Hubbry Logo
Quasistatic approximationQuasistatic approximationMain
Open search
Quasistatic approximation
Community hub
Quasistatic approximation
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Quasistatic approximation
Quasistatic approximation
Quasistatic approximation
View on Wikipedia
from Wikipedia

Quasistatic approximation(s) refers to different domains and different meanings. In the most common acceptance, quasistatic approximation refers to equations that keep a static form (do not involve time derivatives) even if some quantities are allowed to vary slowly with time. In electromagnetism it refers to mathematical models that can be used to describe devices that do not produce significant amounts of electromagnetic waves. For instance, the capacitor and the coil in electrical networks.

Overview

[edit]

The quasistatic approximation can be understood through the idea that the sources in the problem change sufficiently slowly that the system can be taken to be in equilibrium at all times. This approximation can then be applied to areas such as classical electromagnetism, fluid mechanics, magnetohydrodynamics, thermodynamics, and more generally systems described by hyperbolic partial differential equations involving both spatial and time derivatives. In simple cases, the quasistatic approximation is allowed when the typical spatial scale divided by the typical temporal scale is much smaller than the characteristic velocity with which information is propagated. [1] The problem gets more complicated when several length and time scales are involved. In the strict acceptance of the term the quasistatic case corresponds to a situation where all time derivatives can be neglected. However, some equations can be considered as quasistatic while others are not, leading to a system still being dynamic. There is no general consensus in such cases.

Fluid dynamics

[edit]

In fluid dynamics, only quasi-hydrostatics (where no time derivative term is present) is considered as a quasi-static approximation. Flows are usually considered as dynamic as well as acoustic waves propagation.

Thermodynamics

[edit]

In thermodynamics, a distinction between quasistatic regimes and dynamic ones is usually made in terms of equilibrium thermodynamics versus non-equilibrium thermodynamics. As in electromagnetism some intermediate situations also exist; see for instance local equilibrium thermodynamics.

Electromagnetism

[edit]

In classical electromagnetism, there are at least two consistent quasistatic approximations of Maxwell equations: quasi-electrostatics and quasi-magnetostatics depending on the relative importance of the two dynamic coupling terms.[2] These approximations can be obtained using time constants evaluations or can be shown to be Galilean limits of electromagnetism.[3] For example, in applied geophysics (Sea Bed Logging), we can cite quasi-static approximation solvers for reducing computational simulation times and inversion calculation costs (software).[4]

Retarded times point of view

[edit]

In magnetostatics equations such as Ampère's law or the more general Biot–Savart law allow one to solve for the magnetic fields produced by steady electrical currents. Often, however, one may want to calculate the magnetic field due to time varying currents (accelerating charge) or other forms of moving charge. Strictly speaking, in these cases the aforementioned equations are invalid, as the field measured at the observer must incorporate distances measured at the retarded time, that is the observation time minus the time it took for the field (traveling at the speed of light) to reach the observer. The retarded time is different for every point to be considered, hence the resulting equations are quite complicated; often it is easier to formulate the problem in terms of potentials; see retarded potential and Jefimenko's equations.

In this point of view the quasistatic approximation is obtained by using time instead of retarded time or equivalently to assume that the speed of light is infinite. To first order, the mistake of using only Biot–Savart's law rather than both terms of Jefimenko's magnetic field equation fortuitously cancel. [5]

Notes

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Quasistatic approximation

View on Grokipedia
from Grokipedia
The quasistatic approximation is a method in physics used to simplify the analysis of systems undergoing slow changes by treating the system as being in a static equilibrium at each instant, neglecting time derivatives in the governing equations when variations occur on timescales much longer than intrinsic relaxation times. In , this approximation treats time-varying electric and magnetic fields as if they were static at each instant, assuming changes in sources propagate instantaneously without significant or retardation effects. This approach is valid for systems where the characteristic time scale of variations, τ, greatly exceeds the electromagnetic wave transit time across the system's dimensions, L/c (with c the ), typically requiring L/c ≪ τ. The approximation finds applications across various fields, including for processes that maintain internal equilibrium, for quasi-steady flows, and . In practice, within electromagnetism, it manifests in two primary forms: the electroquasistatic (EQS) approximation, which neglects time-varying magnetic induction (∂B/∂t term in Faraday's ) while retaining , and the magnetoquasistatic (MQS) approximation, which neglects the (∂D/∂t term in Ampère's ) while incorporating magnetostatics. Under EQS, the fields are primarily determined by charge distributions using and the electrostatic potential, with error fields scaling as (L/cτ)^2 relative to the dominant fields. Similarly, MQS focuses on current distributions via Ampère's , applying to scenarios like inductors or conductors with alternating currents. These reductions bridge the gap between static field theory and the full , enabling tractable solutions for low-frequency phenomena. Key applications in electromagnetism include analyzing the skin effect, where alternating currents confine to a thin layer near a conductor's surface due to eddy currents, with penetration depth δ = √(2/μσω) (μ permeability, σ conductivity, ω angular frequency). For instance, at 60 Hz in copper, δ ≈ 8.5 mm, but it shrinks to microns at gigahertz frequencies, impacting high-frequency devices like transformers and antennas. The approximation also underpins models in plasma physics, neuromodulation, and atmospheric dynamics, where slow variations allow neglecting wave propagation. Its limitations arise at higher frequencies, where full electrodynamic treatment becomes necessary to account for radiated energy.

Introduction

Definition

The quasistatic approximation is a modeling technique in physics that assumes a process occurs infinitely slowly relative to the system's characteristic relaxation times, ensuring the system remains in thermodynamic or at every instant. This approach idealizes the dynamics such that the system can be treated as traversing a continuous path through equilibrium states, with any perturbations decaying rapidly enough to maintain near-equilibrium conditions. Key characteristics of the quasistatic approximation include the neglect of transient effects, such as rapid fluctuations or non-equilibrium transients, allowing the system's evolution to be described using equilibrium thermodynamic relations or static field equations at each step. Consequently, time-dependent terms in the governing equations, like certain partial derivatives with respect to time, are deemed negligible when the process timescale far exceeds the system's internal response time. The concept originated in the within , where and contemporaries developed it to analyze the efficiency of heat engines through idealized reversible cycles. and cyclic processes in the 1850s and 1860s formalized the idea of processes composed of equilibrium steps, distinguishing them from irreversible ones. In general, the quasistatic approximation finds broad applicability across physics disciplines, simplifying complex dynamic equations by ignoring small time derivatives, thereby bridging static and fully time-dependent analyses in fields ranging from to .

Validity conditions

The quasistatic approximation relies fundamentally on a clear separation of timescales between the duration of the external driving the system, denoted as τprocess\tau_\text{process}, and the intrinsic relaxation timescale τrelax\tau_\text{relax} over which the system returns to equilibrium following a perturbation. This condition, τprocess≫τrelax\tau_\text{process} \gg \tau_\text{relax}, ensures that the system can maintain thermodynamic or at every stage, allowing state variables such as , , or fields to be well-defined throughout the evolution. Without this separation, deviations from equilibrium accumulate, rendering the static-like description invalid. For a characterized by a typical size LL and a relevant speed vv—such as the in or the in electromagnetic contexts—the relaxation timescale is approximately τrelax≈L/v\tau_\text{relax} \approx L / v. The holds when the rate of change imposed by the process is slow relative to the inverse of this timescale, i.e., when variations occur on times much longer than L/vL / v, preventing wave-like effects from dominating. This criterion establishes the boundary of applicability across diverse physical domains, where vv sets the scale for how quickly disturbances equilibrate within the . The errors arising from the quasistatic approximation typically scale as O((τrelax/τprocess)2)\mathcal{O}\left((\tau_\text{relax} / \tau_\text{process})^2\right) in contexts like , which must be much less than unity to ensure quantitative accuracy; for instance, deviations in predicted fields or state variables remain small only if this ratio is sufficiently suppressed. This perturbative error bound underscores the approximation's utility for slow evolutions but highlights its sensitivity to the degree of timescale separation. A key limitation occurs near critical points or during phase transitions, where τrelax\tau_\text{relax} diverges to critical slowing down, eliminating the necessary timescale separation and causing the to exhibit non-equilibrium dynamics even for nominally slow processes. In such regimes, the quasistatic description fails, as the cannot adapt instantaneously to changes, leading to or other irreversible effects.

Thermodynamic applications

Quasistatic processes

In thermodynamics, a quasistatic process refers to an idealized transformation of a thermodynamic system, such as a gas, where the system maintains internal equilibrium at every stage, with state variables like pressure PP, volume VV, and temperature TT changing infinitesimally slowly. This ensures that the process occurs on a timescale much longer than the system's relaxation time, allowing equilibrium thermodynamics to describe the system continuously without significant deviations from balance. On a pressure-volume (PV) diagram or temperature-entropy (TS) diagram, a quasistatic process traces a smooth, continuous path through the state space, differing from irreversible processes that feature abrupt jumps between non-equilibrium states. For an ideal gas undergoing such a process, the work done by the system is calculated as W=∫P dVW = \int P \, dV, integrated along this equilibrium path, while the heat transfer QQ is determined via the first law of thermodynamics: ΔU=Q−W\Delta U = Q - W, where UU is the change in internal energy. A classic example is the slow compression of an confined in a frictionless -cylinder device, where the piston advances at an infinitesimally slow rate, permitting the gas to remain in and exchange with its surroundings to adjust gradually during the volume reduction. This setup highlights the role of controlled in maintaining the quasistatic nature, ensuring the process can be analyzed using equilibrium properties throughout. Quasistatic processes form the basis for understanding reversible thermodynamic transformations, as explored in related contexts.

Relation to reversible processes

A quasistatic process is reversible if it occurs without dissipative effects, such as or , enabling both the system and its surroundings to be restored to their initial states without net change. This equivalence holds because the absence of ensures infinitesimal changes maintain equilibrium, allowing the process to proceed equally well in reverse. In quasistatic reversible processes, the total entropy change of the , ΔStotal\Delta S_\text{total}, equals zero, representing the boundary case between reversible and irreversible behaviors. For irreversible processes, ΔStotal>0\Delta S_\text{total} > 0, indicating due to , whereas quasistatic conditions minimize but do not eliminate this in real scenarios. The exemplifies a quasistatic reversible , consisting of two isothermal and two adiabatic steps, all executed infinitely slowly to maintain equilibrium. This configuration achieves the maximum theoretical for a operating between hot and cold reservoirs, given by η=1−TcoldThot\eta = 1 - \frac{T_\text{cold}}{T_\text{hot}}, where temperatures are in . In practice, real thermodynamic approximate quasistatic behavior only when conducted sufficiently slowly, yet they invariably include some irreversibility from dissipative mechanisms like . Thus, while quasistatic approximations model ideal reversibility for analysis, actual implementations fall short, leading to reduced and generation.

Fluid dynamics applications

Quasi-steady flow approximation

The quasi-steady flow approximation in fluid dynamics applies to situations where the temporal variations in the flow field occur slowly compared to the convective timescale, allowing the neglect of unsteady terms in the Navier-Stokes equations. Specifically, the core assumption is that the partial derivative with respect to time, ∂u/∂t, can be omitted when the Strouhal number St = f L / U ≪ 1, where f is the characteristic frequency, L is the length scale, and U is the flow velocity; this indicates that the time for flow changes is much longer than the time for convection across the domain L/U. Under this approximation, the flow responds nearly instantaneously to changes in boundary conditions or forcing, treating the problem as locally steady at each instant. The governing equations simplify accordingly. For , the unsteady Navier-Stokes momentum equation ρ(∂u∂t+u⋅∇u)=−∇p+μ∇2u+f\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f} reduces to the steady form by ignoring the ∂u/∂t term: ρ(u⋅∇u)≈−∇p+μ∇2u+f,\rho (\mathbf{u} \cdot \nabla \mathbf{u}) \approx -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}, where ρ is , p is , μ is dynamic , and f represents body forces; this yields the steady Euler equations for inviscid cases (μ=0) or steady Navier-Stokes for viscous flows. The remains unchanged as ∇ · u = 0 for incompressibility. This approximation holds for flows with low-frequency oscillations or slowly varying boundary conditions, where the timescale separation ensures unsteady effects are negligible. It is particularly valid in tidal flows, where the quasi-steady balance equates gradients from tidal head differences to frictional forces, neglecting inertial due to the long tidal period relative to flow transit times. Similarly, in creeping flows at low Reynolds numbers (Re ≪ 1), such as the slow motion of small particles in viscous fluids, the quasi-steady assumption applies because viscous equilibrates the flow much faster than boundary changes, allowing treatment of unsteady motions as a sequence of steady Stokes problems. A representative application is for thin-film flows, where the quasi-steady approximation balances pressure gradients with viscous shear forces across the thin gap, neglecting both inertial and unsteady terms to derive the for pressure distribution. In this regime, the flow in narrow channels or bearing films adjusts rapidly to surface motions, enabling efficient computation of load-carrying capacity without resolving full transient dynamics.

Applications in low-speed flows

In low-speed flows, the quasistatic approximation, often termed the quasi-steady approximation, is particularly relevant in the incompressible limit where the ρ\rho remains constant and the Ma≪1\mathrm{Ma} \ll 1. This neglects effects, such as , allowing the to simplify to ∇⋅u=0\nabla \cdot \mathbf{u} = 0, where u\mathbf{u} is the velocity field. The approximation assumes that inertial forces due to fluid acceleration are negligible compared to viscous forces, which holds when the Re≪1\mathrm{Re} \ll 1. A canonical example is , describing quasistatic creeping motion around objects like in viscous fluids at low Reynolds numbers. In this scenario, the Navier-Stokes equations reduce to the linear Stokes equations, ∇[p](/page/Pressure)=η∇2u\nabla [p](/page/Pressure) = \eta \nabla^2 \mathbf{u} and ∇⋅u=0\nabla \cdot \mathbf{u} = 0, where pp is and η\eta is dynamic viscosity. For a of radius RR moving with velocity UU, the drag force is given by : F=6πηRUF = 6 \pi \eta R U. This result, derived for steady translation, extends quasistatically to slowly varying motions where transient effects are minimal. Biological flows provide practical applications of this approximation. In blood circulation through capillaries, where vessel diameters are on the order of 5–10 μ\mum and flow speeds are below 1 mm/s, the is typically Re<0.1\mathrm{Re} < 0.1, enabling quasistatic modeling of red blood cell motion and plasma flow as Stokes-like creeping flows. Similarly, in insect flight, such as that of fruit flies or bees, the quasi-steady approximation captures wing kinematics during hovering or slow maneuvers, where flapping frequencies yield Strouhal numbers St≈0.2–0.4\mathrm{St} \approx 0.2–0.4. Here, aerodynamic forces are estimated by integrating translational and rotational contributions over the wing stroke, aligning well with experimental force measurements despite intermediate s (Re ≈ 100–10^4). Despite its utility, the quasistatic approximation has limitations in low-speed flows involving significant unsteadiness. It fails to capture phenomena like unsteady wakes or vortex shedding, which emerge when oscillatory motions introduce substantial temporal variations. The error increases with the Strouhal number St=fL/U>0.1\mathrm{St} = f L / U > 0.1, where ff is oscillation frequency, LL is a , and UU is flow speed, as this parameter quantifies the ratio of local acceleration to convective acceleration; beyond this threshold, full unsteady Navier-Stokes solutions are required for accuracy.

Electromagnetic applications

Quasi-static fields

In electromagnetism, the quasistatic approximation applies to time-varying fields that evolve slowly enough to be treated as instantaneous static configurations, neglecting wave propagation effects. This regime is valid when the characteristic frequency ω of field variations satisfies ω ≪ c/L, where c is the speed of light and L is the typical system size, ensuring that the time for light to traverse the system is much shorter than the variation timescale. In the quasistatic approximation, depending on whether electric or magnetic effects dominate, either the displacement current term ∂D/∂t is omitted from Ampère's law (magnetoquasistatic case) or the magnetic induction term ∂B/∂t is neglected in Faraday's law (electroquasistatic case). This reduces Maxwell's equations to coupled time-dependent electrostatic and magnetostatic forms. In the electroquasistatic case, applicable to slowly varying charge distributions, the electric field satisfies ∇ · = ρ () and ∇ × = 0 (irrotational field), with = ε in linear media. These equations imply that the φ, defined by = -∇φ, obeys ∇ · (ε ∇φ) = -ρ for regions with ρ, or ∇²φ = 0 in charge-free spaces. This formulation allows the electric field to be computed as if charges were static at each instant, capturing capacitive effects in circuits without radiative losses. For the magnetoquasistatic case, relevant to slowly varying currents, the B obeys ∇ · B = 0 (no magnetic monopoles) and ∇ × H = J (Ampère's law without ), where H = B/μ in linear media and J is the . The A, satisfying B = ∇ × A in the Coulomb gauge ∇ · A = 0, enables solution via the equation ∇ × (μ⁻¹ ∇ × A) = J, often simplified to a Poisson-like form in uniform media. This approach accounts for inductive effects while ignoring propagation delays. A practical example occurs in low-frequency electrical circuits, such as those operating below the range, where the skin depth δ = √(2/ωμσ)—with μ the permeability and σ the conductivity—exceeds the conductor dimensions. This condition ensures uniform current distribution and field penetration, justifying uniform quasistatic fields inside components like inductors and allowing lumped circuit models to predict behavior accurately without full wave analysis.

Retarded time perspective

In the context of , the quasistatic approximation can be understood through the lens of retarded potentials, which represent the exact general solution to in the Lorenz gauge for prescribed charge and current densities. The at position r\mathbf{r} and time tt is expressed as ϕ(r,t)=14πϵ0∫ρ(r′,tr)∣r−r′∣ dV′,\phi(\mathbf{r}, t) = \frac{1}{4\pi \epsilon_0} \int \frac{\rho(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} \, dV', where tr=t−∣r−r′∣/ct_r = t - |\mathbf{r} - \mathbf{r}'|/c is the accounting for the finite propagation speed cc of electromagnetic signals, and ρ(r′,tr)\rho(\mathbf{r}', t_r) is the evaluated at that delayed time. The follows analogously: A(r,t)=μ04π∫J(r′,tr)∣r−r′∣ dV′,\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} \, dV', with J\mathbf{J} the current density at the retarded time. These integrals incorporate the causal structure of electromagnetic interactions, where effects from sources at r′\mathbf{r}' influence the field at r\mathbf{r} only after the light-travel delay ∣r−r′∣/c|\mathbf{r} - \mathbf{r}'|/c. The quasistatic limit emerges when this retardation becomes negligible, specifically when the maximum light-travel time across the system, on the order of L/cL/c for characteristic size LL, is much smaller than the intrinsic time scale τ\tau over which the sources vary significantly (L/c≪τL/c \ll \tau). In this regime, tr≈tt_r \approx t, allowing the potentials to simplify to instantaneous forms: ϕ(r,t)≈14πϵ0∫ρ(r′,t)∣r−r′∣ dV′,\phi(\mathbf{r}, t) \approx \frac{1}{4\pi \epsilon_0} \int \frac{\rho(\mathbf{r}', t)}{|\mathbf{r} - \mathbf{r}'|} \, dV', and similarly for A\mathbf{A}, effectively decoupling the fields from propagation delays and reducing the description to time-dependent but non-retarded integrals akin to static Coulomb and Biot-Savart laws. This viewpoint underscores the approximation's physical basis: it discards the finite-speed corrections inherent in full dynamic solutions, valid for scenarios where electromagnetic signals traverse the system nearly instantaneously relative to source evolution. For instance, in typical laboratory setups with dimensions L≈1L \approx 1 m operating at audio frequencies (f≈1f \approx 1 kHz, so τ≈1\tau \approx 1 ms), the condition holds since cτ≈3×105c \tau \approx 3 \times 10^5 m ≫L\gg L. This perspective bridges the quasistatic approximation to more complete dynamic treatments, revealing when and wave propagation effects—arising from higher-order expansions of the retardation—can be safely ignored. In antenna theory, it manifests as the near-field approximation, where for distances r≪λ/2πr \ll \lambda/2\pi (λ\lambda the ), the fields of a small radiating element like an electric dipole are dominated by quasistatic terms decaying as 1/r31/r^3, neglecting the 1/r1/r components that require full retardation. This regime is crucial for applications such as or , where the system's compactness ensures minimal propagation delays.

Mathematical foundations

Scaling analysis

The quasistatic approximation relies on nondimensional scaling to identify regimes where time-dependent effects can be neglected relative to dominant spatial or equilibrium processes, unifying criteria across , , and through small parameters that quantify timescale separations. In general, these parameters, denoted as ε, represent the ratio of a characteristic relaxation or timescale to the timescale of the external process, with the holding when ε ≪ 1. This scaling approach transforms governing equations into dimensionless form, revealing the of terms involving time derivatives compared to steady-state contributions. In , the small parameter is typically ε = τ_relax / τ_process, where τ_relax is the system's internal relaxation time to equilibrium (e.g., or mechanical equilibration) and τ_process is the duration of the imposed change. When ε ≪ 1, the system remains in near-equilibrium throughout, allowing static thermodynamic relations to approximate the dynamics. Order-of-magnitude estimates show that time derivatives of state variables scale as O(1/τ_process), while relaxation terms balance at O(1/τ_relax); thus, the approximation neglects transients if the evolves much slower than relaxation. This criterion ensures minimal beyond quasistatic limits in finite-time . In , of the Navier-Stokes equations introduces the St = L / (U τ_process) = f L / U, where f = 1/τ_process is the driving frequency, L is a , and U is a convective . The unsteady acceleration term scales as O(U / τ_process) = O(U² / L) · St, which is negligible compared to the convective term O(U² / L) when St ≪ 1, justifying the quasi-steady flow approximation. This holds for low-frequency oscillations where local acceleration is dwarfed by advection, as seen in airfoil pitching or scenarios at low reduced frequencies. In , the key parameter is ε = v / c or equivalently ε = ω L / c, with v the characteristic velocity of sources, ω the , L the system size, and c the . Time derivatives in scale as O(ω) for fields, while spatial gradients scale as O(1/L); the or retardation effects become small when ε ≪ 1, reducing to electroquasistatic or magnetoquasistatic limits. Dimensional analysis confirms that Faraday and Ampère terms balance without radiative corrections if the propagation time L/c ≪ τ_process. This scaling framework unifies the fields by highlighting a common structure: the quasistatic regime emerges when the dimensionless frequency parameter ε = ω L / U_char ≪ 1, where U_char is the characteristic speed (e.g., sound speed in fluids for acoustic limits, akin to c in electromagnetism, or equilibrium velocity in thermodynamics). In acoustic fluid flows, for instance, low St combined with low Mach number mirrors the EM condition, ensuring inertial or wave propagation terms do not dominate over quasistatic balances.

Derivation of approximation criteria

The quasistatic approximation is formalized through asymptotic expansions in perturbation theory, where a small parameter ε quantifies the slowness of temporal variations relative to spatial or other scales, allowing the neglect of time derivatives to leading order. Consider a general governing equation of the form ∂f/∂t = L, where L is a spatial operator and f represents the state variable (e.g., fields, densities, or thermodynamic potentials). Assume an asymptotic solution f = f₀ + ε f₁ + ε² f₂ + ..., with ε ≪ 1 representing the ratio of characteristic time scales, such as the rate of external driving to the system's intrinsic relaxation time. Substituting this expansion yields, at O(1): 0 = L[f₀], recovering the static equilibrium solution f₀. At O(ε): ∂f₀/∂t = L[f₁], where the time derivative of the leading-order solution drives the first correction, but higher-order terms remain small if ε is sufficiently small. This structure ensures that the quasistatic solution approximates the full dynamic behavior with controlled error, as validated in perturbation analyses of slow-varying systems. In , the quasistatic criterion emerges from the second law, dS ≥ đQ / T, with equality for . For a , the system remains in near-equilibrium, allowing the identification đQ ≈ T dS to hold exactly at leading order in the perturbation expansion. To derive this, expand the around a reference state: dS = (đQ / T) + ε δ, where δ ≥ 0 captures dissipative contributions from finite-rate effects, and ε parameterizes the deviation from infinitely slow driving. The du = đQ - p dV implies that, to O(1), the process satisfies the reversible relation du = T dS - p dV, with zero (δ = 0). Higher-order terms introduce small irreversibilities, such as viscous heating or frictional losses, but these are O(ε) and negligible for slow processes, justifying the approximation đQ = T dS for computing state changes. This first-order reversibility underpins applications like slowly compressed gases, where the error in calculation scales with the driving rate. In , the perturbation expansion applies to the unsteady Navier-Stokes equations, ∇ · u = 0 and ∂u/∂t + (u · ∇) u = -∇p/ρ + ν ∇² u, where the time derivative term is scaled by ε = τ / T, with τ the flow relaxation time and T the external variation time (T ≫ τ). Expanding u = u₀ + ε u₁ + ..., p = p₀ + ε p₁ + ..., the leading-order equations (O(1)) reduce to the steady incompressible Navier-Stokes: ∇ · u₀ = 0, (u₀ · ∇) u₀ = -∇p₀/ρ + ν ∇² u₀, treating the flow as quasi-steady despite slow parameter changes (e.g., varying or ). The O(ε) correction incorporates ∂u₀/∂t on the right-hand side, providing unsteady adjustments like or history effects in low-Reynolds-number flows. This expansion is valid when ε ≪ 1, as in creeping flows or at low reduced frequencies, where the steady-state solution dominates. Error bounds in the quasistatic approximation arise from the in the asymptotic series, typically O(ε) for the first correction and higher for subsequent terms. For instance, the neglected unsteady contributions, such as ∂u/∂t in fluids or finite-rate dissipation in , introduce relative errors bounded by ε times the magnitude of the leading-order solution, e.g., |f - f₀| ≤ C ε ||f₀|| for some constant C depending on the operator norms. These bounds confirm the approximation's accuracy for sufficiently slow processes, with higher-order expansions (e.g., including ε f₁) reducing the error to O(ε²).

References

  1. https://web.mit.edu/6.013_book/www/chapter3/3.3.html
  2. https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/06%3A_Electromagnetism/6.03%3A_Quasistatic_Approximation_and_the_Skin_Effect
  3. https://pubs.aip.org/aapt/ajp/article/75/3/230/1056316/Electromagnetics-from-a-quasistatic-perspective
  4. https://farside.ph.utexas.edu/teaching/sm1/Thermalhtml/node37.html
  5. https://pressbooks.online.ucf.edu/phy2048tjb/chapter/3-4-thermodynamic-processes/
  6. https://web.pa.msu.edu/courses/2014fall/PHY842/Lectures/lec1402.pdf
  7. https://sites.pitt.edu/~jdnorton/papers/Reversible.pdf
  8. https://www.sciencedirect.com/science/article/abs/pii/S1355219815300563
  9. https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/03%3A_The_First_Law_of_Thermodynamics/3.05%3A_Thermodynamic_Processes
  10. https://pubs.aip.org/aapt/ajp/article/72/6/799/1062283/Critical-slowing-down-at-a-bifurcation
  11. https://courses.physics.illinois.edu/phys213/su2023/prelecture6.pdf
  12. http://www.physics.smu.edu/scalise/P3374fa15/FirstLaw.pdf
  13. https://mcgreevy.physics.ucsd.edu/s12/lecture-notes/chapter03.pdf
  14. https://astronomy.nmsu.edu/jasonj/565/docs/10_01.pdf
  15. https://www.physics.smu.edu/scalise/P3374fa15/FirstLaw.pdf
  16. https://courses.washington.edu/phy115a/slides/12-115sp14-thermo2.pdf
  17. https://www.usna.edu/Users/physics/mungan/_files/documents/Publications/TPT4.pdf
  18. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node34.html
  19. https://web.mit.edu/16.unified/www/FALL/thermodynamics/chapter_7.htm
  20. https://oyc.yale.edu/physics/phys-200/lecture-23
  21. https://scholar.harvard.edu/files/schwartz/files/5-thermodynamics.pdf
  22. https://www.damtp.cam.ac.uk/user/hinch/teaching/FluidsIInotes.pdf
  23. https://moodle2.units.it/pluginfile.php/464630/mod_resource/content/1/FD_7.pdf
  24. https://royalsocietypublishing.org/doi/10.1098/rsos.180941
  25. https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/unsteady-motion-of-solid-bodies-in-creeping-flows/C6C002A34D584481E4E05D60C2E67C9E
  26. https://link.aps.org/doi/10.1103/PhysRevLett.104.187801
  27. https://www.esaim-m2an.org/articles/m2an/pdf/2005/03/m2an01EDP.pdf
  28. https://gidropraktikum.narod.ru/Happel-Brenner-eng.pdf
  29. https://www.mdpi.com/2073-8994/13/6/1020
  30. https://royalsocietypublishing.org/doi/10.1098/rstb.1984.0049
  31. https://link.aps.org/doi/10.1103/PhysRevFluids.4.110503
  32. https://pubs.aip.org/asa/jasa/article/151/5/3129/2838386/Experimental-study-on-the-quasi-steady
  33. https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism.pdf
Add your contribution
Related Hubs
User Avatar
No comments yet.