechanics continuous medium. WEST. Moscow.
 and mathematics. The necessity of using no
 University. Ser. 1. Mathematics-
 symmetric stress tensor and a vector value
 of pressure in the conservation laws of mechanics.1996. pp. 6-14
 continuum mechanics is substantiated. [9]. V.I. Smirnov, Higher mathematics
 Particular examples from the theory of the course. T.2. M .: Nauka, 1974 , 655.
 boundary layer, the theory of elasticity and [10]. E.V Prozorova . The Effect of Angular
 kinetic theory were given in previous works. Momentum and Ostrogradsky-Gauss
 We suggest an algorithm for taking into Theorem in the Equations of Mechanics.
 account the no symmetric stress tensor by Wseas Transaction on fluid DOI:
 the example of solving the problem of 10.37394/232013.2020.15.2
 calculating a beam with a given distribution [11]. L.G. Loytsyansky, Mechanics of
 of one of the stresses. The ambiguity of liquid and gas. M .: Nauka. 1970, 904.
 using a symmetric stress tensor for closing [12]. Ouadie Koubaiti, Ahmed Elkhalfi,
 and the possibility of using the results with a Jaouad El-Mekkaoui, Nikos Mastorakis,
 symmetric tensor, supplementing the Solving the Problem of Constraints Due to
 solution with iteration, are shown. The Dirichlet Boundary Conditions in the
 proposed model can be applied to a wider Context of the Mini Element Method,
 range of problems in continuum mechanics International Journal of Mechanics, pp. 12-
 than the classical theory. 21, Volume 14, 2020.
 [13]. Tarik Chakkour, Fayssal Benkhaldoun,
 References Slurry Pipeline for Fluid Transients in
 Pressurized Conduits, International Journal
 [1]. Yu.N. Shevchenko, I. V. Prokhorenko, of Mechanics, pp. 1-11, Volume 14, 2020.
 The theory of elastic-plastic shells under [14]. Yu.G. Stepanov, Theoretical
 non-isothermal loading processes. Kiev. hydrodynamics, theory of elasticity, model
 Naukova Dumka. 1981, 296. of a continuous medium with moment
 [2]. R.A. Kayumov, Fundamentals of the stresses. Proceedings of the Krylov State
 theory of elasticity and elements of the Scientific Center. 2016. Issue 93 (377).
 theory of plates and shells: Textbook / R.A.
 pp.163-174
 Kayumov - Kazan: Kazan Publishing House.
 state architect-builds un-that. 2016, 111 [15]. N.G. Kolbasnikov, Theory of metal
 [3]. P.F. Nedorezov, N.M. Sirotkina, pressure treatment. Deformation resistance
 Numerical methods for studying steady-state and ductility. St. Petersburg: St. Petersburg
 vibrations of viscoelastic rectangular plates State University Publishing House, 2000,
 and circular cylindrical shells. Saratov. 314.
 Saratov University Press. [16]. E.A. Bulanov, Moment stresses in the
 1997, 72 mechanics of a solid, bulk and liquid body.
 [4]. A.I. Koshelev, M.A. Narbut, Lectures M.: University book. 2012, 140
 on the mechanics of deformable solids. SPb: [17]. R. Balescu. Equilibrium and
 Publishing House of St. Petersburg nonequilibrium statistic mechanics. A
 University. 2003, 276. Wiley-Intersciences Publication John
 [5].G.Z. Sharafufutdinov, Some plane Willey and Sons. New-Yourk-
 problems of the theory of elasticity. London…1975, 256
 Moscow: Scientific World. 2014, [18]. P. Resibois, M.De. Lener. Classical
 [6]. A.M. Katz, Elasticity theory. St. kinetic theory of fluids. John Wiley and
 Petersburg. 2002, 208 Sons. New-York, London,… 1977, 423

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 [19]. N.G. Van Kampen, Stochastic [22]. V. Ya. Rudyak, Statistical
 processes in physics and chemistry. North- aerohydromechanics of homogeneous and
 Holland. 1984, 376 heterogeneous media. Vol.2 Novosibirsk:
 [20]. Physics of Simple Liquids. Edited by NGASU. 2005, 468
 H.N.V. Temperley, I.S. Rowlinson, G.S. [23]. Yu.V. Klimontovich, Statistical theory
 Rushbrooke. Amsterdam, 1068, 308. of open systems. Vol.2, M.: Janus. K 2019,
 [21]. L. Boltzmann. Selected Works. 438
 Moscow: Nauka, 1984, 146
 17 D.N. Zubarev, Nonequilibrium Statistical Creative Commons Attribution License 4.0
 Thermodynamics, Moscow: Nauka, 1971, (Attribution 4.0 International, CC BY 4.0)
 416
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