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Nonlinear partial differential equation
The porous medium equation, also called the nonlinear heat equation, is a nonlinear partial differential equation taking the form: ∂ u ∂ t = Δ ( u m )
Porous_medium_equation
Partial differential equation describing the evolution of temperature in a region
modeled option price. The equation describing pressure diffusion in a porous medium is identical in form with the heat equation. Diffusion problems dealing
Heat_equation
Equation describing the flow of a fluid through a porous medium
Darcy's law is an equation that describes the flow of a fluid through a porous medium and through a Hele-Shaw cell. The law was formulated by Henry Darcy
Darcy's_law
Measure of the ability of a porous material to allow fluids to pass through it
velocity through the porous medium (i.e., the average flow velocity calculated as if the fluid was the only phase present in the porous medium) (m/s) k {\displaystyle
Permeability_(porous_media)
Transport of dissolved species from the highest to the lowest concentration region
1039/c6sm01153e. PMC 5476231. PMID 27396746. J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, ISBN 0198569033. Stauffer
Diffusion
American-Turkish mathematician (born 1992)
forcing for the incompressible porous medium equation, the 2D Boussinesq system, and the 3D incompressible Euler equations, using Anthropic's Claude and
Levent_Alpöge
Professor of Applied Mathematics
fractional porous medium flow". Journal of the European Mathematical Society. 15 (5): 1701–1746. doi:10.4171/JEMS/401. The Porous Medium Equation. Mathematical
Juan_Luis_Vázquez_Suárez
equation Nonlinear Schrödinger equation in water waves Omega equation Orr–Sommerfeld equation Porous medium equation Potential flow Rayleigh–Bénard convection
List of named differential equations
List_of_named_differential_equations
Representation of water movement in unsaturated soils
three-dimensional problems. Richards equation is derived by first considering the law of mass conservation for an incompressible porous medium and constant liquid density
Richards_equation
Equation used to calculate the electromigration of ions in a fluid
Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It extends Fick's
Nernst–Planck_equation
Equation describing vapor pressure
for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known
Kelvin_equation
Manner in which fluids behave when flowing through a porous medium
fluid mechanics, fluid flow through porous media is the manner in which fluids behave when flowing through a porous medium, for example sponge or wood, or
Fluid flow through porous media
Fluid_flow_through_porous_media
Mathematical descriptions of molecular diffusion
the original (PDF) on 5 February 2009. Vázquez JL (2006). "The Porous Medium Equation". Mathematical Theory. Oxford Univ. Press. Gorban AN, Sargsyan HP
Fick's_laws_of_diffusion
Mathematical system for studying diffusion equations
geometry of dissipative evolution equations: the porous medium equation". Communications in Partial Differential Equations. 26 (1–2): 101–174. doi:10.1081/PDE-100002243
Otto_calculus
Equation that describes density changes of a material that is diffusing in a medium
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian
Diffusion_equation
Phenomenon in physics
nanoparticle size Very similar equations may be applied to the growth and melting of crystals in the confined geometry of porous systems. However the geometry
Gibbs–Thomson_equation
effective and efficiently. Forced convection heat transfer in a confined porous medium has been a subject of intensive studies during the last decades because
Forced convection in porous media
Forced_convection_in_porous_media
Dimensionless quantity associated with free convection of a fluid
number, is different. In a bulk fluid, i.e., not in a porous medium, from the Stokes equation, the falling speed of a domain of size l {\displaystyle
Rayleigh_number
mechanics that studies the behavior of fluid-saturated porous media. A porous medium or a porous material is a solid (referred to as matrix) permeated
Poromechanics
Field in materials science and mechanics
within a linear porous medium and it is an extension of elasticity and porous medium flow (diffusion equation). The deformation of the medium influences the
Poroelasticity
performing fluid substitution—predicting the elastic behaviour of a porous medium under a saturant different to the one in the rock as measured. These
Gassmann's_equation
Israeli mathematician
the porous medium equation, ∂ t u = Δ x u m , m > 1 , {\displaystyle \partial _{t}u=\Delta _{x}u^{m},\,\,m>1,} and to non-linear elliptic equations. Kamenomostskaya
Shoshana_Kamin
Physical law in fluid mechanics
focused on porous mediums, where liquid does not penetrate the grooves on rough surface and leaves air gaps. They devised the Cassie-Baxter equation; cos
Cassie's_law
Ability of porous media to condense liquid from an adsorbed vapor
more condensation will occur. In a porous medium, capillary condensation will always occur if Pv≠0. The Kelvin equation indicates that as Pv/Psat increases
Capillary_condensation
American petroleum engineer
multiphase flow of water, oil and gas in the porous medium of a petroleum reservoir. The generalized flow equation provides the analytical foundation for reservoir
Morris_Muskat
Eigenvalue equation
The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel
Orr–Sommerfeld_equation
Speed of water in soil
Moisture Velocity Equation is particularly useful for calculating the advance of wetting fronts for a liquid invading an unsaturated porous medium under the combined
Soil moisture velocity equation
Soil_moisture_velocity_equation
Ability of a liquid to flow in narrow spaces
medium, in units of m·s−1/2 or mm·min−1/2. This time dependence relation is similar to Washburn's equation for the wicking in capillaries and porous media
Capillary_action
See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations.
List of nonlinear partial differential equations
List_of_nonlinear_partial_differential_equations
Substance-specific relation between two physical quantities
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic
Constitutive_equation
Spanish mathematician
obtained results on the exponential convergence to equilibrium of the porous medium equation by using an analogue of the Bakry-Émery method . He has also worked
José_A._Carrillo
Study of groundwater's movement and distribution
conductivity. The groundwater flow equation, in its most general form, describes the movement of groundwater in a porous medium (aquifers and aquitards). It
Hydrogeology
Dutch soil physicist and hydrologist
Vadose zone Hydraulic conductivity Water retention curve Richards equation Porous medium HYDRUS (software) Groundwater pollution Solute transport Arantes
Martinus Theodorus van Genuchten
Martinus_Theodorus_van_Genuchten
Hypothetical flow velocity
skeleton of the porous medium, etc. present in the channel are disregarded. Superficial velocity is used in many engineering equations because it is the
Superficial_velocity
Mathematical model of how solid objects deform
for. If the medium is homogeneous, then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written
Linear_elasticity
Pressure between two fluids from forces between the fluids and tube walls
cylinder model for ice growth and the following equation to understand the freezing of water in porous media (directly applicable to the formation of needle
Capillary_pressure
2026 scientific priority controversy
smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations". What's new. Retrieved 9 September 2026. Criddle
Navier–Stokes priority controversy
Navier–Stokes_priority_controversy
Term used in composite materials theory
Groundwater flow equation has to be defined in an REV. While RVEs for electromagnetic media can have the same form as those for elastic or porous media, the
Representative elementary volume
Representative_elementary_volume
Probabilistic problem-solving algorithms
(1990). "Large systems of interacting particles and porous medium equation". J. Differential Equations. 88 (2): 294–346. Bibcode:1990JDE....88..294O. doi:10
Mean-field_particle_methods
and tortuosity of pore channels) of a porous medium. Pores are the openings in the surfaces impermeable porous matrix which gases, liquids, or even foreign
Pore_structure
magnetic resonance (NMR) in porous materials covers the application of using NMR as a tool to study the structure of porous media and various processes
Nuclear magnetic resonance in porous media
Nuclear_magnetic_resonance_in_porous_media
Measure of energy loss as sound waves propagate through a medium
range of viscoelastic materials, such as soft tissue, polymers, soil, and porous rock, can be expressed as the following power law with respect to frequency:
Acoustic_attenuation
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Fluid instability
patterns in a morphologically unstable interface between two fluids in a porous medium or in a Hele-Shaw cell, described mathematically by Philip Saffman and
Saffman–Taylor_instability
Gradual loss of flux intensity through a medium
colloquially, damping – is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays,
Attenuation
Measure of electrostatic effect and how far it persists
particles within this medium gives rise to an electric potential Φ ( r ) {\displaystyle \Phi (\mathbf {r} )} that satisfies Poisson's equation: ε ∇ 2 Φ ( r )
Debye_length
{u}}_{g}\right)=0} where ϕ {\displaystyle \phi } is a porosity of the porous medium, S w {\displaystyle S_{w}} is a water saturation, S o , S g {\displaystyle
Black-oil_equations
modeling codes. HydroGeoSphere assumes that the subsurface flow equation in a porous medium is always solved during a simulation, either for fully saturated
HydroGeoSphere
Ratio of void volume and total volume of a porous material
hydrogeology, soil science, and building science, the porosity of a porous medium (such as rock or sediment) describes the fraction of void space in the
Porosity
Branch of mathematical analysis
new equation for groundwater flow. This equation[clarification needed] has been shown useful for modeling contaminant flow in heterogenous porous media
Fractional_calculus
Angle between a liquid–vapor interface and a solid surface
Stefano, Siboni (2006). "Wettability of porous materials. II. Can we obtain the contact angle from the Washburn equation?". In Mittal, K. L. (ed.). Contact
Contact_angle
through porous mediums, J. Appl. Phys., 1(5), 318–333. Ross, P.J., and J.-Y. Parlange (1994). Comparing exact and numerical solutions of Richards equation for
Finite water-content vadose zone flow method
Finite_water-content_vadose_zone_flow_method
Electrokinetic potential in colloidal dispersions
potential of particulates, whereas streaming potential/current is used for porous bodies and flat surfaces. In practice, the zeta potential of dispersion
Zeta_potential
Multivalued function in mathematics
Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium" (PDF). PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir
Lambert_W_function
Diffusion process with a non-linear relationship to time
diffusion process in inhomogeneous or heterogeneous medium, e.g. porous media. Fractional diffusion equations were introduced in order to characterize anomalous
Anomalous_diffusion
Process by which water on the ground surface enters the soil
S2CID 94729063. Richards, L. A. (1931). "Capillary Conduction of Liquids Through Porous Mediums". Physics. 1 (5): 318–333. Bibcode:1931Physi...1..318R. doi:10.1063/1
Infiltration_(hydrology)
Method of studying partial differential equations
physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as ∇ ⋅ ( A ( x → ϵ ) ∇ u ϵ
Asymptotic_homogenization
Propensity of a medium to attract or repel liquid
capillarity. The sorptivity is widely used in characterizing soils and porous construction materials such as brick, stone and concrete. Calculation of
Sorptivity
Type of natural water upwelling
Darcy's law states that the volume of flow of the pore fluid through a porous medium per unit time is proportional to the rate of change of excess fluid
Seepage
Soft, siliceous sedimentary rock
reinforcing filler in plastics and rubber, anti-block in plastic films, porous support for chemical catalysts, cat litter, activator in coagulation studies
Diatomaceous_earth
Concept in mathematics
boundary problems. Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions
Stefan_problem
Gas flow with a relatively large mean free molecular path
the medium vacuum range where λ ≈ d. Gas flow can be grouped in four regimes: For Kn≤0.001, flow is continuous, and the Navier–Stokes equations are applicable
Free_molecular_flow
Relation between seismic amplitude and wave travel distance
{\rho }} = density in medium; Δ ρ {\displaystyle {{\Delta }{\rho }}} = density contrast across interface; In the Shuey equation, R(0) is the reflection
Amplitude_versus_offset
Hydrologic simulation software suite
analysis of water flow, heat and solute transport in variably saturated porous media (e.g., soils). HYDRUS suite of software is supported by an interactive
Hydrus_(software)
Branch of physical chemistry
between the two electrolytes, the liquid junction can be provided through a porous plug that allows ion flow while minimizing electrolyte mixing. To further
Electrochemistry
Concept in fluid mechanics
governing equation of Hele-Shaw flows is identical to that of the inviscid potential flow and to the flow of fluid through a porous medium (Darcy's law)
Hele-Shaw_flow
Aspects of fluid mechanics involving fluid flow
for flow in porous media, and works with variables averaged over several pore-widths. In rotating systems, the quasi-geostrophic equations assume an almost
Fluid_dynamics
Effective diffusion of a substance enhanced by shear flow, studied in fluid dynamics
is of particular relevance for flows in porous media modelled by Darcy's law. One may derive the Taylor equation using method of averages, first introduced
Taylor_dispersion
Dimensionless measure of a porous material's permeability
In multiphase flow in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is
Relative_permeability
Tool for generating heat and flame by burning propane
to store and more dangerous for the user. For example, acetylene needs a porous material mixed with acetone in the tank for safety reasons and cannot be
Propane_torch
Penetration of a liquid, gas, or vapor through a solid
permeability, and the materials' mass diffusivity. Permeation is modeled by equations such as Fick's laws of diffusion, and can be measured using tools such
Permeation
Reactive transport modeling in porous media refers to the creation of computer models integrating chemical reaction with transport of fluids through the
Reactive transport modeling in porous media
Reactive_transport_modeling_in_porous_media
Light scattering by small particles
constant of the medium ϵ ¯ {\displaystyle {\bar {\epsilon }}} , then any incident light will be scattered according to the following equation I = I 0 π 2
Rayleigh_scattering
British mathematician
systems, applications of stochastic differential equations and stochastic partial differential equations, the Bayesian approach to inverse problems, data
Andrew_M._Stuart
Low-Density Gases
describe non-equilibrium phenomena in rarefied gases, the Boltzmann transport equation must be used, which is the appropriate mathematical tool for this purpose
Rarefied_gas_dynamics
Phenomenon of surface adhesion
flat surface equation may be used as a "standard curve" in the normal tradition of comparison curves, with the exception that the porous sample's early
Adsorption
Engineer, inventor, academic and author
heterogeneous biofilm model using balance equations, exploring the effects of changing biofilm surface geometries and porous media conditions. In a paper with
Kambiz_Vafai
chromatographic column. The column is packed with large particles or with a porous medium to be investigated. Subsequently, the column is flushed with the solvent
Particle_deposition
Motion of charged particles in electric field
(2017). Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound. Elsevier. ISBN 978-0-444-63908-0. Anderson, J.L
Electrophoresis
Effect of a material on light
opposed to wave propagation in general. A medium having this common property may be termed a dispersive medium. Although the term is used in the field of
Dispersion_(optics)
Electric effects caused by ultrasound
Newtonian liquid, or complex heterogeneous dispersion, emulsion or even a porous body. There are several different electroacoustic effects depending on the
Electroacoustic_phenomena
"Bouncing back" of waves at an interface
interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light
Reflection_(physics)
Electrode that is pH-sensitive
described by the semi-empirical Nikolsky-Shultz-Eisenman equation, an extension to the Nernst equation. It is given by: E = E 0 + R T z i F ln [ a i + ∑
Glass_electrode
Electronic component
1957 when H. Becker developed a "Low voltage electrolytic capacitor with porous carbon electrodes". He believed that the energy was stored as a charge in
Capacitor
Phenomenon in physics
regarded as incompressible. The unsteady, incompressible boundary-layer equation is ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y − ν ∂ 2 u ∂ y 2 = U ∂ U ∂ x + ∂ U ∂
Acoustic_streaming
particle density, ρo the medium density, λ the specific volume conductivity, and η the viscosity. Smoluchowski developed the equation under five assumptions:
Sedimentation_potential
Norwegian scientist
(1992). "Instability of Buckley-Leverett flow in a heterogeneous medium". Transport in Porous Media. 9 (3): 165–185. Bibcode:1992TPMed...9..165L. doi:10.1007/BF00611965
Hans_Petter_Langtangen
nitrocellulose fibers that transport fluid from an inlet through the porous medium to a desired outlet or region of the device, by means of capillary action
Paper-based_microfluidics
Machine learning framework
complex data like spatial transcriptomics, predicting multiphase flow in porous media, and carbon dioxide migration simulations. Finally, the operator learning
Neural_operators
Artificial structures such as pavements covered with water-tight materials
like naturally pervious soils; examples of such alternative structures are porous pavements, green roofs and infiltration basins. Rainwater from impervious
Impervious_surface
Pressure of groundwater held within a soil or rock, in gaps between particles
velocity head, or with water flow, as exemplified in Bernoulli's energy equations. Hydrostatic water pressure: resulting from the weight of material above
Pore_water_pressure
Class of chemical substance
to organic ligands to form one-, two-, or three-dimensional, typically porous structures. The ligands may be referred to as "struts" or "linkers", such
Metal–organic_framework
High-capacity electrochemical capacitor
extremely porous "spongy" form of carbon with a high specific surface area. In 1957 H. Becker developed a "Low voltage electrolytic capacitor with porous carbon
Supercapacitor
Equivalence between complementary antenna types
and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media, 2022, Elsevier, 4th edition, section 8.15 Listen
Babinet's_principle
Field of scientific study
Damkohler number less than 1. Darcy's Law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy in
Ecohydrology
Synthetic polymer
of machinery. It is used as a coating on catheters, and, in its expanded porous form (ePTFE), as a graft material in surgery and as an implant in cardiac
Polytetrafluoroethylene
Industrial processes using an electric field
oxide fuel cells EPD techniques are widely employed for the fabrication of porous ZrO2 anodes from powder precursors onto conductive substrates. EPD processes
Electrophoretic_deposition
Technique in analytical chemistry
to improve LC particles, for example the historic Zipax, a superficially porous particle. The 1970s brought about many developments in hardware and instrumentation
High-performance liquid chromatography
High-performance_liquid_chromatography
Form of matter
{\displaystyle R} and the volume V {\displaystyle V} is substituted in to the equation above, separation occurs at the moment when R 3 = 3 r γ 2 g ( ρ 2 − ρ 1
Foam
Pattern formation of bacteria colony shapes
waste secretion, substrate stiffness, and cellular growth. In liquid and porous media, bacteria may form dynamic patterns by swimming toward each other
Bacterial_patterns
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POROUS MEDIUM-EQUATION
POROUS MEDIUM-EQUATION
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POROUS MEDIUM-EQUATION
POROUS MEDIUM-EQUATION
POROUS MEDIUM-EQUATION
POROUS MEDIUM-EQUATION
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