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Fluid

A fluid is a state of matter that yields to shearing forces, continuously deforming or flowing when subjected to an applied shear stress, encompassing substances such as liquids, gases, and plasmas but excluding solids that can resist such forces without permanent deformation. Unlike solids, fluids cannot sustain tangential or shear stresses when at rest and instead undergo continuous change in shape under these conditions. This defining characteristic arises from the molecular structure of fluids, where particles have sufficient mobility to flow rather than maintain rigid positions, distinguishing them from the fixed lattice of solids. Fluids are fundamental to numerous natural and engineered phenomena, governing processes from and ocean currents to flow in biological systems and in vehicles. The study of fluids falls under , a branch of physics divided into fluid statics—which examines fluids at rest, such as hydrostatic variations with depth—and , which analyzes motion, including laminar and turbulent flows. Key properties of fluids include (mass per unit volume, varying with and ), (resistance to flow, higher in liquids like than in gases like air), (ability to change volume under , low in liquids and high in gases), and (cohesive force at the surface, enabling phenomena like ). These properties are quantified using assumptions for macroscopic behavior, treating fluids as continuous media despite their molecular composition. In practical applications, fluids exhibit behaviors modeled by equations like the Navier-Stokes equations for viscous flows or for ideal, inviscid motion along streamlines, influencing fields from (e.g., pipeline design) to (e.g., weather prediction). Newtonian fluids, such as , follow linear stress-strain relationships, while non-Newtonian fluids like or display variable under stress. Understanding fluids is essential for addressing real-world challenges, including energy transport, environmental flows, and biomedical innovations.

Fundamentals

Definition

A fluid is a substance that deforms continuously and permanently under the application of a ing , no matter how small the magnitude of that may be. This continuous deformation, known as , distinguishes fluids from , which can resist small es without ongoing deformation. ing refers to per unit area acting parallel to a surface, leading to or relative displacement between layers of the material. In contrast to , which achieve static under by deforming only elastically up to a point, fluids cannot sustain in and instead flow indefinitely to relieve it. This fundamental behavior allows fluids to adapt their shape to conform to the boundaries of their container or applied forces, enabling phenomena such as pouring or movement. Common examples of fluids include liquids like water, which maintain a fixed volume but variable shape, and gases like air, which have neither fixed volume nor shape. Plasmas, consisting of ionized gases, also exhibit fluid-like behavior and are treated as fluids in contexts such as plasma physics, though their charged nature introduces additional electromagnetic interactions.

Classification

Fluids are broadly classified by their phase of matter, which determines their physical behavior under varying conditions of temperature and pressure. Liquids, such as water, are generally incompressible and maintain a fixed volume while conforming to the shape of their container. Gases, like air, are compressible and expand to fill the entire volume of their container. Supercritical fluids occur beyond a substance's critical point, exhibiting gas-like low viscosity combined with liquid-like high density, enabling unique solvent properties without distinct liquid-gas interfaces. Rheological classification distinguishes fluids based on their response to , particularly how varies with the rate of . Newtonian fluids exhibit a linear relationship between and , maintaining constant regardless of applied force; examples include and . Non-Newtonian fluids deviate from this linearity, with changing under : shear-thinning fluids like decrease in viscosity with increasing , shear-thickening fluids such as cornstarch slurries increase in viscosity, and Bingham plastics like require a yield to initiate flow. This classification originated with Isaac Newton's 1687 formulation in , where he described ideal viscous behavior, while observations of deviations in the 19th century, notably by Osborne Reynolds, expanded understanding to encompass non-Newtonian types. Additional categories include ideal versus real fluids and single-phase versus multiphase systems. Ideal fluids are theoretical constructs that are incompressible and possess zero , simplifying analyses in . Real fluids, in contrast, exhibit and to varying degrees, reflecting actual substances like oils or air. Single-phase fluids consist of one uniform , such as pure , while multiphase fluids involve mixtures of phases, exemplified by emulsions (dispersed liquid droplets in another liquid) and foams (gas bubbles dispersed in a liquid).

Physical Properties

Density and Compressibility

Density is a fundamental property of fluids, defined as the mass per unit volume of the substance, expressed as \rho = \frac{m}{V}, where m is and V is . The SI unit of density is kilograms per cubic meter (kg/m³). In fluids, density generally varies with and , with most liquids and gases becoming less dense as temperature increases due to . For gases that approximate ideal behavior, density derives from the PV = nRT, where P is , V is , n is the number of moles, R is the universal , and T is absolute . Substituting n = \frac{m}{M} (with M as ) yields \rho = \frac{PM}{RT}. This relation highlights how gas density increases with and but decreases with . Specific gravity provides a dimensionless measure of a fluid's relative to that of pure at (where water's is approximately 1000 kg/m³), defined as the ratio \frac{\rho_{\text{fluid}}}{\rho_{\text{water}}}. It is particularly useful in contexts for comparing fluid densities without units. describes the ability of a fluid to change volume under at constant , quantified by the isothermal \beta = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T. Liquids exhibit very low , meaning their volume changes minimally with ; for at , \beta \approx 4.5 \times 10^{-10} ^{-1}. In contrast, gases are highly compressible; for an , \beta = \frac{1}{P}. The bulk modulus K, which is the reciprocal of compressibility (K = \frac{1}{\beta} = -V \left( \frac{\partial P}{\partial V} \right)_T), measures a fluid's resistance to uniform compression. For water, K \approx 2.2 GPa, indicating high resistance, whereas for air under isothermal conditions at standard pressure, K \approx 10^5 Pa, reflecting easy compressibility. These properties underpin calculations of buoyant forces in fluid systems.

Viscosity and Rheology

Viscosity represents the measure of a fluid's to , arising from intermolecular forces that oppose the relative motion of fluid layers. It quantifies the "" within the fluid, where higher indicates greater resistance to or deformation. The dynamic viscosity, denoted as μ, is defined in SI units as pascal-seconds (Pa·s), which equals newton-seconds per square meter (N·s/m²). Kinematic viscosity, ν, accounts for and is given by ν = μ / ρ, with units of square meters per second (m²/s); it is particularly useful in analyses involving gravitational flows. For Newtonian fluids, which obey a linear relationship between shear stress and strain rate, Newton's law of viscosity states that the shear stress τ is proportional to the velocity gradient du/dy: \tau = \mu \frac{du}{dy} This law applies to simple shear flows in low-molecular-weight liquids like water or air, where viscosity remains constant regardless of the applied shear rate. Fluids adhering to this behavior are termed Newtonian, contrasting with more complex materials where viscosity varies under stress. The dependence of differs markedly between liquids and gases. In liquids, typically decreases exponentially with increasing , often modeled by the : \mu = A e^{E_a / RT} where A is a , E_a is the for viscous flow, R is the , and T is the absolute ; this reflects reduced intermolecular at higher temperatures. For gases, increases with , approximately proportional to the of T due to enhanced molecular transfer in kinetic theory. Rheology is the scientific study of the and deformation of under applied , encompassing both viscous and elastic responses in materials ranging from simple fluids to complex solids. It extends beyond basic to examine time-dependent behaviors, such as in viscoelastic fluids where deformation involves both energy dissipation (viscous ) and storage ( recovery). is prominent in complex fluids like solutions or melts, where long-chain molecules enable partial reversibility of deformation, leading to phenomena like or . Representative examples illustrate these concepts. exhibits high , often around 10–100 Pa·s at , and displays non-Newtonian shear-thinning behavior, where its decreases under increasing rates due to its complex composition of sugars and . , such as 10W-40 grade, demonstrate strong temperature sensitivity: the "10W" ensures the dynamic at cold-cranking temperatures (e.g., -25°C) does not exceed 7 Pa·s (typically 5–6 Pa·s per cold-cranking simulator tests) to facilitate easy engine startup, while "40" specifies a kinematic at 100°C equivalent to a dynamic of about 0.01–0.015 Pa·s, providing adequate film strength for at operating temperatures.

Fluid Statics

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at rest due to the force of gravity acting on the fluid's weight. In a static fluid, this pressure increases linearly with depth and acts equally in all directions at any given point. The fundamental relationship is described by the hydrostatic equation, P = \rho g h, where P is the pressure, \rho is the fluid density, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the vertical depth below the free surface. This equation holds for incompressible fluids like water under typical conditions and assumes constant density. A key insight from this is the hydrostatic , which demonstrates that the at a given depth depends solely on the height of the fluid column above that point, independent of the container's shape or the total volume of fluid. For instance, vessels of varying geometries—such as a narrow , a wide , or a conical flask—filled to the same depth will exert identical on their bases, despite differing amounts of fluid and thus total weights. This counterintuitive result arises because the is transmitted through the fluid's weight per unit area, not the overall mass supported by the container walls. Pascal's law, also known as Pascal's principle, extends this concept by stating that any external applied to an enclosed fluid is transmitted undiminished and equally in all directions throughout the fluid and to the walls of the containing vessel. Formulated by in the , this principle underpins hydraulic systems, where a small force on a confined area can generate a large force on a larger area due to the equality. It applies strictly to fluids at rest and incompressible conditions, ensuring uniform distribution without loss. Pressure in hydrostatic contexts is commonly measured using manometers and barometers, which rely on the balance between fluid columns under gravity. A manometer typically consists of a U-shaped tube partially filled with a liquid like mercury or water, where the difference in column heights indicates pressure differences between two points, calculated via \Delta P = \rho g \Delta h. Barometers, such as the mercury barometer, measure absolute by supporting a column of mercury against a vacuum, with standard sea-level pressure equivalent to about 760 mmHg or 101.325 kPa. Distinctions between absolute and pressure are essential for accurate hydrostatic applications. Absolute pressure (P_{abs}) is measured relative to (zero pressure), while pressure (P_{gauge}) is relative to local , related by P_{abs} = P_{gauge} + P_{atm}. In , for example, hydrostatic increases by approximately 10 kPa per meter of depth in , so absolute pressure at 10 m is about 201.3 kPa (101.3 kPa atmospheric plus 100 kPa hydrostatic), whereas pressure would read 100 kPa. This differentiation ensures proper accounting of environmental conditions in pressure-sensitive scenarios.

Buoyancy

Buoyancy refers to the upward force exerted by a fluid on an object immersed in it, arising from the pressure difference between the object's upper and lower surfaces. This phenomenon, fundamental to fluid statics, enables objects to float or influences their submersion behavior. The concept traces its origins to , where discovered the principle around 250 BCE while investigating the purity of a gold crown for King ; stepping into a , he observed displacement and realized the buoyant force equals the weight of the displaced fluid, famously exclaiming "!" as described in ' accounts. Archimedes' principle states that the magnitude of the buoyant force F_b on a fully or partially submerged object is equal to the weight of the fluid displaced by the object, given by F_b = \rho_\text{fluid} V_\text{displaced} g, where \rho_\text{fluid} is the fluid density, V_\text{displaced} is the volume of displaced fluid, and g is . This force acts vertically upward through the centroid of the displaced volume, known as the center of buoyancy. The principle derives from integrating the hydrostatic over the object's surface, where pressure increases with depth, resulting in a net upward force equivalent to the displaced fluid's weight; this integration aligns with the hydrostatic pressure distribution detailed in fluid fundamentals. For an object in equilibrium, it floats if its average density is less than that of the surrounding fluid, as the buoyant force then balances the object's weight with partial submersion; conversely, it sinks if denser, fully displacing its volume without sufficient buoyancy. In naval architecture, ship stability depends on the relative positions of the center of gravity and center of buoyancy; when a ship heels, the metacenter—the intersection point of the vertical buoyant force line in the heeled position with the centerline—must lie above the center of gravity for restoring torque and upright stability. A practical application of buoyancy control appears in , which adjust by flooding or emptying tanks with to increase or decrease displaced volume, allowing submersion or surfacing without propulsion.

Fluid Dynamics

Kinematics of Fluid Motion

Fluid kinematics is the branch of that examines the geometric properties of fluid motion, such as , , and particle trajectories, without regard to the forces causing the motion. It focuses on describing how fluid elements move through and time, providing the foundational framework for understanding flow patterns. This description is essential for analyzing phenomena like and , independent of material properties like variations. Fluid motion can be described using two primary approaches: the and Eulerian descriptions. In the description, the observer follows individual fluid particles as they traverse their paths, tracking changes in properties for specific material elements labeled by their initial positions. This method is akin to following a particular parcel in a , where the of the parcel is given by the time of its , \mathbf{V}_L = \frac{\partial \boldsymbol{\xi}}{\partial t}, with \boldsymbol{\xi} denoting the particle's . Conversely, the Eulerian description examines the flow field at fixed points in space, measuring quantities like as functions of and time, \mathbf{V}_E(\mathbf{x}, t). This fixed-frame approach, common in theoretical analyses, relates to the view through the , \frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{V} \cdot \nabla, which accounts for both local changes and convective effects. The Eulerian method is preferred for multidimensional flows due to its mathematical simplicity, while tracking is useful for conservation laws and trajectory computations. Key geometric features of fluid motion include streamlines, pathlines, and streaklines, which illustrate flow direction and particle histories. Streamlines are instantaneous curves tangent to the velocity vector field \mathbf{V}(\mathbf{x}, y, z, t) at every point, satisfying d\mathbf{s} \times \mathbf{V} = 0 or, in two dimensions, \frac{dy}{dx} = \frac{v}{u}. They represent the direction of flow at a given moment and form streamtubes that enclose constant mass flux. Pathlines trace the actual trajectory of a single fluid particle over time, obtained by integrating the velocity field from an initial position, such as x(t) = x_0 + \int_{t_0}^t u \, d\tau. Streaklines connect all particles that have passed through a fixed point at different times, like the smoke trail from a chimney, and evolve dynamically with the flow. In steady flows, where velocity does not vary with time, these three lines coincide, simplifying flow visualization. For incompressible flows, where is constant, the enforces conservation and imposes a kinematic on the : \nabla \cdot \mathbf{v} = 0. This divergence-free condition arises from balancing fluxes across a differential , using Taylor expansions of components around the volume's center; for steady incompressible , it simplifies to \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0. The equation ensures no net accumulation of , implying that the flow is solenoidal and volume-preserving, which is crucial for analyzing potential flows and boundary conditions. Fluid flows are classified based on temporal and spatial variations, as well as flow regime. Steady flows exhibit time-independent at fixed points (\frac{\partial P}{\partial t} = 0), while unsteady flows vary temporally, often characterized by a reduced frequency parameter k = \omega L / V where k < 0.05 approximates quasi-steady conditions. Uniform flows have constant across streamlines, whereas non-uniform flows vary spatially, such as in developing . Laminar flows feature smooth, parallel layers with negligible mixing, predominant at low speeds, while turbulent flows involve chaotic eddies and enhanced mixing at higher speeds. The between laminar and turbulent regimes is governed by the Reynolds number, Re = \frac{\rho V D}{\mu}, a dimensionless ratio of inertial to viscous forces; flows are typically laminar for Re < 2300, transitional for $2300 < Re < 4000, and turbulent for Re > 4000 in . Vorticity quantifies the local rotation within the fluid, defined as the curl of the velocity , \boldsymbol{\omega} = \nabla \times \mathbf{v}. In Cartesian coordinates, its components are \omega_x = \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}, and similarly for other directions, representing twice the angular velocity of a fluid element (\boldsymbol{\omega} = 2 \boldsymbol{\Omega}). Non-zero indicates rotational , such as in vortices, while irrotational has \boldsymbol{\omega} = 0, allowing velocity to be derived from a potential. This vector measure captures the spin and shear in the , essential for understanding circulation and deformation without invoking forces.

Dynamics and Governing Equations

The dynamics of fluid motion arises from applying Newton's second law to fluid elements, accounting for forces such as gradients, viscous stresses, and body forces like . This leads to equations that describe how fields evolve under these influences, bridging the purely kinematic descriptions of patterns to the physical forces driving them. For inviscid flows, where viscous effects are negligible, the governing equations simplify significantly, while viscous flows require more complex formulations. These equations form the foundation for analyzing phenomena ranging from aerodynamics to flow in arteries. The Euler equations represent the fundamental description of inviscid fluid motion. Derived by considering the acceleration of a fluid particle along a streamline and balancing it against the and gravitational force, they take the form \frac{D\mathbf{v}}{Dt} = -\frac{1}{\rho} \nabla [P](/page/Pressure) + \mathbf{g}, where \mathbf{v} is the velocity vector, \rho is , P is , \mathbf{g} is , and D/Dt is the . This equation, originally formulated by Leonhard Euler in 1757, assumes no and is applicable to high-speed flows like those over airfoils where plays a minor role near the surface. For viscous fluids, the Navier-Stokes equations extend Euler's formulation by incorporating diffusive momentum transport due to . In the incompressible form, they are expressed as \rho \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} \right) = -\nabla [P](/page/Pressure) + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} + \mathbf{f}, where \mu is the dynamic and \mathbf{f} represents additional body forces. First derived by in 1822 and refined by George Gabriel Stokes in 1845, these nonlinear partial differential equations capture the full interplay of inertia, , , and external forces. The convective term (\mathbf{v} \cdot \nabla) \mathbf{v} introduces nonlinearity, making analytical solutions rare except for simple cases like Poiseuille flow in pipes; most real-world applications require numerical methods, and the existence of smooth solutions for all time remains an in three dimensions. A key simplification of the Euler equations for steady, inviscid, along a streamline yields Bernoulli's equation: P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}, where h is the height above a reference level and v is the speed. Developed by in 1738, this principle equates pressure energy, per unit volume, and . It explains effects like the Venturi , where fluid speed increases and pressure drops in a constricted pipe, enabling applications in carburetors and aspirators. Limitations arise if flow is rotational or unsteady, violating the assumptions. To analyze forces on macroscopic systems like pumps or nozzles, the form of the equation applies Newton's second law to a fixed : \mathbf{F} = \frac{d}{dt} \int_{CV} \rho \mathbf{v} \, dV + \int_{CS} \rho \mathbf{v} (\mathbf{v} \cdot \mathbf{n}) \, dA, where \mathbf{F} includes surface and body forces, CV is the , and CS is the control surface with outward normal \mathbf{n}. This formulation, rooted in the , allows computation of net forces from inflow and outflow fluxes, as used in calculations for jet engines. For steady flows, the volume accumulation term vanishes, simplifying to surface integrals only.

Applications and Modeling

Engineering Applications

Hydraulic systems leverage the incompressibility of liquids to transmit efficiently over distances, applying Pascal's principle, which states that pressure applied to an enclosed fluid is transmitted undiminished in all directions. This principle enables multiplication in devices such as automotive , where a small input on the brake pedal generates large clamping forces on the wheels via , and hydraulic lifts, which raise heavy loads using pistons of different areas. In engineering design, these systems provide high and precise control, making them essential for construction equipment and industrial machinery. In , fluid principles govern the generation of and wings, where Bernoulli's relates faster over the curved upper surface to lower , contributing to upward , while circulation around the wing enhances this effect. , the resistive opposing motion, is quantified by the C_d, defined as C_d = \frac{F_d}{\frac{1}{2} \rho v^2 A}, where F_d is , \rho is fluid , v is , and A is reference area; engineers optimize shapes to minimize C_d for in design. These concepts are critical for achieving stable flight and reducing operational costs in aviation. Piping systems in engineering transport fluids under pressure, with laminar flow in circular pipes governed by the Hagen-Poiseuille equation for volumetric flow rate Q = \frac{\pi r^4 \Delta P}{8 \mu L}, where r is radius, \Delta P is pressure difference, \mu is viscosity, and L is length; this relation guides the sizing of pipelines to ensure adequate throughput without excessive energy loss. Pumps maintain flow in these systems but risk cavitation, where low local pressures cause vapor bubble formation and collapse, leading to erosion of impeller surfaces, vibration, and reduced efficiency; mitigation involves maintaining sufficient net positive suction head to prevent such damage in water supply and chemical processing plants. Turbomachinery, including turbines and compressors, converts fluid to mechanical work or vice versa, with defined as \eta = \frac{\text{[power](/page/Power) output}}{\text{[power](/page/Power) input}}, a key metric for optimizing transfer in gas turbines where compressors pressurize incoming air before drives the . In generation and , axial-flow compressors achieve high ratios through multiple stages, while turbines extract from high-velocity exhaust gases, with designs focusing on blade to maximize \eta and minimize losses. These components enable efficient in jet engines and electricity production in combined-cycle .

Computational Modeling

Computational modeling of fluids involves numerical techniques to approximate solutions to the governing equations of fluid motion, particularly the Navier-Stokes equations, when analytical solutions are infeasible due to complex geometries, nonlinearities, or transient behaviors. These methods discretize the continuous domain into a finite number of points or elements, enabling simulations on computers to predict flow patterns, pressure distributions, and other phenomena in engineering and scientific applications. Key discretization approaches in (CFD) include , , and . The approximates derivatives using expansions on a structured , suitable for simple geometries but less flexible for irregular domains. , widely used in CFD for their conservation properties, integrate the governing equations over control volumes and apply the to fluxes at cell faces, ensuring local and global conservation of , , and . divide the domain into elements and use variational principles or Galerkin weighting to solve weak forms of the equations, offering advantages in handling complex, unstructured meshes for multiphysics problems like fluid-structure interactions. Turbulence modeling is essential in CFD as direct resolution of all scales is computationally prohibitive for most practical flows. Reynolds-Averaged Navier-Stokes (RANS) approaches time-average the equations, modeling unresolved Reynolds stresses via eddy viscosity closures, with the k-ε model being a seminal two-equation method that solves transport equations for turbulent kinetic energy (k) and its dissipation rate (ε) to estimate effective viscosity. Large Eddy Simulation (LES) resolves large-scale eddies while modeling subgrid-scale effects, often using the Smagorinsky model for dynamic viscosity based on local strain rates. Direct Numerical Simulation (DNS) resolves all turbulent scales without modeling, providing benchmark data but requiring immense computational resources limited to low Reynolds number flows. Prominent CFD software includes ANSYS Fluent, a commercial solver supporting finite volume discretization for multiphase, reacting, and turbulent flows across industries. OpenFOAM, an open-source C++ , enables customizable finite volume simulations for a wide range of problems using unstructured meshes. These tools are applied in weather prediction, where CFD enhances numerical models by simulating microscale atmospheric flows around terrain for improved forecast accuracy, and in vehicle design, optimizing to reduce drag and enhance through iterative simulations of external flows. Challenges in CFD include ensuring , often governed by the Courant-Friedrichs-Lewy (CFL) condition, which requires the product of flow speed (v), time step (Δt), and inverse grid spacing (1/Δx) to be less than 1 to prevent information propagation exceeding the numerical domain. Boundary conditions, such as the for viscous flows at solid walls—enforcing zero fluid velocity relative to the wall to capture shear layers and development—pose implementation difficulties, particularly in resolving thin layers without excessive grid refinement.