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Physics of Fluids
Physics of Fluids
Physics of Fluids
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Physics of Fluids
DisciplineFluid dynamics
LanguageEnglish
Edited byAndré Anders (interim)
Publication details
History1958–present
Publisher
AIP Publishing (United States)
FrequencyMonthly
4.1 (2023)
Standard abbreviations
ISO 4Phys. Fluids
Indexing
CODENPHFLE6
ISSN1070-6631 (print)
1089-7666 (web)
Links

Physics of Fluids is a monthly peer-reviewed scientific journal covering fluid dynamics, established by the American Institute of Physics in 1958, and is published by AIP Publishing. The journal focus is the dynamics of gases, liquids, and complex or multiphase fluids—and the journal contains original research resulting from theoretical, computational, and experimental studies.[1]

History

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From 1958 through 1988, the journal included plasma physics. From 1989 until 1993, the journal split into Physics of Fluids A covering fluid dynamics, and Physics of Fluids B, on plasma physics. In 1994, the latter was renamed Physics of Plasmas, and the former continued under its original name, Physics of Fluids.

The journal was originally published by the American Institute of Physics in cooperation with the American Physical Society's Division of Fluid Dynamics. In 2016, the American Institute of Physics became the sole publisher. From 1985 to 2015, Physics of Fluids published the Gallery of Fluid Motion, containing award-winning photographs, images, and visual streaming media of fluid flow.

With funding from the American Institute of Physics the annual "François Naftali Frenkiel Award" was established by the American Physical Society in 1984 to reward a young scientist who published a paper containing significant contributions to fluid dynamics during the previous year. The award-winning paper was chosen from Physics of Fluids until 2016, but is presently chosen from Physical Review Fluids. Similarly, the invited papers from plenary talks at the annual American Physical Society Division of Fluid Dynamics were formerly published in Physics of Fluids but, since 2016, are now published in either this journal[2][3] or Physical Review Fluids.

Reception

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Physics of Fluids A, Physics of Fluids B, and Physics of Fluids were ranked 3, 4, and 6, respectively, based on their "impact from 1981-2004" within the category of journals on the physics of fluids and plasmas.[4][failed verification] According to the Journal Citation Reports, the journal has a 2023 impact factor of 4.1.[5]

Editors-in-chief

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The following persons are or have been editors-in-chief:

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Physics of Fluids

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from Grokipedia
The physics of fluids, commonly referred to as fluid mechanics, is a fundamental branch of physics that examines the macroscopic behavior of fluids—substances such as liquids and gases that deform continuously under applied shear stress, lacking a fixed shape while often maintaining a fixed volume in the case of liquids. This field addresses both fluid statics, the study of fluids at rest where forces like pressure and buoyancy dominate, and fluid dynamics, which analyzes fluids in motion, including phenomena like flow velocity, turbulence, and wave propagation. Central to the discipline is the continuum assumption, treating fluids as continuous media rather than discrete molecular collections, which holds for most engineering and natural applications where molecular spacing is negligible compared to flow scales. Key properties defining fluid behavior include density (mass per unit volume, ρ = m/V, with typical values like 1000 kg/m³ for water and 1.225 kg/m³ for air at sea level), pressure (force per unit area, p = F/A, measured in Pascals where 1 Pa = 1 N/m²), temperature (influencing molecular motion and often linked via the ideal gas law), viscosity (resistance to shear, e.g., 1.789 × 10⁻⁵ kg/(m·s) for air at sea level), and speed of sound (a = √(γRT) for ideal gases, where γ is the adiabatic index, R the gas constant, and T temperature). These properties are interdependent; for instance, pressure in a static fluid increases linearly with depth (p = p₀ + ρgh, where g is gravity and h depth), a principle formalized by Blaise Pascal in the 17th century. Additional characteristics like surface tension (force per unit length causing liquid surfaces to minimize area) and compressibility (more pronounced in gases than liquids) further influence behaviors such as capillary action and shock waves. The behavior of fluids is governed by three primary conservation laws derived from Newtonian mechanics: the for mass conservation (∂ρ/∂t + ∇·(ρU) = 0, where U is the velocity vector), the momentum equation (Newton's second law in integral or differential form, ∂(ρU)/∂t + ∇·(ρUU) = -∇p + ∇·τ + ρg, incorporating viscous stresses τ), and the energy equation (∂(ρe)/∂t + ∇·(ρeU) = -∇·(pU) + ∇·(τ·U) + ∇·(k∇T) + ρq, balancing e, heat conduction k, and sources q). For ideal fluids, simplifications like the Navier-Stokes equations emerge, combining these into nonlinear partial differential equations that describe viscous, incompressible flows. These principles, rooted in 19th-century developments by scientists like and George Gabriel Stokes, enable predictions of complex flows from laminar to turbulent regimes. Applications of the physics of fluids span natural phenomena, such as and ocean currents, to engineered systems including for lift, cardiovascular flow, and systems like turbines and pipelines. In , for example, understanding compressible flows around vehicles at high speeds is essential for designing , while in , it informs pollutant dispersion models. The field's interdisciplinary nature extends to , , and , underscoring its role in advancing technology and scientific understanding.

Overview

Definition and Scope

The physics of fluids, commonly referred to as , is a branch of that investigates the behavior of matter in its liquid and gaseous states, treating fluids as continuous media capable of undergoing deformation, flow, and response to internal and external forces. This field primarily examines liquids and gases, with extensions to plasmas in certain contexts where they exhibit fluid-like properties under electromagnetic influences. Fluids are distinguished from solids by their inability to sustain without continuous deformation; under applied shear, a fluid element will flow indefinitely, establishing internal motion patterns, whereas a solid maintains its shape through elastic resistance. The scope of physics of fluids delineates the boundaries within by encompassing both static and dynamic phenomena: addresses fluids at rest, focusing on equilibrium states and distributions, while analyzes motion under forces such as , gradients, and . It includes multiphase flows involving interactions between immiscible fluids, like gas-liquid mixtures in pipelines or suspensions in , but excludes the deformation of solids, which falls under . The field overlaps with , particularly in compressible flows where energy equations couple with momentum, and with , which provides the foundational assumption of treating fluids as smooth continua rather than discrete molecules. Representative examples illustrate the field's breadth, such as the flow of water through hydraulic systems, atmospheric air circulation driving weather patterns, and the dynamics of blood as a non-Newtonian fluid in cardiovascular vessels. These cases highlight fluids' practical manifestations, from engineering designs like aircraft wings to natural processes like ocean currents. Interdisciplinary connections extend to mechanical and chemical engineering, where fluid principles optimize pumps and reactors, and to biology via biomechanics, modeling phenomena like pulmonary airflow or microvascular perfusion to understand physiological transport.

Historical Development

The study of the physics of fluids traces its origins to ancient civilizations, where early observations laid the groundwork for understanding and basic fluid behavior. In the third century BCE, formulated the principle of , stating that a body immersed in a fluid experiences an upward force equal to the weight of the displaced fluid, which provided foundational insights into . Around the first century CE, conducted pioneering studies on , exploring the principles of air and water flow in devices such as the , an early that demonstrated concepts. The 17th and 18th centuries marked a shift toward experimental and theoretical advancements in and early hydrodynamics. In 1643, invented the , demonstrating and the behavior of under vacuum conditions, which challenged prevailing Aristotelian views. During the 1640s, performed key experiments on hydrostatic , establishing that in a is transmitted equally in all directions and varies linearly with depth, as detailed in his treatise Traité de l'équilibre des liqueurs. By 1738, published , introducing the principle of in flow for steady, incompressible, inviscid conditions along a streamline, linking , , and . The 19th century saw the formal establishment of as a mathematical discipline, with significant progress in modeling viscous effects and flow instabilities. In the 1820s, extended Euler's inviscid equations by incorporating viscous terms, laying the groundwork for the Navier-Stokes equations that describe momentum conservation in viscous fluids. George Gabriel Stokes refined these equations in the 1840s, providing solutions for low-Reynolds-number flows, such as the drag on small spheres, which became essential for understanding . In the 1860s, developed the theory of vortex motion in inviscid fluids through his 1858 paper on hydrodynamic integrals corresponding to vortex movements, while built upon this work to explore vortex dynamics, including the stability and interaction of vortex filaments in three-dimensional flows. The late 19th and 20th centuries brought experimental and theoretical breakthroughs addressing real-world complexities like and boundary effects, alongside the rise of computational methods. In 1883, Osborne Reynolds conducted pipe flow experiments using dye injection, identifying the transition from laminar to turbulent regimes and introducing the dimensionless to characterize flow stability based on inertial and viscous forces. revolutionized the field in 1904 with his theory, presented at the Third , which explained drag in viscous flows by positing a thin layer near solid surfaces where viscosity dominates, resolving . solidified as a distinct during the 19th century through these mathematical formulations, enabling applications in engineering. Post-World War II, particularly from the onward, emerged, driven by advances in numerical methods and computing power for solving the Navier-Stokes equations in contexts like flow simulations.

Fundamental Principles

Fluid Properties

Fluids are characterized by several intrinsic properties that determine their mechanical and thermodynamic behavior under various conditions. , denoted as ρ\rho, is defined as the per unit of the . In incompressible fluids, such as most liquids, remains constant irrespective of applied , enabling simplifications in flow analyses. Compressible fluids, predominantly gases, exhibit variations with changes in and , which is critical in high-speed or pressurized flows. Viscosity, represented by μ\mu, quantifies the fluid's resistance to shear or internal between adjacent layers in motion. Newtonian fluids maintain a constant regardless of the applied , as described by the linear relationship τ=μγ˙\tau = \mu \dot{\gamma}, where τ\tau is and γ˙\dot{\gamma} is . In contrast, non-Newtonian fluids display that varies with ; shear-thinning examples include , where decreases under higher shear, facilitating flow in narrow vessels. Surface tension, denoted σ\sigma, arises from unbalanced cohesive forces at the fluid's surface or interface with another phase, manifesting as per unit length that minimizes surface area. This property drives capillary action, where rises or depresses in a narrow tube due to the balance between surface tension and adhesive forces to the tube walls, as seen in water climbing plant xylem or mercury depression in glass. Compressibility measures a fluid's susceptibility to volume reduction under pressure and is inversely related to the KK, defined as K=−V(∂P∂V)TK = -V \left( \frac{\partial P}{\partial V} \right)_T, where VV is volume and PP is pressure. For gases, is pronounced, and the PV=nRTPV = nRT—with nn as moles, RR the , and TT —provides a foundational relation for density and pressure interdependence under isothermal conditions. Many fluid properties vary with , influencing overall behavior. The volumetric thermal expansion coefficient α=1V(∂V∂T)P\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P quantifies the fractional volume increase per unit temperature rise at constant pressure. Specific heat capacities, cpc_p at constant pressure and cvc_v at constant volume, represent the energy required to elevate the of one unit mass by one degree, essential for analyses in fluids. Representative examples illustrate these properties: water at 20°C and standard atmospheric pressure has a density ρ≈1000\rho \approx 1000 kg/m³ and dynamic viscosity μ≈10−3\mu \approx 10^{-3} Pa·s, behaving as a nearly incompressible Newtonian fluid. Air under the same conditions exhibits ρ≈1.2\rho \approx 1.2 kg/m³ and μ≈1.8×10−5\mu \approx 1.8 \times 10^{-5} Pa·s, highlighting its lower density and viscosity as a compressible Newtonian gas.

Continuum Assumption

The continuum assumption in posits that fluids can be modeled as continuous media, where macroscopic properties such as , , , and vary smoothly and continuously across space and time, rather than exhibiting discrete molecular behavior. This approximation is justified when the characteristic length scale LL of the flow is significantly larger than the molecular scales, allowing statistical averaging over a representative volume to define local properties without resolving individual particle motions. The assumption underpins the derivation of governing equations in classical and is valid primarily for dense fluids where intermolecular interactions maintain local . A key parameter quantifying the applicability of this assumption is the , defined as \Kn=λ/L\Kn = \lambda / L, where λ\lambda is the —the a travels between successive collisions—and LL is the of the system, such as the of a pipe or the scale of flow features. For \Kn≪1\Kn \ll 1 (typically \Kn<0.01\Kn < 0.01), the continuum model holds robustly, as the is negligible compared to macroscopic dimensions; for air at standard conditions, λ≈65\lambda \approx 65 nm, ensuring validity in most terrestrial flows. However, the assumption breaks down in rarefied gases where \Kn>0.1\Kn > 0.1, such as in the upper atmosphere (e.g., at altitudes above 100 km, where λ≈0.1 m\lambda \approx 0.1 \, \mathrm{m}), leading to non-continuum effects like velocity slip at walls. Within the continuum framework, fluid flows are described using either Eulerian or Lagrangian perspectives. The Eulerian description treats the fluid as a field of properties fixed in space, with variables like velocity u(x,t)\mathbf{u}(\mathbf{x}, t) defined at spatial points x\mathbf{x} over time tt, facilitating the analysis of continuum fields through partial derivatives and enabling the formulation of conservation laws in a fixed grid. In contrast, the Lagrangian description follows individual fluid parcels labeled by their initial positions, tracking their trajectories x(a,t)\mathbf{x}( \mathbf{a}, t ) where a\mathbf{a} identifies the parcel, with the D/Dt=∂/∂t+u⋅∇D/Dt = \partial / \partial t + \mathbf{u} \cdot \nabla capturing changes along paths; this approach aligns naturally with the conservation principles inherent to but is computationally intensive for complex three-dimensional flows. Both viewpoints assume a continuous medium and are interconvertible via the fundamental relating parcel motions to field variables. The continuum assumption has notable limitations in regimes where molecular effects dominate, such as microfluidics or near-vacuum conditions, where \Kn\Kn becomes order unity or greater, invalidating smooth property variations and local equilibrium. In microchannels with dimensions on the order of micrometers, rarefaction effects manifest as slip flows or transition regimes, requiring modifications like second-order slip boundary conditions to extend Navier-Stokes applicability up to \Kn≈0.25\Kn \approx 0.25, beyond which non-continuum modeling is essential. For highly rarefied gases, the paradigm shifts to kinetic theory, governed by the Boltzmann equation (∂t+v⋅∇+Fm⋅∇v)f=Q(f,f)\left( \partial_t + \mathbf{v} \cdot \nabla + \frac{\mathbf{F}}{m} \cdot \nabla_v \right) f = Q(f, f), which describes the molecular distribution function f(x,v,t)f(\mathbf{x}, \mathbf{v}, t) and collision integral QQ, capturing nonequilibrium phenomena absent in continuum descriptions. This assumption is validated by the successful application of continuum-based Navier-Stokes equations to the vast majority of engineering flows, where low \Kn\Kn ensures accurate predictions of phenomena like around aircraft or pipe flows in chemical processing, with errors typically below 1% in dense conditions.

Governing Equations

Conservation of Mass and Momentum

The and forms the cornerstone of , expressing the fundamental physical principles that govern the behavior of continuous media under the assumption of no internal sources or sinks of mass. These laws are derived from the application of Newton's laws to fluid elements or control volumes, providing the basis for all subsequent governing equations in the field. The integral forms account for arbitrary volumes and surfaces, while the differential forms describe local behavior at a point in the fluid.

Conservation of Mass

The principle of conservation states that the within a fixed changes only due to the net of across its boundary, assuming no creation or destruction of within the . For a VV bounded by surface SS, the form of conservation is given by ddt∫Vρ dV+∫Sρv⋅dA=0,\frac{d}{dt} \int_V \rho \, dV + \int_S \rho \mathbf{v} \cdot d\mathbf{A} = 0, where ρ\rho is the , v\mathbf{v} is the vector, and dAd\mathbf{A} is the outward-pointing area element. This equation balances the time rate of change of inside VV with the out through SS. To obtain the differential form, apply the (also known as Gauss's theorem), which converts the surface integral into a : ∫SF⋅dA=∫V∇⋅F dV\int_S \mathbf{F} \cdot d\mathbf{A} = \int_V \nabla \cdot \mathbf{F} \, dV for any F\mathbf{F}. Substituting F=ρv\mathbf{F} = \rho \mathbf{v} yields ddt∫Vρ dV+∫V∇⋅(ρv) dV=0.\frac{d}{dt} \int_V \rho \, dV + \int_V \nabla \cdot (\rho \mathbf{v}) \, dV = 0. Since this holds for arbitrary VV, the integrand must vanish , leading to the : ∂ρ∂t+∇⋅(ρv)=0.\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0. This partial differential equation was first derived by Leonhard Euler in 1757 through geometric considerations of fluid particle motion. The continuity equation can also be obtained by applying the Leibniz rule for differentiating under the integral sign to a material volume following the fluid, assuming density uniformity across infinitesimal elements. For incompressible fluids, where ρ\rho is constant, the simplifies to ∇⋅v=0\nabla \cdot \mathbf{v} = 0, implying that the field is divergence-free and is preserved under flow. This form is widely used in low-speed flows, such as in . A representative example is steady, through a pipe of varying cross-section: mass conservation requires A1v1=A2v2A_1 v_1 = A_2 v_2, where AA is the cross-sectional area and vv is the average , ensuring constant despite geometric changes.

Conservation of Momentum

The conservation of follows from Newton's second law applied to a element, equating the rate of change of to the on it. For a moving with the , the relates the time derivative of an extensive property (here, ∫ρv dV\int \rho \mathbf{v} \, dV) between a system and a fixed control : ddt∫VmB dV=∂∂t∫VB dV+∫SBv⋅dA,\frac{d}{dt} \int_{V_m} B \, dV = \frac{\partial}{\partial t} \int_V B \, dV + \int_S B \mathbf{v} \cdot d\mathbf{A}, where B=ρvB = \rho \mathbf{v} is the momentum density and VmV_m is the material volume. This theorem, formulated by Osborne Reynolds in 1903, bridges Lagrangian (following the fluid) and Eulerian (fixed in space) descriptions. Applying it to momentum and incorporating surface and body forces leads to the integral momentum equation for a control volume. The differential form, known as the , is obtained via the and is expressed as ρ(∂v∂t+(v⋅∇)v)=−∇p+∇⋅τ+ρf,\rho \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} \right) = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{f}, or compactly using the DvDt=∂v∂t+(v⋅∇)v\frac{D\mathbf{v}}{Dt} = \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v}: ρDvDt=−∇p+∇⋅τ+ρf.\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{f}. Here, pp is the isotropic pressure, τ\boldsymbol{\tau} is the , and f\mathbf{f} represents body forces per unit mass (e.g., ). This equation was introduced by in 1823 as part of the general equations for continuum motion. The left side represents the substantial of the fluid particle, while the right side accounts for , viscous stresses, and body forces. To derive the Cauchy equation from Newton's second law, consider a small fluid element of volume δV\delta V. The net momentum change δmDvDt\delta m \frac{D\mathbf{v}}{Dt} (with δm=ρδV\delta m = \rho \delta V) equals the surface forces ∫∂(δV)σ⋅dA\int_{\partial (\delta V)} \boldsymbol{\sigma} \cdot d\mathbf{A} plus body forces ρfδV\rho \mathbf{f} \delta V, where σ=−pI+τ\boldsymbol{\sigma} = -p \mathbf{I} + \boldsymbol{\tau} is the total stress tensor. Applying the to the surface integral and taking the limit δV→0\delta V \to 0 yields the local form. The assumption of no mass sources or sinks ensures the validity of this balance. For Newtonian fluids, the is linear in the velocity gradients, given by τ=μ(∇v+(∇v)T)−23μ(∇⋅v)I,\boldsymbol{\tau} = \mu \left( \nabla \mathbf{v} + (\nabla \mathbf{v})^T \right) - \frac{2}{3} \mu (\nabla \cdot \mathbf{v}) \mathbf{I}, where μ\mu is the dynamic viscosity (a fluid property reflecting resistance to shear, as briefly referenced in discussions of properties). This constitutive relation assumes the fluid is isotropic and the stress depends only on the symmetric rate-of-strain tensor, originating from Isaac Newton's 1687 postulate of proportionality between and in viscous materials. The Navier-Stokes equations constitute the fundamental mathematical framework describing the motion of viscous, Newtonian fluids under the continuum assumption. These partial differential equations arise from the conservation laws of , , and , incorporating viscous effects through the stress tensor derived from Newton's law of viscosity. Originally formulated in the early , they provide a closed system for predicting fluid behavior in a wide range of and natural phenomena, from pipe flows to . The complete set of Navier-Stokes equations for a compressible, viscous includes the for conservation, the equation, and the energy equation. The equation, expressed in conservative form, is ∂(ρv)∂t+∇⋅(ρvv)=−∇p+∇⋅τ+ρf,\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{f}, where ρ\rho is , v\mathbf{v} is , pp is , τ\boldsymbol{\tau} is the (for Newtonian fluids, τ=μ(∇v+(∇v)T)+λ(∇⋅v)I\boldsymbol{\tau} = \mu (\nabla \mathbf{v} + (\nabla \mathbf{v})^T) + \lambda (\nabla \cdot \mathbf{v}) \mathbf{I}, with μ\mu as dynamic , λ\lambda as the second viscosity coefficient, and I\mathbf{I} the identity tensor), and f\mathbf{f} represents body forces per unit . The energy equation, accounting for , viscous dissipation, and heat sources, takes the form ρDhDt=∇⋅(k∇T)+Φ+ρq,\rho \frac{Dh}{Dt} = \nabla \cdot (k \nabla T) + \Phi + \rho q, where hh is specific , TT is , kk is thermal conductivity, Φ=τ:∇v\Phi = \boldsymbol{\tau} : \nabla \mathbf{v} is the viscous dissipation rate, and qq is the heat source per unit mass. This formulation closes the system when combined with an relating pp, ρ\rho, and TT, such as for an . For incompressible flows, where density ρ\rho is constant and the speed of sound is effectively infinite (valid for low Mach numbers, Ma≪1Ma \ll 1), the equations simplify significantly. The reduces to the divergence-free condition ∇⋅v=0\nabla \cdot \mathbf{v} = 0, and the momentum equation becomes ρ(∂v∂t+v⋅∇v)=−∇p+μ∇2v+ρf.\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{f}. Here, acts as a to enforce incompressibility, and the energy equation decouples if temperature variations do not affect or . This form is widely used in simulations of liquids and low-speed gases. To analyze solutions, the Navier-Stokes equations are often nondimensionalized by introducing characteristic scales: length LL, velocity UU, time L/UL/U, density ρ0\rho_0, pressure ρ0U2\rho_0 U^2, and temperature ΔT\Delta T. The resulting dimensionless momentum equation highlights key dimensionless groups, notably the Re=ρ0UL/μRe = \rho_0 U L / \mu, which quantifies the ratio of inertial to viscous forces, and the Fr=U/gLFr = U / \sqrt{g L}
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