In surface chemistry, **disjoining pressure** (symbol Π_{d}) according to an IUPAC definition^{ [1] } arises from an attractive interaction between two surfaces. For two flat and parallel surfaces, the value of the disjoining pressure (i.e., the force per unit area) can be calculated as the derivative of the Gibbs energy of interaction per unit area in respect to distance (in the direction normal to that of the interacting surfaces). There is also a related concept of **disjoining force**, which can be viewed as disjoining pressure times the surface area of the interacting surfaces.

The concept of disjoining pressure was introduced by Derjaguin (1936) as the difference between the pressure in a region of a phase adjacent to a surface confining it, and the pressure in the bulk of this phase.^{ [2] }^{ [3] }

Disjoining pressure can be expressed as:^{ [4] }

where (in SI units):

- Π
_{d}- disjoining pressure (N/m^{2}) - A - the surface area of the interacting surfaces (m
^{2}) - G - total Gibbs energy of the interaction of the two surfaces (J)
- x - distance (m)
- indices T, V and A signify that the temperature, volume, and the surface area remain constant in the derivative.

Using the concept of the disjoining pressure, the pressure in a film can be viewed as:^{ [4] }

where:

- P - pressure in a film (Pa)
*P*_{0}- pressure in the bulk of the same phase as that of the film (Pa)

Disjoining pressure is interpreted as a sum of several interactions: dispersion forces, electrostatic forces between charged surfaces, interactions due to layers of neutral molecules adsorbed on the two surfaces, and the structural effects of the solvent.

Classic theory predicts that the disjoining pressure of a thin liquid film on a flat surface as follows,^{ [5] }

where:

- A
_{H}- Hamaker constant (J) *δ*_{0}- liquid film thickness (m)

For a solid-liquid-vapor system where the solid surface is structured, the disjoining pressure is affected by the solid surface profile, *ζ*_{S}, and the meniscus shape, *ζ*_{L}^{ [6] }

where:

*ω*(*ρ*,*z*) - solid-liquid potential (J/m^{6})

The meniscus shape can be by minimization of total system free energy as follows ^{ [7] }

where:

*W*_{total}- total system free energy including surface excess energy and free energy due to solid-liquid interactions (J/m^{2})*ζ*_{L}- meniscus shape (m)*ζ'*_{L}- slope of meniscus shape (1)

In the theory of liquid drops and films, the disjoining pressure can be shown to be related to the equilibrium liquid-solid contact angle θ_{e} through the relation^{ [8] }

where γ is the liquid-vapor surface tension and *h*_{0} is the precursor film thickness.

In physics, the **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).

In fluid mechanics, the **Grashof number** is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number.

In surface science, **surface free energy** quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less energetically favorable than the bulk of the material, otherwise there would be a driving force for surfaces to be created, removing the bulk of the material. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. There is "excess energy" as a result of the now-incomplete, unrealized bonding at the two surfaces.

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In materials science, **shear modulus** or **modulus of rigidity**, denoted by *G*, or sometimes *S* or *μ*, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:

In mathematics, the **von Mangoldt function** is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.

In mathematics, the **explicit formulae for L-functions** are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.

In physics, precisely in the study of the theory of general relativity and many alternatives to it, the **post-Newtonian formalism** is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

**Local-density approximations** (**LDA**) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space. Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems.

The **Kelvin equation** describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.

**Contact mechanics** is the study of the deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces and frictional stresses acting tangentially between the surfaces. **Normal contact mechanics** or **frictionless contact mechanics** focuses on normal stresses caused by applied normal forces and by the adhesion present on surfaces in close contact, even if they are clean and dry. *Frictional contact mechanics* emphasizes the effect of friction forces.

**Diffusiophoresis** is the spontaneous motion of colloidal particles or molecules in a fluid, induced by a concentration gradient of a different substance. In other words, it is motion of one species, A, in response to a concentration gradient in another species, B. Typically, A is colloidal particles which are in aqueous solution in which B is a dissolved salt such as sodium chloride, and so the particles of A are much larger than the ions of B. But both A and B could be polymer molecules, and B could be a small molecule. For example, concentration gradients in ethanol solutions in water move 1 μm diameter colloidal particles with diffusiophoretic velocities of order 0.1 to 1 μm/s, the movement is towards regions of the solution with lower ethanol concentration. Both species A and B will typically be diffusing but diffusiophoresis is distinct from simple diffusion: in simple diffusion a species A moves down a gradient in its own concentration.

In mathematics, **Reidemeister torsion** is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by Wolfgang Franz (1935) and Georges de Rham (1936). **Analytic torsion** is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer as an analytic analogue of Reidemeister torsion. Jeff Cheeger and Werner Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.

**Acoustic streaming** is a steady flow in a fluid driven by the absorption of high amplitude acoustic oscillations. This phenomenon can be observed near sound emitters, or in the standing waves within a Kundt's tube. Acoustic streaming was explained first by Lord Rayleigh in 1884. It is the less-known opposite of sound generation by a flow.

In nonideal fluid dynamics, the **Hagen–Poiseuille equation**, also known as the **Hagen–Poiseuille law**, **Poiseuille law** or **Poiseuille equation**, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846. The theoretical justification of the Poiseuille law was given by George Stokes in 1845.

In fluid dynamics, the **mild-slope equation** describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

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In fluid dynamics, a **cnoidal wave** is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function *cn*, which is why they are coined *cn*oidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

**Flotation of flexible objects** is a phenomenon in which the bending of a flexible material allows an object to displace a greater amount of fluid than if it were completely rigid. This ability to displace more fluid translates directly into an ability to support greater loads, giving the flexible structure an advantage over a similarly rigid one. Inspiration to study the effects of elasticity are taken from nature, where plants, such as black pepper, and animals living at the water surface have evolved to take advantage of the load-bearing benefits elasticity imparts.

In physics and engineering, the **thin-film equation** is a partial differential equation that approximately predicts the time evolution of the thickness *h* of a liquid film that lies on a surface. The equation is derived via lubrication theory which is based on the assumption that the length-scales in the surface directions are significantly larger than in the direction normal to the surface. In the non-dimensional form of the Navier-Stokes equation the requirement is that terms of order and are negligible, where is the aspect ratio and is the Reynold's number. This significantly simplifies the governing equations. However lubrication theory, as the name suggests, is typically derived for flow between two solid surfaces hence the liquid forms a lubricating layer. The thin-film equation holds when there is a single free surface. With two free surfaces the flow must be treated as a viscous sheet.

- ↑ ""Disjoining pressure". Entry in the IUPAC Compendium of Chemical Terminology ("The Gold Book"), the International Union of Pure and Applied Chemistry".
- ↑ See:
- Дерягин, Б. В. and Кусаков М. М. (Derjaguin, B. V. and Kusakov, M. M.) (1936) "Свойства тонких слоев жидкостей" (The properties of thin layers of liquids),
*Известия Академии Наук СССР, Серия Химическая*(Proceedings of the Academy of Sciences of the USSR, Chemistry series),**5**: 741-753. - Derjaguin, B. with E. Obuchov (1936) "Anomalien dünner Flussigkeitsschichten. III. Ultramikrometrische Untersuchungen der Solvathüllen und des "elementaren" Quellungsaktes" (Anomalies of thin liquid layers. III. Investigations via ultramicroscope measurements of solvation shells and of the "elementary" act of imbibition),
*Acta Physicochimica U.R.S.S.*,**5**: 1-22.

- Дерягин, Б. В. and Кусаков М. М. (Derjaguin, B. V. and Kusakov, M. M.) (1936) "Свойства тонких слоев жидкостей" (The properties of thin layers of liquids),
- ↑ A. Adamson, A. Gast, "Physical Chemistry of Surfaces", 6th edition, John Wiley and Sons Inc., 1997, page 247.
- 1 2 Hans-Jürgen Butt, Karlheinz Graf, Michael Kappl,"Physics and chemistry of interfaces", John Wiley & Sons Canada, Ltd., 1 edition, 2003, page 95 (Google books)
- ↑ Jacob N. Israelachvili,"Intermolecular and Surface Forces", Academic Press, Revised Third edition, 2011, page 267-268 (Google books)
- ↑ Robbins, Mark O.; Andelman, David; Joanny, Jean-François (April 1, 1991). "Thin liquid films on rough or heterogeneous solids".
*Physical Review A*.**43**(8): 4344–4354. doi:10.1103/PhysRevA.43.4344. PMID 9905537. - ↑ Hu, Han; Weinberger, Christopher R.; Sun, Ying (December 10, 2014). "Effect of Nanostructures on the Meniscus Shape and Disjoining Pressure of Ultrathin Liquid Film".
*Nano Letters*.**14**(12): 7131–7137. doi:10.1021/nl5037066. PMID 25394305. - ↑ Churaev, N. V.; Sobolev, V. D. (January 1, 1995). "Prediction of contact angles on the basis of the Frumkin-Derjaguin approach".
*Advances in Colloid and Interface Science*.**61**: 1–16. doi:10.1016/0001-8686(95)00257-Q.

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