Stokes' Law describes the force on objects passing through a fluid. It includes which property of the fluid usually represented by the Greek letter eta?
"viscosity of the fluid is a much stronger effect than the inertia of the particles. The particles do not interact strongly with each other. By considering the forces on a particle in a fluid in the Stokes regime, it can be shown that the presence of the particle simply increases the 'effective viscosity' of the fluid. At high shear rates, the inertia of the particles is the dominant effect, and the suspension's behaviour is governed by collisions between particles. In his 1954 paper, Bagnold justified the quadratic relationship by collisional arguments. He considered an idealised situation in which layers of"
"force, formula_28, and torque, formula_57 on a sphere, they took the following form: where formula_43 is the dynamic viscosity, formula_41 is the particle radius, formula_61 is the ambient flow, formula_62 is the speed of the particle, formula_63 is the angular velocity of the background flow, and formula_64 is the angular velocity of the particle. The Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops. Stokes flow Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces"
"mass of the electron. The field of fluid dynamics contains in Stokes' law, which approximates the frictional force exerted on small, spherical objects of radius , moving with velocity in a fluid with dynamic viscosity : In electromagnetics, the vacuum permeability constant ""μ"" appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation. It is defined as exactly A relation for the speed of light in vacuum, can be derived from Maxwell's equations in the medium of classical vacuum using a relationship between ""μ"" and the electric constant (vacuum permittivity), in SI units: Under"
"unit area σ, exerted by the flow on the surface on the sphere, is in the ""z""–direction everywhere. More strikingly, it has also the same value everywhere on the sphere: with e the unit vector in the ""z""–direction. For other shapes than spherical, σ is not constant along the body surface. Integration of the viscous force per unit area σ over the sphere surface gives the frictional force ""F"" according to Stokes' law. Stokes' law In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on"
"that in a steady state flow of a viscous fluid through a rigid spherical body immersed in the fluid, where we assume the inertia is negligible in such a flow, it is believed that the downward gravitational force of the particle is balanced by the viscous drag force. From this force balance the speed of fall can be shown to be given by Stokes' law Where ""a"" is the particle radius, ""ρ"", ""ρ"" are the respectively particle and fluid density ""μ"" is the fluid viscosity, ""g"" is the gravitational acceleration. From the above equation we can see that the sedimentation"
"and the subsidence of ripples and waves in water, but also to the solution of practical problems, such as the flow of water in rivers and channels, and the skin resistance of ships. His work on fluid motion and viscosity led to his calculating the terminal velocity for a sphere falling in a viscous medium. This became known as Stokes's law. He derived an expression for the frictional force (also called drag force) exerted on spherical objects with very small Reynolds numbers. His work is the basis of the falling sphere viscometer, in which the fluid is stationary in a"
"Stokes number The Stokes number (Stk), named after George Gabriel Stokes, is a dimensionless number characterising the behavior of particles suspended in a fluid flow. The Stokes number is defined as the ratio of the characteristic time of a particle (or droplet) to a characteristic time of the flow or of an obstacle, or where formula_2 is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), formula_3 is the fluid velocity of the flow well away from the obstacle and formula_4 is the characteristic dimension of the obstacle (typically"
"the radius of the particle and ""μ"" is the dynamic viscosity of the fluid. Stokes' law applies when the Reynolds number, Re, of the particle is less than 0.1. Experimentally Stokes' law is found to hold within 1% for formula_2, within 3% for formula_3 and within 9% formula_4. With increasing Reynolds numbers, Stokes law begins to break down due to the increasing importance of fluid inertia, requiring the use of empirical solutions to calculate drag forces. Defining a drag coefficient, formula_5, as the ratio of the force experienced by the particle divided by the impact pressure of the fluid, a"
"(roundness and sphericity) and density of the grains, as well as to the viscosity and density of the fluid. For dilute suspensions, Stokes' law predicts the settling velocity of small spheres in fluid, either air or water. This originates due to the strength of viscous forces at the surface of the particle providing the majority of the retarding force. Stokes' law finds many applications in the natural sciences, and is given by: where ""w"" is the settling velocity, ""ρ"" is density (the subscripts ""p"" and ""f"" indicate particle and fluid respectively), ""g"" is the acceleration due to gravity, ""r"" is"
"Stokes flow Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. formula_1. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint,"
"Coriolis–Stokes force In fluid dynamics, the Coriolis–Stokes force is a forcing of the mean flow in a rotating fluid due to interaction of the Coriolis effect and wave-induced Stokes drift. This force acts on water independently of the wind stress. This force is named after Gaspard-Gustave Coriolis and George Gabriel Stokes, two nineteenth-century scientists. Important initial studies into the effects of the Earth's rotation on the wave motion – and the resulting forcing effects on the mean ocean circulation – were done by , and . The Coriolis–Stokes forcing on the mean circulation in an Eulerian reference frame was first"
"to describe the motion of fluid around a spherical particle with prescribed surface flow, a so-called squirmer, or to describe the flow inside a spherical drop of fluid. For interior flows, the terms with formula_38 are dropped, while for exterior flows the terms with formula_39 are dropped (often the convention formula_40 is assumed for exterior flows to avoid indexing by negative numbers). The Stokes' law for the resistance to a moving sphere, also known as Stokes' drag, is a relation describing the drag force on a sphere exerted by the surrounding fluid in a Stokes flow. Given a sphere of"
"the object. At very low Reynolds numbers, without flow separation, the drag force formula_32 is proportional to formula_33 instead of formula_34; for a sphere this is known as Stokes law. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids. A formula_1 equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up stagnation pressure over the whole front surface. The top figure shows a flat plate with the fluid coming from the right and stopping at the plate. The graph to the"
"slowly through a viscous fluid (and thus at small Reynolds number), George Gabriel Stokes derived an expression for the drag constant: where: The resulting expression for the drag is known as Stokes' drag: For example, consider a small sphere with radius formula_52 = 0.5 micrometre (diameter = 1.0 µm) moving through water at a velocity formula_56 of 10 µm/s. Using 10 Pa·s as the dynamic viscosity of water in SI units, we find a drag force of 0.09 pN. This is about the drag force that a bacterium experiences as it swims through water. In aerodynamics, aerodynamic drag is the"
"in solution. Hydrodynamic radius, ""R"", can refer to the Stokes radius of a polymer or other macromolecule. According to Stokes’ law, a perfect sphere traveling through a viscous liquid feels a drag force proportional to the frictional coefficient formula_1: formula_2 where formula_3 is the liquid's viscosity, formula_4 is the sphere's drift speed, and formula_5 is its radius. Because ionic mobility formula_6 is directly proportional to drift speed, it is inversely proportional to the frictional coefficient: formula_7 where formula_8 represents ionic charge in integer multiples of electron charges. In 1905, Albert Einstein found the diffusion coefficient formula_9 of an ion to"
"Navier–Stokes equations In physics, the Navier–Stokes equations (), named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances. These balance equations arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ""viscous flow"". The main difference between them and the simpler Euler equations for inviscid flow is that Navier–Stokes equations also factor in the Froude limit (no external field) and are not conservation equations, but rather"
": Stokes drift For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation. More generally, the Stokes drift velocity is the difference between the average Lagrangian flow velocity of a fluid parcel, and the average Eulerian flow velocity of the fluid at a fixed position. This nonlinear phenomenon is named after George Gabriel Stokes, who derived expressions"
"Stokes drift For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation. More generally, the Stokes drift velocity is the difference between the average Lagrangian flow velocity of a fluid parcel, and the average Eulerian flow velocity of the fluid at a fixed position. This nonlinear phenomenon is named after George Gabriel Stokes, who derived expressions for"
"called Stokes' law. When the value of formula_5 is substituted in the equation (5), we obtain the expression for terminal speed of a spherical object moving under creeping flow conditions: The creeping flow results can be applied in order to study the settling of sediments near the ocean bottom and the fall of moisture drops in the atmosphere. The principle is also applied in the falling sphere viscometer, an experimental device used to measure the viscosity of highly viscous fluids, for example oil, paraffin, tar etc. When the buoyancy effects are taken into account, an object falling through a fluid"
"on a sphere in Stokes flow, known as Stokes' law. Stokes flow is the low Reynolds-number limit of the Navier–Stokes equations describing the motion of a viscous liquid. However, when the flow problem is put into a non-dimensional form, the viscous Navier–Stokes equations converge for increasing Reynolds numbers towards the inviscid Euler equations, suggesting that the flow should converge towards the inviscid solutions of potential flow theory – having the zero drag of the d'Alembert paradox. Of this, there is no evidence found in experimental measurements of drag and flow visualisations. This again raised questions concerning the applicability of fluid"
"and formula_14 is in the vertical direction, the Coriolis–Stokes forcing is perpendicular to the wave propagation direction (i.e. in the direction parallel to the wave crests). In deep water the Stokes drift velocity is formula_15 with formula_16 the wave's phase velocity, formula_17 the wavenumber, formula_18 the wave amplitude and formula_19 the vertical coordinate (positive in the upward direction opposing the gravitational acceleration). Coriolis–Stokes force In fluid dynamics, the Coriolis–Stokes force is a forcing of the mean flow in a rotating fluid due to interaction of the Coriolis effect and wave-induced Stokes drift. This force acts on water independently of the"
"its diameter). A particle with a low Stokes number follows fluid streamlines (perfect advection), while a particle with a large Stokes number is dominated by its inertia and continues along its initial trajectory. In the case of Stokes flow, which is when the particle (or droplet) Reynolds number is less than unity. Then the particle drag coefficient is inversely proportional to the Reynolds number itself, the characteristic time of the particle can be defined as where formula_6 is the particle density, formula_7 is the particle diameter and formula_8 is the gas dynamic viscosity. In experimental fluid dynamics, the Stokes number"
"formula_25, then the velocity field is given by The frictional force per unit area on the moving plane is formula_32 and on the fixed plane is formula_33. Stokes problem In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations. In turbulent flow, this is still named a Stokes boundary layer, but"
"introduction to interactive fluid simulation can be found in the 2007 ACM SIGGRAPH course, Fluid Simulation for Computer Animation. Navier–Stokes equations In physics, the Navier–Stokes equations (), named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances. These balance equations arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ""viscous flow"". The main difference between them and the simpler Euler equations for inviscid flow is"
"is commonly neglected for practical reasons; however, it can be substantially large when the body is accelerated at a high rate. This force in an accelerating Stokes flow has been proposed by Joseph Valentin Boussinesq in 1885 and Alfred Barnard Basset in 1888. Consequently, it is also referred to as the Boussinesq–Basset force. Consider an infinitely large plate started impulsively with a step change in velocity—from 0 to ""u""—in the direction of the plate–fluid interface plane. The equation of motion for the fluid—Stokes flow at low Reynolds number—is where ""u""(""y"",""t"") is the velocity of the fluid, at some time ""t"","
"the motion of the sphere does not disturb that reference parcel of fluid. The density and viscosity are those belonging to the fluid. Note that purely laminar flow only exists up to = 10 under this definition. Under the condition of low , the relationship between force and speed of motion is given by Stokes' law. The equation for a rectangular object is identical to that of a sphere, with the object being approximated as an ellipsoid and the axis of length being chosen as the characteristic length scale. Such considerations are important in natural streams, for example, where there"
"given by : to be added to the common Coriolis forcing formula_2 Here formula_3 is the mean flow velocity in an Eulerian reference frame and formula_4 is the Stokes drift velocity – provided both are horizontal velocities (perpendicular to formula_5). Further formula_6 is the fluid density, formula_7 is the cross product operator, formula_8 where formula_9 is the Coriolis parameter (with formula_10 the Earth's rotation angular speed and formula_11 the sine of the latitude) and formula_5 is the unit vector in the vertical upward direction (opposing the Earth's gravity). Since the Stokes drift velocity formula_4 is in the wave propagation direction,"
"the well-known Stokes's law for the friction force in fluid motion. Stokes's law applies to sound propagation in an isotropic and homogeneous Newtonian medium. Consider a plane sinusoidal pressure wave that has amplitude formula_7 at some point. After traveling a distance formula_8 from that point, its amplitude formula_9 will be The parameter formula_1 is dimensionally the reciprocal of length. In the International System of Units (SI), it is expressed in neper per meter or simply reciprocal of meter (formula_12). That is, if formula_13, the wave's amplitude decreases by a factor of formula_14 for each meter traveled. The law is amended"
"function of strain rate. A consequence of this behavior is Pascal's law which describes the role of pressure in characterizing a fluid's state. Depending on the relationship between shear stress, and the rate of strain and its derivatives, fluids can be characterized as one of the following: The behavior of fluids can be described by the Navier–Stokes equations—a set of partial differential equations which are based on: The study of fluids is fluid mechanics, which is subdivided into fluid dynamics and fluid statics depending on whether the fluid is in motion. Fluid In physics, a fluid is a substance that"
"has been patented. The hydrodynamic resistance force is evaluated following the Stokes’ law. The electrophoretic force is evaluated following the Coulomb’s law. In these equations ""r"" presents the hydrodynamic radius of the colloids, formula_3 – the speed of electrophoretic migration, formula_4 – the dynamic viscosity of the solutions, formula_5 – dielectric constant in vacuum, formula_6 is water’s relative dielectric constant at 298 K, formula_7 is the zeta potential, ""E"" is the electric field. The hydrodynamic radius is the sum of particles’ radiuses and the stationary solvent interface. By steady state electrophoretic migration of charged colloids the electrophoretic force and the"
"through some kind of fluid, possibly bodily fluid, in which case the aforementioned equation can be modified to: Based on these equations, there will be the greatest force in the direction of the largest positive slope of the energy density scalar field. Another important consideration is the force acting against the magnetic force. As iron oxide nanoparticles translate toward the magnetic field source, they experience Stokes' drag force in the opposite direction. The drag force is expressed below. In this equation, η is the fluid viscosity, R is the hydrodynamic radius of the particle, and 𝑣 is the velocity of"
"coefficient that can be considered as the transfer of available fluid force into drag is established. In this region the inertia of the impacting fluid is responsible for the majority of force transfer to the particle. For a spherical particle in the Stokes regime this value is not constant, however in the Newtonian drag regime the drag on a sphere can be approximated by a constant, 0.44. This constant value implies that the efficiency of transfer of energy from the fluid to the particle is not a function of fluid velocity. As such the terminal velocity of a particle in"
"ft, and is given by the equation: Momentum is one of the most important basic definitions in Fluid Mechanics. The conservation of momentum is one of the three fundamental physical principles in both Fluid Mechanics and [Open-channel flow | open channel flow] (the other two are mass conservation and energy conservation). This principle leads to the momentum equation set in three dimensions (x, y and z). With different assumptions, these momentum equations can be simplified to several widely applied forms: With Newton’s second law, Newtonian fluids assumption and Stokes hypothesis, the original fluid momentum equations are derived as the Navier–Stokes"