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Solution to partial differential equation
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the
Viscosity_solution
French mathematician (born 1956)
prestigious Fields Medal. He was cited for his contributions to viscosity solutions, the Boltzmann equation, and the calculus of variations. He has also
Pierre-Louis_Lions
limiting value is the intrinsic viscosity, which is a good measure of the polymerization degree. "Dilute Solution Viscosity of Polymers".{{cite web}}: CS1
Inherent_viscosity
Optimality condition in optimal control theory
classical solutions only for a sufficiently smooth value function, which is not guaranteed in most situations. Instead, the notion of a viscosity solution is
Hamilton–Jacobi–Bellman equation
Hamilton–Jacobi–Bellman_equation
Resistance of a fluid to shear deformation
In continuum mechanics, viscosity is a property of a fluid that quantifies the resistance force acting on fluids when there is relative motion between
Viscosity
Relative viscosity ( η r e l {\displaystyle \eta _{rel}} ) (a synonym of "viscosity ratio") is the ratio of the viscosity of a solution ( η {\displaystyle
Relative_viscosity
up to 30% mass percentage, after which viscosity increases). The increase in viscosity for sucrose solutions is particularly dramatic, and explains in
List_of_viscosities
Type of fluid
not follow Newton's law of viscosity; that is, it has variable viscosity dependent on stress. In particular, the viscosity of non-Newtonian fluids can
Non-Newtonian_fluid
Measure of solute's contribution to viscosity
solute's contribution to the viscosity η {\displaystyle \eta } of a solution. If η 0 {\displaystyle \eta _{0}} is the viscosity in the absence of the solute
Intrinsic_viscosity
Mathematical solution
Hamilton–Jacobi equation, there is a very different definition of weak solution called viscosity solution. Evans, L. C. (1998). Partial Differential Equations. Providence:
Weak_solution
Caustic soda, with formula NaOH
natural skin oils. Concentrated (50%) aqueous solutions of sodium hydroxide have a characteristic viscosity, 78 mPa·s, that is much greater than that of
Sodium_hydroxide
Maximized objective function of an optimization problem
derivative with respect to time. The value function is the unique viscosity solution to the Hamilton–Jacobi–Bellman equation. In an online closed-loop
Value_function
Problem in continuous-time finance
its viscosity solutions. When there are fixed transaction costs the problem was addressed by Eastman and Hastings in 1988. A numerical solution method
Merton's_portfolio_problem
Equations of motion for viscous fluids
arbitrary constants. This solution is valid in the domain r ≥ 1 and for A < −2ν. In Cartesian coordinates, when the viscosity is zero (ν = 0), this is:
Navier–Stokes_equations
Capillary-based viscometer
capillary based method of measuring viscosity. It is recommended for higher viscosity cellulosic polymer solutions. The advantage of this instrument is
Ubbelohde_viscometer
Elastic fluids
elastic fluids exhibit shear thinning (viscosity decreases as shear strain is applied), because they are solutions containing polymers. But Boger fluids
Constant viscosity elastic fluid
Constant_viscosity_elastic_fluid
American mathematician
made significant contributions to the development of the theory of viscosity solutions of nonlinear equations, to the understanding of the Hamilton–Jacobi–Bellman
Lawrence_C._Evans
Type of fluid
describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow
Newtonian_fluid
State of matter
properties, such as viscosity. An important physical property characterizing the flow of liquids is viscosity. Intuitively, viscosity describes the resistance
Liquid
describes the relationship between the viscosity of a solution and the concentration of solute within the solution (at a fixed temperature and pressure)
Jones–Dole_equation
Chemistry subdiscipline
and melting temperature. Polymers in solution have special characteristics with respect to solubility, viscosity, and gelation. Illustrative of the quantitative
Polymer_chemistry
Non-linear partial differential equation encountered in problems of wave propagation
techniques are necessary to prove this. This led to the development of viscosity solutions in the 1980s by Pierre-Louis Lions and Michael G. Crandall, and Lions
Eikonal_equation
Material in which viscosity increases with the rate of shear strain
(/daɪˈleɪtənt/, /dɪ-/) (also termed shear thickening) material is one in which viscosity increases with the rate of shear strain. Such a shear thickening fluid
Dilatant
Chemical compound
solutions. Comparison of the physical properties of NH3 with those of water shows NH3 has the lower melting point, boiling point, density, viscosity,
Ammonia
Sauce used as a condiment
This equation represents apparent viscosity where apparent viscosity is the shear stress divided by shear rate. Viscosity is dependent on stress. This is
Ketchup
equation used to relate the reduced viscosity of a dilute polymer solution to the concentration of the polymer in solution. It is named after Maurice L. Huggins
Huggins_equation
Rayleigh and G. I. Taylor had addressed special cases, such as flows with viscosity but no heat conduction, or vice versa. In a frame fixed with a planar
Becker–Morduchow–Libby solution
Becker–Morduchow–Libby_solution
Non-Newtonian fluid behavior
rheology, shear thinning is the non-Newtonian behavior of fluids whose viscosity decreases under shear strain. It is sometimes considered synonymous with
Shear_thinning
Law describing the pressure drop in an incompressible and Newtonian fluid
equation. Poiseuille's equation describes the pressure drop due to the viscosity of the fluid; other types of pressure drops may still occur in a fluid
Hagen–Poiseuille_equation
Chemical compound
results in greater viscosity compared to water. It is used as an oxidizer, bleaching agent, and antiseptic, usually as a dilute solution (3%–6% by weight)
Hydrogen_peroxide
Lions–Magenes lemma John von Neumann Prize Japan Prize Pierre-Louis Lions Viscosity solution Fields Medal Takashi Ono Work in Number theory Algebraic groups Ken
List of second-generation mathematicians
List_of_second-generation_mathematicians
Italian mathematician (born 1970)
co-authored a paper that led to the solution of the long-standing problem of stability and convergence of vanishing viscosity approximations. "EMS Prizes".
Stefano_Bianchini
Cellulose ether used as emulsifier or thickening agent to disperse colloids in water
ranges of error, viscosity is typically given as a mean, or as a range. Typical viscosity test will specify the following: Solution concentration (1%
Hypromellose
Substance whose dissolved ions conduct electricity
Kaminsky M (1957). "Ion-solvent interaction and the viscosity of strong-electrolyte solutions". Discussions of the Faraday Society. 24: 171. doi:10
Electrolyte
Thickening agent
polysaccharides of natural origin, capable of causing a large increase in a solution's viscosity, even at small concentrations. They are mostly botanical gums, found
Natural_gum
Polymer solution parameter
extensional stress. It is often used for characterizing polymer solutions. Extensional viscosity can be measured using rheometers that apply extensional stress
Extensional_viscosity
Japanese mathematician
differential equations (PDEs) such as Hamilton–Jacobi equation, viscosity solutions of PDEs, optimal control theory, differential games, and evolution
Hitoshi_Ishii
Turkish American mathematician
equations; asymptotic analysis of Ginzburg-Landau type systems, viscosity solutions, and mathematical finance. Currently he is working on mean field
Halil_Mete_Soner
Vegetable gum from the guar bean, Cyamopsis tetragonoloba
in solution over pH range 5–7. Strong acids cause hydrolysis, and loss of viscosity and alkalis in strong concentration also tend to reduce viscosity. It
Guar_gum
Method of industrial polymerization
Moreover, the viscosity of the reaction mixture is reduced, preventing autoacceleration at high monomer concentrations. A decrease in viscosity of the reaction
Solution_polymerization
Branch of physics
Evangelista Torricelli (invented the barometer), Isaac Newton (investigated viscosity) and Blaise Pascal (researched hydrostatics, formulated Pascal's law)
Fluid_mechanics
Thick, viscous solution of sugar in water
concentrated form, its consistency is similar to that of molasses. The viscosity arises from the multiple hydrogen bonds between the dissolved sugar, which
Syrup
Ratio of inertial to viscous forces acting on a liquid
surrounding flow and its fall velocity. Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. The
Reynolds_number
Egyptian-French mathematician
Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton–Jacobi equation. The Weak KAM Theorem in Lagrangian
Albert_Fathi
Probabilistic optimal control
Fleming, W.; Soner, M. (2006). Controlled Markov Processes and Viscosity Solutions. Springer. Stein, J. L. (2012). Stochastic Optimal Control and the
Stochastic_control
Study of the flow of matter, primarily in a fluid state
be characterized by a single coefficient of viscosity for a specific temperature. Although this viscosity will change with temperature, it does not change
Rheology
Study of flow properties of blood and its elements of plasma and cells
properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit (volume fraction of red blood cell, which
Hemorheology
For this reason the correct notion of solutions is that given by the viscosity solutions. Viscosity solutions to the equation Δ ∞ u = 0 {\displaystyle
Infinity_Laplacian
American mathematician
Hamilton–Jacobi equations. With Pierre-Louis Lions he did research on the viscosity solutions of partial differential equations. In 2000 he was elected a member
Michael_G._Crandall
Branch of engineering and mathematics
principle and the bang-bang principle. Pierre-Louis Lions developed viscosity solutions into stochastic control and optimal control methods. Rudolf E. Kálmán
Control_theory
Canned processed cheese product
concentration within the cheese matrix is directly proportional to the viscosity of the solution due to their interactions with hydrated protein molecules. Therefore
Easy_Cheese
Viscous water-insoluble liquid
Oil is a liquid with varying degrees of viscosity depending on temperature. Oil is any nonpolar chemical substance that is composed primarily of hydrocarbons
Oil
American mathematician (1942–2017)
Mather theory. This theory turned out to be deeply related to the viscosity solution theory of Michael G. Crandall, Pierre-Louis Lions et al. for Hamilton–Jacobi
John_N._Mather
Greek-American mathematician
The main parts of his work are qualitative properties of viscosity and entropy solutions, front propagation and asymptotic behavior of reaction diffusion
Panagiotis_E._Souganidis
Class of partial differential equations
G.; Ishii, Hitoshi; Lions, Pierre-Louis (1992). "User's guide to viscosity solutions of second order partial differential equations". Bulletin of the
Elliptic partial differential equation
Elliptic_partial_differential_equation
Chemical data page
is viscosity, all at 20 °C; Teq is the equilibrium temperature between two phases: ice/liquid solution for Teq < 0–0.1 °C and NaCl/liquid solution for
Sodium_chloride_(data_page)
Physics calculation for collisions
equal-size particles at a concentration n {\displaystyle n} in a solution of viscosity η {\displaystyle \eta } , an expression for collision frequency
Collision_frequency
American award for mathematical analysis
20. Oxford University Press, Oxford, 2000. xii+250 pp. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. of Math. (2) 161 (2005), no
Bôcher_Memorial_Prize
Equations for calculations of the Darcy friction factor
the pipe diameter, and where ν is the kinematic viscosity μ / ρ, with μ the fluid's Dynamic viscosity, and ρ the fluid's density. The pipe's relative
Darcy friction factor formulae
Darcy_friction_factor_formulae
Awarded every year by the American Mathematical Society
Zbl 0394.14008. Crandall, Michael G.; Lions, Pierre-Louis (1983). "Viscosity solutions of Hamilton-Jacobi equations" (PDF). Transactions of the American
Leroy_P._Steele_Prize
High density solution
A heavy liquid is a solution or liquid chemical substance with a high density and a relatively low viscosity. Heavy liquids are often used for determination
Heavy_liquid
Italian-born American mathematician (1923–2023)
generalized notion of differential inequalities, predating the later viscosity solutions introduced by Michael Crandall and Pierre-Louis Lions. By extending
Eugenio_Calabi
Measure of the ability of a solution containing electrolytes to conduct electricity
viscosity of the solvent. As the name suggests, this is an extension of the Debye–Hückel theory, due to Onsager. It is very successful for solutions at
Conductivity_(electrolytic)
Repeating pattern of swirling vortices
viscosity parameter of the fluid, which in turn is the ratio between: ρ0 = the reference fluid density. μ0 = the free stream fluid dynamic viscosity For
Kármán_vortex_street
Mathematical optimization function
formulation and vice versa. By Hopf–Lax formula, the Moreau envelope is a viscosity solution to a Hamilton–Jacobi equation. Stanley Osher and co-authors used this
Moreau_envelope
Concept in partial differential equations
space or time and the other dimensional quantities present—such as the viscosity ν {\displaystyle \nu } . These constructs are not 'guessed' but are derived
Self-similar_solution
Polymer
process conditions. The molecular weight of PET is measured by solution viscosity. Viscosity is highly dependent on molecular parameters such as chain length
Polyethylene_terephthalate
Equation for the velocity of a body in viscous fluid
the effects this has on the viscosity. Industrial methods include many different oils, and polymer liquids such as solutions. The importance of Stokes's
Stokes's_law
Mathematical models for calculating viscosity
The shear viscosity (or "viscosity" for short) of a fluid is a material property that describes the friction between internal neighboring fluid surfaces
Viscosity_models_for_mixtures
Component of stress coplanar with a material cross section
{\frac {\partial u}{\partial y}}\right|_{y=0},} where μ is the dynamic viscosity, u is the flow velocity, and y is the distance from the wall. It is used
Shear_stress
Structure formed by diffuse material in orbital motion around a massive central body
it is being carried inward by accretion of matter. A simple solution is assuming a viscosity much larger than the magnetic diffusivity in the disk. However
Accretion_disk
Homogeneous mixture containing dissolved polymers
temperature, yield temperature, and melt viscosity. Understanding the thermodynamics of a polymer solution is critical in manufacturing processes. For
Polymer_solution
Phenomenon of attracting and holding water molecules
can become physically changed, e.g. changing in volume, boiling point, viscosity or some other physical characteristic or property of the substance. For
Hygroscopy
Millennium Prize Problem
properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Medication applied to body surfaces
desired.[citation needed] Topical solutions can be marketed as drops, rinses, or sprays, are generally of low viscosity, and often use alcohol or water
Topical_medication
Type of fluid flow
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
Stokes_flow
Layer of fluid in the immediate vicinity of a bounding surface
{\displaystyle \rho =} density and μ = {\displaystyle \mu =} dynamic viscosity). The Blasius solution uses boundary conditions in a dimensionless form: v x − v S
Boundary_layer
British-Canadian mathematician
1972) Stochastic Calculus and Applications (Springer-Verlag, 1982) Viscosity Solutions and Optimal Control (Longman, 1987) Stokasticheski Analiz i evo Prilozeniya
Robert_J._Elliott
Cellulose derivative grafted with carboxymethyl groups
a viscosity modifier or thickener and to stabilize emulsions in both food and non-food products. It is used primarily because it has high viscosity, is
Carboxymethyl_cellulose
Substitute for blood in a theatrical or cinematic performance
unsuitable for repeated takes without freshening; the longer-lasting viscosity of stage blood makes it far easier to work with on the set. Fake blood
Theatrical_blood
Solid or semisolid lubricant
initial viscosities, which upon the application of shear, drop to give the effect of an oil-lubricated bearing of approximately the same viscosity as the
Grease_(lubricant)
Oscillating boundary layer over a plate
^{2}u}{\partial y^{2}}}} where ν {\displaystyle \nu } is the kinematic viscosity. The pressure gradient does not enter into the problem. The initial, no-slip
Stokes_problem
Fiber production method
distribution and architecture (branched, linear etc.) of the polymer Solution properties (viscosity, conductivity, and surface tension) Electric potential, flow
Electrospinning
Material which is solid at low stress but becomes viscous at high stress
stress is achieved. For the Newtonian fluid the slope of this line is the viscosity, which is the only parameter needed to describe its flow. By contrast
Bingham_plastic
Scientific instrument used to measure viscosity
instrument used to measure the viscosity of a fluid. Viscometers can only measure constant viscosity, that is, viscosity that does not change with flow
Viscometer
Atmospheric model
static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations
International Standard Atmosphere
International_Standard_Atmosphere
American mathematician
Mathematicians in Berlin. Barron, E.N; Evans, L.C; Jensen, R. (1984). "Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls"
Robert_R._Jensen
Type of generalized Newtonian fluid
have a constant viscosity, μ, across all shear rates and include many of the most common fluids, such as water, most aqueous solutions, oils, corn syrup
Power-law_fluid
Equation in polymer science
Ichiro (2012). "Shape of threadlike molecules in solution, and relationship between solution viscosity and molecular weight". Polymer Journal. 44 (1):
Mark–Houwink_equation
Aqueous solution of hydrogen chloride
muriatic acid or spirits of salt, is an aqueous solution of hydrogen chloride (HCl). It is a colorless solution with a distinctive pungent smell. It is classified
Hydrochloric_acid
Aqueous solution of hydrogen iodide
1326 European Chemicals Bureau Viscosities of Aqueous Hydrochloric Acid Solutions, and Densities and Viscosities of Aqueous Hydroiodic Acid Solutions
Hydroiodic_acid
Motion of a curve based on its curvature
Klaus (2000), "Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow", Interfaces and Free Boundaries, 2 (2): 117–142
Curve-shortening_flow
sucrose or Percoll. This application requires a solution with high density and yet relatively low viscosity, and CsCl suits it because of its high solubility
Buoyant density centrifugation
Buoyant_density_centrifugation
Innermost part of Earth, a solid ball of iron-nickel alloy
In 2009, B. Buffett estimated the viscosity of the inner core at 1018 Pa·s, which is a sextillion times the viscosity of water, and more than a billion
Earth's_inner_core
Sweet and viscous substance made by bees
content generally has a viscosity around 400 poise, while a honey containing 20% water has a viscosity around 20 poise. Viscosity increases very slowly
Honey
Thickening agent used for culinary purposes
of the solution. Potato starch contains approximately 800 ppm phosphate bound to the starch; this increases the viscosity and gives the solution a slightly
Potato_starch
Mixture of an insoluble substance microscopically dispersed throughout another substance
effectively "soluble", they change the rheology of water by raising the viscosity and/or inducing gelation. They may provide other interactive effects with
Colloid
deposit crystals. Its consistency is similar to that of molasses. The viscosity arises from the hydrogen bonds between the dissolved sugar, which has
List_of_syrups
Polysaccharide gum used as a food additive and thickener
and permeation enhancing properties can be introduced. The viscosity of xanthan gum solutions decreases with higher shear rates. This is called shear thinning
Xanthan_gum
Mathematical model in fluid dynamics
{\displaystyle F(t)=e^{-2\nu k^{2}t}} , ν {\displaystyle \nu } being the kinematic viscosity of the fluid. Following the analysis of Taylor and Green for the two-dimensional
Taylor–Green_vortex
VISCOSITY SOLUTION
VISCOSITY SOLUTION
Girl/Female
Tamil
Good or Happy condition, Solution
Boy/Male
Gujarati, Hindu, Indian, Sanskrit
Space; God of Tech; Bliss Solutions; Wise
Girl/Female
Hindu
Good or Happy condition, Solution
Girl/Female
Hindu
Good or Happy condition, Solution, Fortune
Girl/Female
Tamil
Trick, Power, Strategy, Solution by logic, By reasoning
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Telugu
Strategy; Idea; Trick; Solution; A Device; Planning
Girl/Female
Hindu
Trick, Power, Strategy, Solution by logic, By reasoning
Girl/Female
Gujarati, Hindu, Indian, Kannada, Tamil
Trick; Power; Strategy; Solution by Logic; By Reasoning
Girl/Female
Tamil
Good or Happy condition, Solution, Fortune
Girl/Female
Tamil
Trick, Power, Strategy, Solution by logic, By reasoning
VISCOSITY SOLUTION
VISCOSITY SOLUTION
Boy/Male
German
Flame.
Girl/Female
Hindu
Name of a Raga
Boy/Male
Gujarati, Hindu, Indian, Kannada
Lord Shiva
Girl/Female
American, British, Christian, Dutch, English, French, German, Gujarati, Hindu, Indian, Jamaican, Netherlands, Sanskrit, Telugu, Vietnamese
From the Royal Fortress Meadow; Chief War; Armlet; Regal Hill; Golden; Noble or Brave; What
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
The Sky and the Earth
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Sweet Smell; Fragrance; Divine; Celestial
Boy/Male
Indian
Crown
Girl/Female
Muslim
Example, Copy, Torch, Light, Lightened, Sparkling, Shining
Girl/Female
Tamil
Tehihya | தேஹீஹà¯à®¯à®¾
Boy/Male
Muslim
Felicity. Good fortune. Good luck.
VISCOSITY SOLUTION
VISCOSITY SOLUTION
VISCOSITY SOLUTION
VISCOSITY SOLUTION
VISCOSITY SOLUTION
n.
That quality of bodies which makes them adhere to other bodies; adhesiveness; viscosity.
n.
The quality or state of being vinous.
n.
Viciousness; depravity.
n.
The quality or state of being viscous.
n.
The quality or state of being viscid; also, that which is viscid; glutinous concretion; stickiness.
n.
Mossiness.
n.
A coating of long, slender hairs.
n.
Quality of being ropy; viscosity.
n.
Viscidity.
a.
Tenacity; viscidity, as of fluids.
n.
An instrument for measuring the degree of viscosity of liquids, as solutions of gum.
n.
Vitiosity.
n.
A villus.
n.
A quality analogous to that of a viscous fluid, supposed to be caused by internal friction, especially in the case of gases.
n.
One of the minute papillary processes on certain vascular membranes; a villosity; as, villi cover the lining of the small intestines of many animals and serve to increase the absorbing surface.
n.
Alt. of Viscounty
n.
State of being villous.
n.
The quality, rank, or office of a viscount.