Fluid dynamics is the study of the behavior of fluids (liquids and gases) in motion. The equations that govern fluid dynamics describe how fluids flow and how they interact with forces and boundaries. The main equations governing fluid dynamics are derived from the principles of fluid mechanics and encompass several fundamental concepts, including conservation of mass, momentum, and energy. Here are the key equations in fluid dynamics: 1. **Continuity Equation**: This equation is based on the principle of conservation of mass.
The Allen-Cahn equation is a partial differential equation that describes the evolution of phase interfaces in materials science and represents the dynamics of gas-liquid phase transitions typically in the context of, but not limited to, crystallization processes. It is an example of a conserved order parameter system and is derived from the principles of thermodynamics and variational calculus.
The Basset–Boussinesq–Oseen (BBO) equation is a mathematical model that describes the motion of small particles suspended in a viscous fluid. This equation accounts for the effects of inertial and viscous forces acting on the particles, along with the interaction between the particles and the surrounding fluid. It is particularly important in the fields of fluid mechanics and particle dynamics, especially in scenarios where the Reynolds number is low.
The Batchelor–Chandrasekhar equation is a fundamental equation in the field of fluid dynamics, specifically in the study of turbulence and the behavior of suspensions of small particles in a fluid. It describes the way that particles, such as bubbles or solid particles, interact with the surrounding fluid flow, particularly under conditions of sedimentation or dispersion.
The Benedict–Webb–Rubin (BWR) equation is a thermodynamic model used to describe the behavior of gases, particularly mixtures and non-ideal gas mixtures. It is a more complex equation of state compared to the ideal gas law, allowing for the incorporation of molecular interactions and the effects of pressure and temperature on gas behavior.
The Benjamin–Bona–Mahony (BBM) equation is a mathematical model that describes wave propagation in shallow water. It is a simplified equation derived from the full water wave equations, particularly focusing on long waves with small amplitude. The BBM equation incorporates both nonlinearity and dispersion, making it a significant model in the study of wave phenomena.
Bernoulli's principle is a fundamental concept in fluid dynamics that describes the behavior of a fluid moving along a streamline. Formulated by the Swiss mathematician Daniel Bernoulli in the 18th century, the principle states that in a steady flow of an incompressible, non-viscous fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy in that flow.
The term "black oil equations" refers to a set of mathematical relations used in reservoir engineering and petroleum production to model the behavior of black oil, a type of crude oil characterized by its relatively high viscosity and the presence of dissolved gases and lighter hydrocarbon components. Black oil models help in understanding and predicting the behavior of oil reservoirs during production.
The Borda-Carnot equation describes the relationship between the temperature, pressure, and specific properties of a fluid in a thermodynamic context, particularly for a fluid undergoing adiabatic (no heat transfer) expansion or compression. It is commonly associated with the performance of turbines and compressors. The equation itself typically relates how the enthalpy, pressure, and temperature of the fluid change during these processes.
The Bosanquet equation is a mathematical expression used in the field of fluid dynamics and rheology to model the steady-state flow of non-Newtonian fluids. It is particularly relevant for describing the flow behavior of viscoelastic fluids, which exhibit both viscous and elastic characteristics.
The Boussinesq approximation is a mathematical simplification used in fluid dynamics, particularly in the study of weakly non-linear and dispersive wave phenomena, such as water waves. Named after the French physicist Joseph Boussinesq, this approximation is particularly useful for analyzing the behavior of surface waves in fluids where the amplitude of the waves is small compared to the wavelength.
The Buckley–Leverett equation is a fundamental equation in petroleum engineering and reservoir engineering that describes the movement of two-phase fluids (typically oil and water) in porous media. It models the flow behavior of immiscible fluids in a reservoir when one fluid displaces another, commonly used to analyze waterflooding operations during oil recovery. The equation is derived from the conservation of mass principle and reflects the dynamics of the interfaces between the two fluids.
Burgers' equation is a fundamental partial differential equation in fluid mechanics and mathematics. It is named after the Dutch physicist Johannes Burgers, who introduced it in his study of turbulence and other fluid dynamics phenomena. The equation can be seen as a simplification of the Navier-Stokes equations, which govern fluid motion.
The Cahn–Hilliard equation is a partial differential equation that describes the phase separation and motion of interfaces in a binary mixture or alloy. It is particularly important in materials science, as it models the process by which two phases of a material (such as solid and liquid) separate from each other, leading to the formation of distinct microstructures over time. The equation was introduced by John W. Cahn and John E.
The Camassa-Holm equation is a nonlinear partial differential equation that describes the dynamics of shallow water waves. It was first introduced by Roberta Camassa and Darryl Holm in their 1993 paper. The equation models unidirectional wave propagation and is noteworthy for its ability to describe solitary waves, which can maintain their shape while traveling at constant speeds.
The continuity equation is a fundamental principle in fluid dynamics and other fields that describes the transport of some quantity (such as mass, energy, or charge) in a system. It expresses the idea that, in a closed system, the rate at which a quantity enters a volume must equal the rate at which it leaves that volume, plus any accumulation of that quantity within the volume.
The Darcy friction factor, often denoted as \( f \), is a key component in the Darcy-Weisbach equation, which is used to calculate pressure loss (or head loss) due to friction in a pipe or duct.
The Darcy–Weisbach equation is used in fluid mechanics to calculate the pressure loss (or head loss) due to friction in a pipeline or duct. It is an essential equation for engineers and designers working with fluid flow systems to assess the efficiency and performance of piping and ductwork.
The Davey-Stewartson equation is a nonlinear partial differential equation that arises in the study of wave phenomena, particularly in the context of two-dimensional surface water waves. It is a generalization of the nonlinear Schrödinger equation and describes the evolution of complex wave packets in a two-dimensional setting.
The drag equation is a mathematical formula used to calculate the drag force experienced by an object moving through a fluid, such as air or water. The drag force (F_d) is the resistance experienced by the object due to the fluid surrounding it.
The Euler–Tricomi equation is a second-order partial differential equation (PDE) that arises in various fields, including fluid dynamics and mathematical physics. It is named after the mathematicians Leonhard Euler and Francesco Tricomi.
The Fanning friction factor is a dimensionless quantity used in fluid mechanics to characterize the frictional resistance to flow in a pipe or duct. It is defined as the ratio of the wall shear stress to the dynamic pressure of the fluid. The Fanning friction factor (\(f\)) is commonly used in the analysis of laminar and turbulent flow regimes and plays a crucial role in the calculation of pressure losses due to friction in piping systems.
Faxén's law describes the force experienced by a spherical particle suspended in a fluid when it is subjected to an external oscillating field, such as a pressure gradient or a fluid flow. It is particularly relevant in the study of colloidal suspensions and the behavior of particles in non-Newtonian fluids.
The Hadamard–Rybczynski equations describe the motion of a fluid in a gravitational field, particularly in the context of fluid dynamics. These equations are important in studying the behavior of inviscid and incompressible fluids, especially when analyzing potential flow around bodies. The Hadamard–Rybczynski equations relate the velocity potential or stream function to the shape of the body and the flow conditions around it.
The Hasegawa–Mima equation is a nonlinear partial differential equation that arises in the study of plasma physics, particularly in the context of magnetically confined plasmas, such as those found in fusion reactors. It describes the evolution of certain wave phenomena in a magnetized plasma, specifically the dynamics of plasma turbulence and the behavior of density perturbations.
The Hazen–Williams equation is an empirical formula used to calculate the flow of water through pipes, specifically in civil engineering and hydraulics. It estimates the head loss (pressure loss due to friction) in a pipe based on the flow rate, pipe diameter, and the roughness of the pipe's interior surface. The equation is particularly applicable for water flow in pipes where the flow is turbulent. The general form of the Hazen–Williams equation is: \[ h_f = 0.
A Herschel–Bulkley fluid is a type of non-Newtonian fluid that exhibits both yield stress and shear-thinning (or shear-thickening) behavior. The defining characteristic of such fluids is that they do not begin to flow until a certain threshold stress, known as the yield stress, is exceeded. Once this yield stress is surpassed, the fluid flows according to a power-law relationship that describes its viscosity.
The Kadomtsev–Petviashvili (KP) equation is a fundamental nonlinear partial differential equation (PDE) that describes the propagation of waves in a quasi-one-dimensional medium. It arises in various fields such as fluid dynamics, plasma physics, and nonlinear optics. The equation serves as a higher-dimensional generalization of the Korteweg–de Vries (KdV) equation, which describes solitons in one dimension.
Kelvin's circulation theorem is a fundamental principle in fluid dynamics, particularly in the study of inviscid (non-viscous) and irrotational flows. It states that the circulation around a closed contour moving with the fluid is constant in time, provided the flow is conservative and the fluid is incompressible and inviscid.
The Korteweg–de Vries (KdV) equation is a third-order nonlinear partial differential equation that describes the evolution of waves in shallow water. It is significant in various fields, including fluid dynamics, nonlinear wave theory, and mathematical physics.
The Kozeny-Carman equation is a mathematical model that describes the flow of fluids through porous media. It relates the permeability of a porous material to its porosity and specific surface area. It is widely used in fields such as hydrogeology, petroleum engineering, and soil science to analyze how fluids move through soils and rocks.
The Kármán–Howarth equation is a fundamental relation in fluid dynamics, particularly in the study of turbulence. It describes the evolution of the second-order velocity correlation function in an incompressible flow. The equation provides insight into the relationships between different scales of motion in turbulent flows. In turbulent fluid mechanics, the velocity field can be characterized using correlation functions, which measure the statistical relationships between the velocities at different points in space.
The mild-slope equation is a mathematical representation used in coastal engineering and fluid dynamics to describe the propagation of surface water waves over varying bathymetry (the underwater equivalent of topography). It is especially useful for analyzing wave behavior in coastal areas, where the depth of the water changes gradually.
The Milne-Thomson circle theorem is a concept in fluid dynamics and potential flow theory. It relates to the flow of an ideal fluid (inviscid and incompressible) around obstacles and is particularly useful for analyzing flow patterns around circular objects.
The Morison equation is a mathematical model used in engineering, particularly in the fields of civil and ocean engineering, to estimate the wave forces on structures such as offshore oil platforms, wind turbines, and coastal structures. It accounts for both the inertia and drag forces acting on a structure submerged in a fluid, such as water.
The Orr–Sommerfeld equation is a fundamental equation in fluid dynamics that describes the stability of an incompressible flow, particularly in the context of boundary layer theory. It is named after William Richard Orr and Arnold Sommerfeld, who contributed to its development. The equation arises when analyzing small disturbances or perturbations in a basic flow profile. It is particularly important in studying the stability of laminar flows and understanding transition to turbulence.
The Oseen equations are a set of equations that describe the steady-state flow of a viscous fluid. They can be seen as a linearization of the Navier-Stokes equations, which govern the motion of fluid substances. The Oseen equations are particularly useful in the study of low Reynolds number flows, where inertial forces are negligible compared to viscous forces.
The Rankine–Hugoniot conditions are a set of mathematical conditions used in fluid dynamics and gas dynamics to describe the behavior of shock waves and discontinuities in a medium. These conditions relate the values of physical quantities (such as pressure, density, and velocity) on either side of a discontinuity, which can be a shock wave or a contact discontinuity.
Rayleigh's equation in fluid dynamics refers to a fundamental principle that describes the stability of a fluid flow. It is often associated with the stability analysis of boundary layers and the onset of turbulence and instabilities in various fluid flow situations. One common context in which Rayleigh's equation is discussed is in the study of stability of various flow regimes, particularly in relation to the growth of instabilities in a shear flow. The equation is typically derived from the Navier-Stokes equations under specific assumptions and conditions.
The Rayleigh–Plesset equation is a fundamental equation in the field of fluid dynamics, particularly in the study of cavitation—a phenomenon where vapor bubbles form and collapse in a liquid. The equation describes the dynamics of a spherical gas bubble in an incompressible liquid, accounting for the effects of pressure, surface tension, and viscous forces.
The relativistic Euler equations are a set of equations that describe the dynamics of perfect fluids in the context of relativistic physics. They extend the classical Euler equations, which govern the flow of inviscid (non-viscous), incompressible fluids, to situations where the speeds involved approach the speed of light, or in contexts where relativistic effects are significant, such as in astrophysics or cosmology.
The Shallow Water Equations (SWE) are a set of hyperbolic partial differential equations that describe the flow of a thin layer of fluid, such as water in rivers, lakes, and coastal areas. These equations are particularly useful in hydraulic and environmental engineering for modeling phenomena like flooding, tsunami propagation, and sediment transport. The SWE are derived under the assumption that the horizontal length scale of the fluid flow is much larger than the vertical scale of the fluid depth.
Stokes' paradox refers to a phenomenon in fluid dynamics that highlights an apparent inconsistency in the flow of a viscous fluid around an object. The paradox is named after the British mathematician and physicist George Gabriel Stokes who analyzed the flow of a viscous (incompressible) fluid around a cylinder. The paradox arises when considering a two-dimensional flow of a viscous fluid past an infinitely long, solid cylinder.
Stokes flow refers to the flow of an incompressible viscous fluid at low Reynolds numbers, where inertial forces can be neglected in comparison to viscous forces. This type of flow is governed by the Stokes equations, which are simplified forms of the Navier-Stokes equations. These equations assume that the fluid is Newtonian, meaning its viscosity is constant and the stress is linearly proportional to the rate of strain.
Stream thrust averaging is a method used in fluid dynamics and aerodynamics to analyze and predict the performance of airfoils, wings, or propellers by averaging the thrust output over a certain stream-wise length or area. This technique is particularly useful in assessing the overall efficiency and behavior of a propulsion system, such as jet engines or helicopters, as it helps to understand how thrust is distributed and how it varies with different operating conditions.
The Taylor–Goldstein equation is a fundamental equation in fluid dynamics and hydrodynamic stability theory, particularly in the study of parallel flows and stability analyses of shear flows. It derives from the linear stability analysis of a basic state in a fluid that is affected by small disturbances.
The Taylor–von Neumann–Sedov (TNNS) blast wave is a theoretical model describing the propagation of a shock wave resulting from an explosion in a homogeneous medium. It is named after three scientists who contributed to the understanding of this phenomenon: G.I. Taylor, J. von Neumann, and L.I. Sedov. The TNNS blast wave model provides a framework for understanding the dynamics of the shock wave and the resulting flow fields in the vicinity of the explosion.
The thin-film equation describes the evolution of a thin liquid film, typically on a solid substrate. This equation is important in fluid dynamics and materials science and is often used in contexts such as coatings, wetting, and thin-film flow dynamics. The thin-film equation can be derived from the Navier-Stokes equations under certain assumptions, specifically when considering a thin film with small thickness compared to its other dimensions.
Washburn's equation describes the capillary action of liquids in porous media or thin tubes. It quantifies the rate at which a liquid will diffuse into a porous material due to capillary forces. The equation is often used in the context of materials science, fluid mechanics, and petroleum engineering, among other fields.
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