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Solution for a moving sphere in incompressible fluid. |
Consider the case of a solid sphere moving in a stationary liquid with a constant velocity. |
The liquid is modeled as an incompressible fluid (i.e. with constant density), and being stationary means that its velocity tends towards zero as the distance from the sphere approaches infinity. |
For a real body there will be a transient effect due to its acceleration as it begins its motion; however after enough time it will tend towards zero, so that the fluid velocity everywhere will approach the one obtained in the hypothetical case in which the body is already moving for infinite time. |
Thus we assume a sphere of radius "a" moving at a constant velocity formula_79, in an incompressible fluid that is at rest at infinity. We will work in coordinates formula_80 that move along with the sphere with the coordinate center located at the sphere's center. We have: |
Since these boundary conditions, as well as the equation of motions, are time invariant (i.e. they are unchanged by shifting the time formula_82) when expressed in the formula_80 coordinates, the solution depends upon the time only through these coordinates. |
The equations of motion are the Navier-Stokes equations defined in the resting frame coordinates formula_84. While spatial derivatives are equal in both coordinate systems, the time derivative that appears in the equations satisfies: |
where the derivative formula_86 is with respect to the moving coordinates formula_80. We henceforth omit the "m" subscript. |
Oseen's approximation sums up to neglecting the term non-linear in formula_88. Thus the incompressible Navier-Stokes equations become: |
for a fluid having density ρ and kinematic viscosity ν = μ/ρ (μ being the dynamic viscosity). "p" is the pressure. |
Due to the continuity equation for incompressible fluid formula_90, the solution can be expressed using a vector potential formula_91. This turns out to be directed at the formula_92 direction and its magnitude is equivalent to the stream function used in two-dimensional problems. It turns out to be: |
where formula_94 is Reynolds number for the flow close to the sphere. |
Note that in some notations formula_95 is replaced by formula_96 so that the derivation of formula_88 from formula_98 is more similar to its derivation from the stream function in the two-dimensional case (in polar coordinates). |
The vector Laplacian of a vector of the type formula_104 reads: |
where we have used the vanishing of the divergence of formula_91 to relate the vector laplacian and a double curl. |
The equation of motion's left hand side is the curl of the following: |
We calculate the derivative separately for each term in formula_95. |
Taking the curl, we find an expression that is equal to formula_116 times the gradient of the following function, which is the pressure: |
where formula_118 is the pressure at infinity, formula_119.is the polar angle originated from the opposite side of the front stagnation point (formula_120 where is the front stagnation point). |
Also, the velocity is derived by taking the curl of formula_91: |
These "p" and "u" satisfy the equation of motion and thus constitute the solution to Oseen's approximation. |
One may question, however, whether the correction term was chosen by chance, because in a frame of reference moving with the sphere, the fluid near the sphere is almost at rest, and in that region inertial force is negligible and Stokes' equation is well justified. Far away from the sphere, the flow velocity approaches "u" and Oseen's approximation is more accurate. But Oseen's equation was obtained applying the equation for the entire flow field. This question was answered by Proudman and Pearson in 1957, who solved the Navier-Stokes equations and gave an improved Stokes' solution in the neighborhood of the sphere and an improved Oseen's solution at infinity, and matched the two solutions in a supposed common region of their validity. They obtained: |
The method and formulation for analysis of flow at a very low Reynolds number is important. The slow motion of small particles in a fluid is common in bio-engineering. Oseen's drag formulation can be used in connection with flow of fluids under various special conditions, such as: containing particles, sedimentation of particles, centrifugation or ultracentrifugation of suspensions, colloids, and blood through isolation of tumors and antigens. The fluid does not even have to be a liquid, and the particles do not need to be solid. It can be used in a number of applications, such as smog formation and atomization of liquids. |
Blood flow in small vessels, such as capillaries, is characterized by small Reynolds and Womersley numbers. A vessel of diameter of with a flow of , viscosity of for blood, density of and a heart rate of , will have a Reynolds number of 0.005 and a Womersley number of 0.0126. At these small Reynolds and Womersley numbers, the viscous effects of the fluid become predominant. Understanding the movement of these particles is essential for drug delivery and studying metastasis movements of cancers. |
In fluid dynamics, stream thrust averaging is a process used to convert three-dimensional flow through a duct into one-dimensional uniform flow. It makes the assumptions that the flow is mixed adiabatically and without friction. However, due to the mixing process, there is a net increase in the entropy of the system. Although there is an increase in entropy, the stream thrust averaged values are more representative of the flow than a simple average as a simple average would violate the second Law of Thermodynamics. |
Solving for formula_5 yields two solutions. They must both be analyzed to determine which is the physical solution. One will usually be a subsonic root and the other a supersonic root. If it is not clear which value of velocity is correct, the second law of thermodynamics may be applied. |
The values formula_10 and formula_11 are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive. |
One possible unreal solution for the stream thrust averaged velocity yields a negative entropy. Another method of determining the proper solution is to take a simple average of the velocity and determining which value is closer to the stream thrust averaged velocity. |
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics. |
There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term "dynamics" refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations. |
However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (), initial velocity (), final velocity (), acceleration (), and time (). |
Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these. |
A differential equation of motion, usually identified as some physical law and applying definitions of physical quantities, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, which fixes the values of the constants. |
To state this formally, in general an equation of motion is a function of the position of the object, its velocity (the first time derivative of , ), and its acceleration (the second derivative of , ), and time . Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in is a second-order ordinary differential equation (ODE) in , |
where is time, and each overdot denotes one time derivative. The initial conditions are given by the "constant" values at , |
The solution to the equation of motion, with specified initial values, describes the system for all times after . Other dynamical variables like the momentum of the object, or quantities derived from and like angular momentum, can be used in place of as the quantity to solve for from some equation of motion, although the position of the object at time is by far the most sought-after quantity. |
Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how "sensitive" the system is to the initial conditions. |
Kinematics, dynamics and the mathematical models of the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know. In antiquity, priests, astrologers and astronomers predicted solar and lunar eclipses, the solstices and the equinoxes of the Sun and the period of the Moon. But they had nothing other than a set of algorithms to guide them. Equations of motion were not written down for another thousand years. |
Medieval scholars in the thirteenth century — for example at the relatively new universities in Oxford and Paris — drew on ancient mathematicians (Euclid and Archimedes) and philosophers (Aristotle) to develop a new body of knowledge, now called physics. |
At Oxford, Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, who were of similar stature to the intellectuals at the University of Paris. Thomas Bradwardine extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time. Nicholas Oresme further extended Bradwardine's arguments. The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. |
Discourses such as these spread throughout Europe, shaping the work of Galileo Galilei and others, and helped in laying the foundation of kinematics. Galileo deduced the equation in his work geometrically, using the Merton rule, now known as a special case of one of the equations of kinematics. |
The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.) |
Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope. |
Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum. |
More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum. |
Thus we arrive at René Descartes, Isaac Newton, Gottfried Leibniz, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones. |
Later the equations of motion also appeared in electrodynamics, when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field. With the advent of special relativity and general relativity, the theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light, and curvature of spacetime. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations. |
However, the equations of quantum mechanics can also be considered "equations of motion", since they are differential equations of the wavefunction, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields. |
From the instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions; |
Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature. |
The rotational analogues are the "angular vector" (angle the particle rotates about some axis) , angular velocity , and angular acceleration : |
where is a unit vector in the direction of the axis of rotation, and is the angle the object turns through about the axis. |
The following relation holds for a point-like particle, orbiting about some axis with angular velocity : |
where is the position vector of the particle (radial from the rotation axis) and the tangential velocity of the particle. For a rotating continuum rigid body, these relations hold for each point in the rigid body. |
The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below. |
Constant translational acceleration in a straight line. |
These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. |
Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; |
which breaks into the radial acceleration , centripetal acceleration , Coriolis acceleration , and angular acceleration . |
Special cases of motion described be these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration. |
In 3D space, the equations in spherical coordinates with corresponding unit vectors , and , the position, velocity, and acceleration generalize respectively to |
In the case of a constant this reduces to the planar equations above. |
The first general equation of motion developed was Newton's second law of motion. In its most general form it states the rate of change of momentum of an object equals the force acting on it, |
The force in the equation is "not" the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as |
since is a constant in Newtonian mechanics. |
Newton's second law applies to point-like particles, and to all points in a rigid body. They also apply to each point in a mass continuum, like deformable solids or fluids, but the motion of the system must be accounted for; see material derivative. In the case the mass is not constant, it is not sufficient to use the product rule for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with conservation of momentum; see variable-mass system. |
It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex. |
The momentum form is preferable since this is readily generalized to more complex systems, such as special and general relativity (see four-momentum). It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces. |
For a number of particles (see many body problem), the equation of motion for one particle influenced by other particles is |
where is the momentum of particle , is the force on particle by particle , and is the resultant external force due to any agent not part of system. Particle does not exert a force on itself. |
Euler's laws of motion are similar to Newton's laws, but they are applied specifically to the motion of rigid bodies. The Newton–Euler equations combine the forces and torques acting on a rigid body into a single equation. |
Newton's second law for rotation takes a similar form to the translational case, |
by equating the torque acting on the body to the rate of change of its angular momentum . Analogous to mass times acceleration, the moment of inertia tensor depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity, |
Again, these equations apply to point like particles, or at each point of a rigid body. |
Likewise, for a number of particles, the equation of motion for one particle is |
where is the angular momentum of particle , the torque on particle by particle , and is resultant external torque (due to any agent not part of system). Particle does not exert a torque on itself. |
Some examples of Newton's law include describing the motion of a simple pendulum, |
and a damped, sinusoidally driven harmonic oscillator, |
For describing the motion of masses due to gravity, Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of mass thrown in the air, in air currents (such as wind) described by a vector field of resistive forces , |
where is the gravitational constant, the mass of the Earth, and is the acceleration of the projectile due to the air currents at position and time . |
The classical -body problem for particles each interacting with each other due to gravity is a set of nonlinear coupled second order ODEs, |
where labels the quantities (mass, position, etc.) associated with each particle. |
Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has degrees of freedom, then one can use a set of generalized coordinates , to define the configuration of the system. They can be in the form of arc lengths or angles. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. The time derivatives of the generalized coordinates are the "generalized velocities" |
where the "Lagrangian" is a function of the configuration and its time rate of change (and possibly time ) |
Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled second order ODEs in the coordinates are obtained. |
is a function of the configuration and conjugate ""generalized" momenta" |
in which is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example matrix calculus for this denominator notation), and possibly time , |
Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled first order ODEs in the coordinates and momenta are obtained. |
is "Hamilton's principal function", also called the "classical action" is a functional of . In this case, the momenta are given by |
Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order "non-linear" PDE, in variables. The action allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any differentiable symmetry of the action of a physical system has a corresponding conservation law, a theorem due to Emmy Noether. |
All classical equations of motion can be derived from the variational principle known as Hamilton's principle of least action |
stating the path the system takes through the configuration space is the one with the least action . |
In electrodynamics, the force on a charged particle of charge is the Lorentz force: |
Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle: |
The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass and charge : |
where and are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by: |
instead of just , implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation. |
Alternatively the Hamiltonian (and substituting into the equations): |
The above equations are valid in flat spacetime. In curved spacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a "geodesic" of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor , the metric provides the notion of arc length (see line element for details). The differential arc length is given by: |
and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics: |
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