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Introduction to the Theory of Gyroscopic Fields: the unification of solid and fluid mechanics through an extension of Fluid Mechanics
In this book, we aim to obtain and develop an alternative formulation to Euler’s equations for describing Rigid Body Dynamics and, at the same time, we postulate the unification of solid and fluid dynamics.
For this unification, it is necessary to review the Navier-Stokes (NS) equations adding a new term corresponding to the variation of the translational velocity v0 respect to time. We understand as translational velocity of the fluid particles that one which is common to all the particles of the fluid when they are translated together in the same container. Imagine, for example, a truck with velocity v0 and acceleration a0 = მv0/მt that contains a glass box filled with water which flow we want to simulate using the framework of theory of fields in Continuum Media. We know that in this framework, the material or convective derivative is defined respect to an observer at rest. Another example would be a laboratory at rest where a swimming-pool is situated on a platform which is moving with velocity v0.
Then, we consider as an axiom of the theory that the translational velocity v0 is not contained in the velocity of the flow particles v that appears in the Navier-Stokes equations. Thus, a0 must be added by hand to the convective derivative Dv/Dt. For solids, all the terms concerning the velocity v can be ignored, recovering the second law of Newtonian mechanics after spatial integration of our extended NS equations over a control volume that coincides with the body: m a0= ∑ F, where m is the mass of the body and F are the applied forces.
Thus, we have added a new vector field and, therefore, we need more equations (coupled with v0) to solve the system. This is the first step for describing an extension of Fluid Mechanics. This changes everything and it could explain why the NS equations do not have an exact solution and make it possible now to find the exact one if we were able to resolve the extended system, fixing one of the millennium problems. These supplementary equations correspond to the ones that account for rotations, defining an independent system of PDEs as in Classical Mechanics, where the equations that account for the translation of the body are independent from that accounting for rotations which are described in Rigid Body Dynamics.
It is important to remark that in our extended model of Fluid Mechanics described by the extended NS equations, there appear two different velocity fields: the usual velocity field v of the flow, and a new velocity field representing the translational velocity v0. So, the question is: why should there be two differentiated velocities in the same equation rather than a unique velocity? The only logical explanation is that the two vector fields are not applied at exactly the same point of the space in the interior of the fluid particle, so they cannot be added algebraically using the vector sum whereas, due to the superposition principle, the accelerations are applied at the same point of space, so it can be summed. Then, with this hypothesis, it makes sense to define different velocity fields in our theory.
Then, we have two different rotational motions:
On one hand, the usual vorticity of Fluid Mechanics which corresponds to the curl of the convective velocity v, where we postulate that vorticity is generated by the action of internal molecular forces and gravity.
On the other hand, the classical rotational motion composed of intrinsic rotations and precession, where they are generated only by torques (for example, the torque exerted by a spoon in the cofee when mixing sugar with the fluid). In our model, the latter motions must be represented by two independent vector fields: w and Ω, completely different from vorticity, and their dynamics are described in two independent systems of differential equations. As we know, the dynamics of the vorticity field is described using the Navier-Stokes equations:
ρ მv0 / მt + ρ [მv / მt + (v. grad) v] = – grad (p) + ρ g + μ Δ v, (1)
where we have added a new term, ρ მv0 / მt corresponding to the translational acceleration ρ მv0 / მt.
As we know, the dynamics of the vorticity field is described by using (1) after applying the curl operator.
Regarding solids, the orthodox Rigid Body Dynamics theory is based in a mathematical model that implicitly allows for the presence of an inadmissible “action at a distance”, due to the fact that the angular momentum L appearing in the equations is a vector that is supposed to be common to all the constituent particles of the body.
Then, according to the classical description, when a torque is exerted at a point of the body from rest, all these particles start to rotate with the same L, no matter how far away was each particle from the point of the body where the torque was applied.
This is obviously an incorrect description of the motion: there should be a transitory description to account for the behavior of the particles when we study the problem from rest to a stable and permanent regime of rotation with L as a good approximation for the angular momentum of all the particles.
The best example of a model like that can be found in the theory of Electromagnetism with the Maxwell equations as the mathematical model where, taking the curl to one of the vector equations and substituting in the other, gives automatically a wave equation for the propagation of a certain field.
Then, the idea is to express the Rigid Body Dynamics equation: dL/dt = ∑ r x F
in a relativistic framework, where “action at a distance” is not possible to occur. A framework similar to that of Maxwell equation: The Minkowski space-time.
The presently accepted model of Rigid Body Dynamics presents two separated mathematical blocks:
On one hand, a differential equation dL/dt = ∑ r x F which is equivalent to the conservation of the angular momentum of the body in a Euclidean space. On the other hand, a kinematic definition of the velocity of a constituent particle of the body: v = w x r.
We postulate a compact vector structure, where the above equalities could be derived from. And this is exactly what we get when you use the structure of Maxwell’s equations for continuum media as an analogy necessary to describe the usual equations for rotation. But now we obtain some new terms that are absent in the orthodox formulation. These extra-terms account for the transient regime of the motion, from rest to rotational motion.
One of them represent a field of internal torques or dynamic moments that propagate from the point where the external torque was exerted to the rest of the body particles, thus explaining the angular momentum propagating wave obtained from the analogy with electromagnetism. To understand this analogy more vividly, we must remind that, whereas Maxwell’s equations are used for a model that represents the dynamics of a flow of electrons, our equations of gyroscopic fields represent the dynamics of a flow of protons which would carry the rotational velocity w x r and a different set of Maxwell-like equations to represent the dynamics of a flow of neutrons that would carry the precesional velocity Ω x r.
Then, we have defined two independent vectors: one for the angular velocity of intrinsic rotation w, and the other for the angular velocity of precession Ω.
As we know, the Euler equations for rotation which are obtained when identifying the three components of angular velocity which the variation respect to time of the Euler angles that represent the dynamical variables of the rotational motion of an object in the Euclidean space with symmetry group SO(3).
We claim that this representation, being mathematically correct, does not describe the real motion because it is not compatible with the Minkowski framework where action at a distance is not possible and also it is not invariant under a Lorentz transformation with symmetry group SO(3, 1). In other words, with the orthodox approach based on a unique vector w where the three components of w correspond to the derivatives respect to time of the intrinsic rotation angle ψ, the precession angle φ and the nutation angle θ respectively, is not possible to get an element of metric distance ds2 as the sum of the squared angular velocities plus the squared time.
However, postulating two independent (2-dimensional) vectors, one corresponding to the angular velocity of intrinsic rotation and the other corresponding to the angular velocity of precession, it is possible to get two differenciated Minkowski-like elements of metric distance ds2.
The Maxwell-like equations for intrinsic rotational motion are:
div GR = ρ / ε0R (2) curl GR = J2R + (vR2/c’2) მLR / მt (3)
div LR = 0 (4) curl LR = J1R – (1/vR2) მGR / მt (5)
After developing these expressions, we recover the classical equations of Rigid Body Dynamics:
dLR/dt = ∑ r x F
In this book, we aim to obtain and develop an alternative formulation to Euler’s equations for describing Rigid Body Dynamics and, at the same time, we postulate the unification of solid and fluid dynamics.
For this unification, it is necessary to review the Navier-Stokes (NS) equations adding a new term corresponding to the variation of the translational velocity v0 respect to time. We understand as translational velocity of the fluid particles that one which is common to all the particles of the fluid when they are translated together in the same container. Imagine, for example, a truck with velocity v0 and acceleration a0 = მv0/მt that contains a glass box filled with water which flow we want to simulate using the framework of theory of fields in Continuum Media. We know that in this framework, the material or convective derivative is defined respect to an observer at rest. Another example would be a laboratory at rest where a swimming-pool is situated on a platform which is moving with velocity v0.
Then, we consider as an axiom of the theory that the translational velocity v0 is not contained in the velocity of the flow particles v that appears in the Navier-Stokes equations. Thus, a0 must be added by hand to the convective derivative Dv/Dt. For solids, all the terms concerning the velocity v can be ignored, recovering the second law of Newtonian mechanics after spatial integration of our extended NS equations over a control volume that coincides with the body: m a0= ∑ F, where m is the mass of the body and F are the applied forces.
Thus, we have added a new vector field and, therefore, we need more equations (coupled with v0) to solve the system. This is the first step for describing an extension of Fluid Mechanics. This changes everything and it could explain why the NS equations do not have an exact solution and make it possible now to find the exact one if we were able to resolve the extended system, fixing one of the millennium problems. These supplementary equations correspond to the ones that account for rotations, defining an independent system of PDEs as in Classical Mechanics, where the equations that account for the translation of the body are independent from that accounting for rotations which are described in Rigid Body Dynamics.
It is important to remark that in our extended model of Fluid Mechanics described by the extended NS equations, there appear two different velocity fields: the usual velocity field v of the flow, and a new velocity field representing the translational velocity v0. So, the question is: why should there be two differentiated velocities in the same equation rather than a unique velocity? The only logical explanation is that the two vector fields are not applied at exactly the same point of the space in the interior of the fluid particle, so they cannot be added algebraically using the vector sum whereas, due to the superposition principle, the accelerations are applied at the same point of space, so it can be summed. Then, with this hypothesis, it makes sense to define different velocity fields in our theory.
Then, we have two different rotational motions:
On one hand, the usual vorticity of Fluid Mechanics which corresponds to the curl of the convective velocity v, where we postulate that vorticity is generated by the action of internal molecular forces and gravity.
On the other hand, the classical rotational motion composed of intrinsic rotations and precession, where they are generated only by torques (for example, the torque exerted by a spoon in the cofee when mixing sugar with the fluid). In our model, the latter motions must be represented by two independent vector fields: w and Ω, completely different from vorticity, and their dynamics are described in two independent systems of differential equations. As we know, the dynamics of the vorticity field is described using the Navier-Stokes equations:
ρ მv0 / მt + ρ [მv / მt + (v. grad) v] = – grad (p) + ρ g + μ Δ v, (1)
where we have added a new term, ρ მv0 / მt corresponding to the translational acceleration ρ მv0 / მt.
As we know, the dynamics of the vorticity field is described by using (1) after applying the curl operator.
Regarding solids, the orthodox Rigid Body Dynamics theory is based in a mathematical model that implicitly allows for the presence of an inadmissible “action at a distance”, due to the fact that the angular momentum L appearing in the equations is a vector that is supposed to be common to all the constituent particles of the body.
Then, according to the classical description, when a torque is exerted at a point of the body from rest, all these particles start to rotate with the same L, no matter how far away was each particle from the point of the body where the torque was applied.
This is obviously an incorrect description of the motion: there should be a transitory description to account for the behavior of the particles when we study the problem from rest to a stable and permanent regime of rotation with L as a good approximation for the angular momentum of all the particles.
The best example of a model like that can be found in the theory of Electromagnetism with the Maxwell equations as the mathematical model where, taking the curl to one of the vector equations and substituting in the other, gives automatically a wave equation for the propagation of a certain field.
Then, the idea is to express the Rigid Body Dynamics equation: dL/dt = ∑ r x F
in a relativistic framework, where “action at a distance” is not possible to occur. A framework similar to that of Maxwell equation: The Minkowski space-time.
The presently accepted model of Rigid Body Dynamics presents two separated mathematical blocks:
On one hand, a differential equation dL/dt = ∑ r x F which is equivalent to the conservation of the angular momentum of the body in a Euclidean space. On the other hand, a kinematic definition of the velocity of a constituent particle of the body: v = w x r.
We postulate a compact vector structure, where the above equalities could be derived from. And this is exactly what we get when you use the structure of Maxwell’s equations for continuum media as an analogy necessary to describe the usual equations for rotation. But now we obtain some new terms that are absent in the orthodox formulation. These extra-terms account for the transient regime of the motion, from rest to rotational motion.
One of them represent a field of internal torques or dynamic moments that propagate from the point where the external torque was exerted to the rest of the body particles, thus explaining the angular momentum propagating wave obtained from the analogy with electromagnetism. To understand this analogy more vividly, we must remind that, whereas Maxwell’s equations are used for a model that represents the dynamics of a flow of electrons, our equations of gyroscopic fields represent the dynamics of a flow of protons which would carry the rotational velocity w x r and a different set of Maxwell-like equations to represent the dynamics of a flow of neutrons that would carry the precesional velocity Ω x r.
Then, we have defined two independent vectors: one for the angular velocity of intrinsic rotation w, and the other for the angular velocity of precession Ω.
As we know, the Euler equations for rotation which are obtained when identifying the three components of angular velocity which the variation respect to time of the Euler angles that represent the dynamical variables of the rotational motion of an object in the Euclidean space with symmetry group SO(3).
We claim that this representation, being mathematically correct, does not describe the real motion because it is not compatible with the Minkowski framework where action at a distance is not possible and also it is not invariant under a Lorentz transformation with symmetry group SO(3, 1). In other words, with the orthodox approach based on a unique vector w where the three components of w correspond to the derivatives respect to time of the intrinsic rotation angle ψ, the precession angle φ and the nutation angle θ respectively, is not possible to get an element of metric distance ds2 as the sum of the squared angular velocities plus the squared time.
However, postulating two independent (2-dimensional) vectors, one corresponding to the angular velocity of intrinsic rotation and the other corresponding to the angular velocity of precession, it is possible to get two differenciated Minkowski-like elements of metric distance ds2 in terms of the Euler angles:
ds2 = dr2 + dψ2 + dθ2 – c’2 dt2
and
ds’2 = dr2 + dϕ 2 + dα2 – c’2 dt2
The Maxwell-like equations for intrinsic rotational motion are:
div GR = ρ / ε0R (2) curl GR = J2R + (vR2/c’2) მLR / მt (3)
div LR = 0 (4) curl LR = J1R – (1/vR2) მGR / მt (5)
After developing these expressions, we recover the classical equations of Rigid Body Dynamics only for intrinsic rotation:
dLR/dt = ∑ r x FR
vR = w x r
plus some additional terms which are derivates respect to time and, thus, they are zero at the steady state.
The Maxwell-like equations for the precession motion are:
div GP = ρ / ε0P (6) curl GP = J2P + (vR2/c’2) მLP / მt (7)
div LP = 0 (8) curl LP = J1P – (1/vR2) მGP / მt (9)
As in the previous case, after expanding these expressions, we recover the classical equations of Rigid Body Dynamics only for precession:
dLP/dt = ∑ rcp x FP
vP = v0 + Ω x rcp
plus some additional terms.
Thus, adding the two obtained equations for the variation of angular momentum, we get;
dL/dt =dLR/dt + dLP/dt = ∑ r x FR + ∑ rcp x FP
If the centre of gyration of intrinsic rotation and the centre of gyration of presession are the same, then r = rcp and we recover the classical equation of Rigid Body Dynamics:
dL/dt = r x FT
where is FT, the total force, is the sum of external and internal forces exerted on the body. Apparently, we have recovered the classical approach, but this is only the consequence of ignoring the stationary terms. This would mean that the classical approach is a good approximation to the problem. but it is not the final description.
These equations can explain the turbulence phenomena appearing at the exit stream of a wing, as studied in Aeronautics. In our approach, the eddies generated are a consequence of internal torques created by the Coriolis force. The torque is:
ρ r x (v0 x Ω),
where v0 is the translational velocity of the flow and Ω is the angular velocity generated near the wing surface.
Expanding the previous expression:
ρ r x (v0 x Ω) = ρ((r . v0) Ω – (r . Ω) v0)
But r . Ω is always zero and the expression of the field of eddies corresponding to the turbulent regime is:
ρ((r . v0) Ω
In the laminar regime, there are not eddies due to r . v0 = 0, because r is transversal to the translational velocity v0.
So, our theory of gyroscopic fields gives an analytical description of turbulence without needing constitutive equations.
(Draft #5, Arthur Palenz, 2023, the nineteenth of january)
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