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We can then write it in the symbolic operational form: |
There are mass forces acting on the inside of the control volume. We can write them using the acceleration field formula_30 (e.g. gravitational acceleration): |
Let us calculate momentum of the cube: |
Because we assume that tested mass (cube) formula_33 is constant in time, so |
Divide both sides by formula_38, and because formula_39 we get: |
Applying Newton's second law (th component) to a control volume in the continuum being modeled gives: |
Then, based on the Reynolds transport theorem and using material derivative notation, one can write |
where represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes . |
The Cauchy momentum equation can also be put in the following form: |
where is the momentum density at the point considered in the continuum (for which the continuity equation holds), is the flux associated to the momentum density, and contains all of the body forces per unit volume. is the dyad of the velocity. |
Here and have same number of dimensions as the flow speed and the body acceleration, while , being a tensor, has . |
In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations. |
A significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. |
Regardless of what kind of continuum is being dealt with, convective acceleration is a nonlinear effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping flow (also called Stokes flow). Convective acceleration is represented by the nonlinear quantity , which may be interpreted either as or as , with the tensor derivative of the velocity vector . Both interpretations give the same result. |
The convection term formula_44 can be written as , where is the advection operator. This representation can be contrasted to the one in terms of the tensor derivative. |
The tensor derivative is the component-by-component derivative of the velocity vector, defined by , so that |
where the Feynman subscript notation is used, which means the subscripted gradient operates only on the factor . |
Lamb in his famous classical book Hydrodynamics (1895), used this identity to change the convective term of the flow velocity in rotational form, i.e. without a tensor derivative: |
where the vector formula_48 is called the Lamb vector. The Cauchy momentum equation becomes: |
In fact, in case of an external conservative field, by defining its potential : |
And by projecting the momentum equation on the flow direction, i.e. along a "streamline", the cross product disappears due to a vector calculus identity of the triple scalar product: |
If the stress tensor is isotropic, then only the pressure enters: formula_55 (where is the identity tensor), and the Euler momentum equation in the steady incompressible case becomes: |
that is, "the mass conservation for a steady incompressible flow states that the density along a streamline is constant". This leads to a considerable simplification of the Euler momentum equation: |
in fact, the above equation can be simply written as: |
That is, "the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant". |
The Lamb form is also useful in irrotational flow, where the curl of the velocity (called vorticity) is equal to zero. In that case, the convection term in formula_44 reduces to |
The effect of stress in the continuum flow is represented by the and terms; these are gradients of surface forces, analogous to stresses in a solid. Here is the pressure gradient and arises from the isotropic part of the Cauchy stress tensor. This part is given by the normal stresses that occur in almost all situations. The anisotropic part of the stress tensor gives rise to , which usually describes viscous forces; for incompressible flow, this is only a shear effect. Thus, is the deviatoric stress tensor, and the stress tensor is equal to: |
where is the identity matrix in the space considered and the shear tensor. |
All non-relativistic momentum conservation equations, such as the Navier–Stokes equation, can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a constitutive relation. By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. By assuming inviscid flow, the Navier–Stokes equations can further simplify to the Euler equations. |
The divergence of the stress tensor can be written as |
The effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure. |
As written in the Cauchy momentum equation, the stress terms and are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the flow motion. For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density. |
The vector field represents body forces per unit mass. Typically, these consist of only gravity acceleration, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "inertial accelerations" associated with rotating coordinates may arise. |
Often, these forces may be represented as the gradient of some scalar quantity , with in which case they are called conservative forces. Gravity in the direction, for example, is the gradient of . Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force . The pressure and force terms on the right-hand side of the Navier–Stokes equation become |
It is also possible to include external influences into the stress term formula_66 rather than the body force term. This may even include antisymmetric stresses (inputs of angular momentum), in contrast to the usually symmetrical internal contributions to the stress tensor. |
In order to make the equations dimensionless, a characteristic length and a characteristic velocity need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained: |
Substitution of these inverted relations in the Euler momentum equations yields: |
and by dividing for the first coefficient: |
and the coefficient of skin-friction or the one usually referred as 'drag' co-efficient in the field of aerodynamics: |
by passing respectively to the conservative variables, i.e. the momentum density and the force density: |
the equations are finally expressed (now omitting the indexes): |
2 \nabla \cdot \boldsymbol \tau + \frac 1 {\mathrm{Fr}} \mathbf g |
Cauchy equations in the Froude limit (corresponding to negligible external field) are named free Cauchy equations: |
and can be eventually conservation equations. The limit of high Froude numbers (low external field) is thus notable for such equations and is studied with perturbation theory. |
Finally in convective form the equations are: |
For asymmetric stress tensors, equations in general take the following forms: |
Below, we write the main equation in pressure-tau form assuming that the stress tensor is symmetrical (formula_75): |
Plastic limit theorems in continuum mechanics provide two bounds that can be used to determine whether material failure is possible by means of plastic deformation for a given external loading scenario. According to the theorems, to find the range within which the true solution must lie, it is necessary to find both a stress field that balances the external forces and a velocity field or flow pattern that corresponds to those stresses. If the upper and lower bounds provided by the velocity field and stress field coincide, the exact value of the collapse load is determined. |
The two plastic limit theorems apply to any elastic-perfectly plastic body or assemblage of bodies. |
If an equilibrium distribution of stress can be found which balances the applied load and nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure. |
The body (or bodies) will collapse if there is any compatible pattern of plastic deformation for which the rate of work done by the external loads exceeds the internal plastic dissipation. |
In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as |
where "n"q is the number of quarks, and "n" is the number of antiquarks. Baryons (three quarks) have a baryon number of +1, mesons (one quark, one antiquark) have a baryon number of 0, and antibaryons (three antiquarks) have a baryon number of −1. Exotic hadrons like pentaquarks (four quarks, one antiquark) and tetraquarks (two quarks, two antiquarks) are also classified as baryons and mesons depending on their baryon number. |
Quarks carry not only electric charge, but also charges such as color charge and weak isospin. Because of a phenomenon known as "color confinement", a hadron cannot have a net color charge; that is, the total color charge of a particle has to be zero ("white"). A quark can have one of three "colors", dubbed "red", "green", and "blue"; while an antiquark may be either "anti-red", "anti-green" or "anti-blue". |
For normal hadrons, a white color can thus be achieved in one of three ways: |
The baryon number was defined long before the quark model was established, so rather than changing the definitions, particle physicists simply gave quarks one third the baryon number. Nowadays it might be more accurate to speak of the conservation of quark number. |
In theory, exotic hadrons can be formed by adding pairs of quarks and antiquarks, provided that each pair has a matching color/anticolor. For example, a pentaquark (four quarks, one antiquark) could have the individual quark colors: red, green, blue, blue, and antiblue. In 2015, the LHCb collaboration at CERN reported results consistent with pentaquark states in the decay of bottom Lambda baryons (). |
Particles without any quarks have a baryon number of zero. Such particles are |
The baryon number is conserved in all the interactions of the Standard Model, with one possible exception. 'Conserved' means that the sum of the baryon number of all incoming particles is the same as the sum of the baryon numbers of all particles resulting from the reaction. The one exception is the hypothesized Adler–Bell–Jackiw anomaly in electroweak interactions; however, sphalerons are not all that common and could occur at high energy and temperature levels and can explain electroweak baryogenesis and leptogenesis. Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa). No experimental evidence of sphalerons has yet been observed. |
The hypothetical concepts of grand unified theory (GUT) models and supersymmetry allows for the changing of a baryon into leptons and antiquarks (see "B" − "L"), thus violating the conservation of both baryon and lepton numbers. Proton decay would be an example of such a process taking place, but has never been observed. |
The conservation of baryon number is not consistent with the physics of black hole evaporation via Hawking radiation. It is expected in general that quantum gravitational effects violate the conservation of all charges associated to global symmetries. The violation of conservation of baryon number led John Archibald Wheeler to speculate on a principle of mutability for all physical properties. |
In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry) while its spatial coordinates are inverted ("mirror" or P symmetry). The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch. |
It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present universe, and in the study of weak interactions in particle physics. |
Until the 1950s, parity conservation was believed to be one of the fundamental geometric conservation laws (along with conservation of energy and conservation of momentum). After the discovery of parity violation in 1956, CP-symmetry was proposed to restore order. However, while the strong interaction and electromagnetic interaction seem to be invariant under the combined CP transformation operation, further experiments showed that this symmetry is slightly violated during certain types of weak decay. |
Only a weaker version of the symmetry could be preserved by physical phenomena, which was CPT symmetry. Besides C and P, there is a third operation, time reversal T, which corresponds to reversal of motion. Invariance under time reversal implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one and occurs at the same rate forwards and backwards. |
The combination of CPT is thought to constitute an exact symmetry of all types of fundamental interactions. Because of the CPT symmetry, a violation of the CP-symmetry is equivalent to a violation of the T symmetry. CP violation implied nonconservation of T, provided that the long-held CPT theorem was valid. In this theorem, regarded as one of the basic principles of quantum field theory, charge conjugation, parity, and time reversal are applied together. Direct observation of the time reversal symmetry violation without any assumption of CPT theorem was done in 1998 by two groups, CPLEAR and KTeV collaborations, at CERN and Fermilab, respectively. Already in 1970 Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell-Steinberger unitarity relation. |
The idea behind parity symmetry was that the equations of particle physics are invariant under mirror inversion. This led to the prediction that the mirror image of a reaction (such as a chemical reaction or radioactive decay) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang revealed that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. |
The first test based on beta decay of cobalt-60 nuclei was carried out in 1956 by a group led by Chien-Shiung Wu, and demonstrated conclusively that weak interactions violate the P symmetry or, as the analogy goes, some reactions did not occur as often as their mirror image. However, parity symmetry still appears to be valid for all reactions involving electromagnetism and strong interactions. |
Overall, the symmetry of a quantum mechanical system can be restored if another approximate symmetry "S" can be found such that the combined symmetry "PS" remains unbroken. This rather subtle point about the structure of Hilbert space was realized shortly after the discovery of "P" violation, and it was proposed that charge conjugation, "C", which transforms a particle into its antiparticle, was the suitable symmetry to restore order. |
In 1956 Reinhard Oehme in a letter to Yang and shortly after, Ioffe, Okun and Rudik showed that the parity violation meant that charge conjugation invariance must also be violated in weak decays. |
Charge violation was confirmed in the Wu experiment and in experiments performed by Valentine Telegdi and Jerome Friedman and Garwin and Lederman who observed parity non-conservation in pion and muon decay and found that C is also violated. Charge violation was more explicitly shown in experiments done by John Riley Holt at the University of Liverpool. |
Oehme then wrote up a paper with Lee and Yang in which they discussed the interplay of non-invariance under P, C and T. The same result was also independently obtained by B.L. Ioffe, Okun and A.P. Rudik. Both groups also discussed possible CP violations in neutral kaon decays. |
Lev Landau proposed in 1957 "CP-symmetry", often called just "CP" as the true symmetry between matter and antimatter. "CP-symmetry" is the product of two transformations: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the mirror image of the original proces and so the combined CP symmetry would be conserved in the weak interaction. |
In 1962, a group of experimentalists at Dubna, on Okun's insistence, unsuccessfully searched for CP-violating kaon decay. |
In 1964, James Cronin, Val Fitch and coworkers provided clear evidence from kaon decay that CP-symmetry could be broken. This work won them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity, but also their combination. The discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today. The lack of an exact CP-symmetry, but also the fact that it is so close to a symmetry, introduced a great puzzle. |
The kind of CP violation discovered in 1964 was linked to the fact that neutral kaons can transform into their antiparticles (in which each quark is replaced with the other's antiquark) and vice versa, but such transformation does not occur with exactly the same probability in both directions; this is called "indirect" CP violation. |
Despite many searches, no other manifestation of CP violation was discovered until the 1990s, when the NA31 experiment at CERN suggested evidence for CP violation in the decay process of the very same neutral kaons ("direct" CP violation). The observation was somewhat controversial, and final proof for it came in 1999 from the KTeV experiment at Fermilab and the NA48 experiment at CERN. |
Starting in 2001, a new generation of experiments, including the BaBar experiment at the Stanford Linear Accelerator Center (SLAC) and the Belle Experiment at the High Energy Accelerator Research Organisation (KEK) in Japan, observed direct CP violation in a different system, namely in decays of the B mesons. A large number of CP violation processes in B meson decays have now been discovered. Before these "B-factory" experiments, there was a logical possibility that all CP violation was confined to kaon physics. However, this raised the question of why CP violation did "not" extend to the strong force, and furthermore, why this was not predicted by the unextended Standard Model, despite the model's accuracy for "normal" phenomena. |
In 2011, a hint of CP violation in decays of neutral D mesons was reported by the LHCb experiment at CERN using 0.6 fb−1 of Run 1 data. However, the same measurement using the full 3.0 fb−1 Run 1 sample was consistent with CP symmetry. |
In 2013 LHCb announced discovery of CP violation in strange B meson decays. |
In March 2019, LHCb announced discovery of CP violation in charmed formula_1 decays with a deviation from zero of 5.3 standard deviations. |
In 2020, the T2K Collaboration reported some indications of CP violation in leptons for the first time. |
In this experiment, beams of muon neutrinos () and muon antineutrinos () were alternately produced by an accelerator. By the time they got to the detector, a significantly higher proportion of electron neutrinos () were detected from the beams, than electron antineutrinos () were from the beams. The results were not yet precise enough to determine the size of the CP violation, relative to that seen in quarks. In addition, another similar experiment, NOvA sees no evidence of CP violation in neutrino oscillations and is in slight tension with T2K. |
"Direct" CP violation is allowed in the Standard Model if a complex phase appears in the CKM matrix describing quark mixing, or the PMNS matrix describing neutrino mixing. A necessary condition for the appearance of the complex phase is the presence of at least three generations of quarks. If fewer generations are present, the complex phase parameter can be absorbed into redefinitions of the quark fields. A popular rephasing invariant whose vanishing signals absence of CP violation and occurs in most CP violating amplitudes is the "Jarlskog invariant", |
The reason why such a complex phase causes CP violation is not immediately obvious, but can be seen as follows. Consider any given particles (or sets of particles) formula_3 and formula_4, and their antiparticles formula_5 and formula_6. Now consider the processes formula_7 and the corresponding antiparticle process formula_8, and denote their amplitudes formula_9 and formula_10 respectively. Before CP violation, these terms must be the "same" complex number. We can separate the magnitude and phase by writing formula_11. If a phase term is introduced from (e.g.) the CKM matrix, denote it formula_12. Note that formula_10 contains the conjugate matrix to formula_9, so it picks up a phase term formula_15. |
Physically measurable reaction rates are proportional to formula_18, thus so far nothing is different. However, consider that there are "two different routes": formula_19 and formula_20 or equivalently, two unrelated intermediate states: formula_21 and formula_22. Now we have: |
Thus, we see that a complex phase gives rise to processes that proceed at different rates for particles and antiparticles, and CP is violated. |
From the theoretical end, the CKM matrix is defined as = . , where and are unitary transformation matrices which diagonalize the fermion mass matrices and , respectively. |
Thus, there are two necessary conditions for getting a complex CKM matrix: |
There is no experimentally known violation of the CP-symmetry in quantum chromodynamics. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as the strong CP problem. |
QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory in which the gauge fields couple to chiral currents constructed from the fermionic fields, the gluons couple to vector currents. Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create the electric dipole moment of the neutron which would be comparable to 10−18 e·m while the experimental upper bound is roughly one trillionth that size. |
This is a problem because at the end, there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry. |
For a nonzero choice of the θ angle and the chiral phase of the quark mass θ′ one expects the CP-symmetry to be violated. One usually assumes that the chiral quark mass phase can be converted to a contribution to the total effective formula_27 angle, but it remains to be explained why this angle is extremely small instead of being of order one; the particular value of the θ angle that must be very close to zero (in this case) is an example of a fine-tuning problem in physics, and is typically solved by physics beyond the Standard Model. |
There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory, involving new scalar particles called axions. A newer, more radical approach not requiring the axion is a theory involving two time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev. |
The universe is made chiefly of matter, rather than consisting of equal parts of matter and antimatter as might be expected. It can be demonstrated that, to create an imbalance in matter and antimatter from an initial condition of balance, the Sakharov conditions must be satisfied, one of which is the existence of CP violation during the extreme conditions of the first seconds after the Big Bang. Explanations which do not involve CP violation are less plausible, since they rely on the assumption that the matter–antimatter imbalance was present at the beginning, or on other admittedly exotic assumptions. |
The Big Bang should have produced equal amounts of matter and antimatter if CP-symmetry was preserved; as such, there should have been total cancellation of both—protons should have cancelled with antiprotons, electrons with positrons, neutrons with antineutrons, and so on. This would have resulted in a sea of radiation in the universe with no matter. Since this is not the case, after the Big Bang, physical laws must have acted differently for matter and antimatter, i.e. violating CP-symmetry. |
If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new physics beyond the Standard Model would be required to explain additional sources of CP violation. Adding new particles and/or interactions to the Standard Model generally introduces new sources of CP violation since CP is not a symmetry of nature. |
Sakharov proposed a way to restore CP-symmetry using T-symmetry, extending spacetime "before" the Big Bang. He described complete "CPT reflections" of events on each side of what he called the "initial singularity". Because of this, phenomena with an opposite arrow of time at "t" < 0 would undergo an opposite CP violation, so the CP-symmetry would be preserved as a whole. The anomalous excess of matter over antimatter after the Big Bang in the orthochronous (or positive) sector, becomes an excess of antimatter before the Big Bang (antichronous or negative sector) as both charge conjugation, parity and arrow of time are reversed due to CPT reflections of all phenomena occurring over the initial singularity: |
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: "positive" and "negative" (commonly carried by protons and electrons respectively). Like charges repel each other and unlike charges attract each other. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects. |
Electric charges produce electric fields. A moving charge also produces a magnetic field. The interaction of electric charges with an electromagnetic field (combination of electric and magnetic fields) is the source of the electromagnetic (or Lorentz) force, which is one of the four fundamental forces in physics. The study of photon-mediated interactions among charged particles is called quantum electrodynamics. |
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