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Blasius Boundary Layer

Posted on October 14, 2025 by user

The Blasius boundary layer is the canonical steady, two‑dimensional laminar solution describing the viscous shear region that develops along a semi‑infinite flat plate placed parallel to a uniform free stream. Under the assumptions of steady flow, two‑dimensionality, incompressibility, laminar viscosity and a constant external velocity U aligned with the plate, a thin layer adjacent to the wall forms in which the streamwise velocity increases from zero at the no‑slip surface to U far from the wall; this layer defines the boundary layer and its characteristic thickness grows downstream.

Mathematically the Blasius profile arises from a self‑similar reduction of Prandtl’s boundary‑layer equations. Introducing the similarity coordinate η = y √(U/(ν x)) (with x the streamwise and y the wall‑normal coordinate and ν the kinematic viscosity) collapses the governing partial differential equations to the ordinary differential equation
f‴ + (1/2) f f″ = 0,
subject to f(0) = 0, f′(0) = 0 and f′(∞) = 1. The dimensionless streamwise velocity is given by u/U = f′(η). The solution implies a downstream growth of the layer of order δ ∼ √(ν x / U), equivalently δ/x ∼ Re_x^−1/2 where Re_x = U x / ν.

The Falkner–Skan family generalizes Blasius by admitting wedge‑type outer flows and nonzero streamwise pressure gradients; varying the wedge parameter produces boundary layers under favorable or adverse pressure gradients and recovers the Blasius case as the special parallel‑plate limit.

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Prandtl’s boundary‑layer formulation isolates the thin viscous region adjacent to a solid surface in high‑Reynolds‑number flows. In this region the streamwise velocity profile, when expressed in terms of an appropriate similarity coordinate η, collapses onto a single curve u(η)/U: profiles at different streamwise stations differ only by scaling, so the normalized streamwise velocity depends primarily on η rather than on x and y separately.

Prandtl obtained this reduction by a scaling analysis of the Navier–Stokes equations, showing that, except very near a leading edge singularity, roughly half of the terms are asymptotically small inside a thin boundary layer. For steady, incompressible flow with constant kinematic viscosity ν and density ρ the resulting boundary‑layer equations are
Continuity: ∂u/∂x + ∂v/∂y = 0
x‑momentum: u ∂u/∂x + v ∂u/∂y = −(1/ρ) ∂p/∂x + ν ∂²u/∂y²
y‑momentum: 0 = −∂p/∂y

Here x is the coordinate along the plate in the flow direction, y is normal to the plate, u and v are the corresponding velocity components, and p denotes pressure. The y‑momentum reduction implies that pressure is effectively uniform across the layer thickness: p = p(x), so the pressure within the boundary layer is imposed by the outer inviscid flow at the same x.

When the outer pressure distribution and geometry permit, one can introduce similarity scalings that remove x‑dependence from the partial differential system and reduce it to a nonlinear ordinary differential problem. Blasius, following Prandtl’s ideas, derived the canonical similarity solution for a steady boundary layer developing on a semi‑infinite flat plate by assuming the streamwise pressure gradient is negligible (so the pressure along the plate equals the constant free‑stream pressure). Under these assumptions the boundary‑layer PDEs collapse to the classical Blasius ODE governing the universal flat‑plate profile.

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The boundary‑layer approximation and the Blasius similarity solution have limited locality: they fail in a small region close to the plate leading edge where the neglected terms are not small, and the assumption of negligible ∂p/∂x must be validated against the specific external flow.

Blasius equation — first‑order boundary layer

For a uniform free stream (∂p/∂x = 0) the boundary‑layer equations admit a scaling symmetry x → c^2 x, y → c y, u → u, v → v/c, which allows reduction of the two‑dimensional partial‑differential problem to a single ordinary differential equation via a self‑similar ansatz. Introducing the similarity variable η = y/δ(x) with δ(x) ∝ √(ν x / U) (conveniently written η = y √(U/(ν x))) and a stream function ψ = √(ν U x) f(η) yields a dimensionless profile f(η) that completely determines the velocity field. In this representation the tangential and normal velocity components become
u(x,y) = U f′(η),
v(x,y) = (1/2) √(ν U / x) [η f′(η) − f(η)],
where primes denote d/dη.

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Substitution into the x‑momentum equation produces the classical Blasius ordinary differential equation, commonly written
f‴ + (1/2) f f″ = 0,
subject to no‑slip and impermeability at the wall and matching to the free stream:
f(0) = 0, f′(0) = 0, f′(∞) = 1.
This third‑order nonlinear boundary‑value problem is solved numerically (e.g., by shooting). The physical solution requires the wall curvature f″(0) ≈ 0.33206, which also appears as the leading coefficient in the near‑wall Taylor expansion f(η) = (1/2) α η^2 + O(η^5) with α ≈ 0.33206, implying the familiar linear increase of u with y close to the plate.

Far from the wall the Blasius profile approaches the free stream with an offset:
f(η) = η − β + smaller, rapidly decaying corrections,
with the displacement constant β ≈ 1.7207876575; f′(η) → 1 as η → ∞ so u → U. Corresponding scaled quantities attain finite asymptotes: the nondimensional normal velocity and a suitably scaled transverse pressure gradient approach constant values (approximately 0.86 and 0.43, respectively), while the scaled y‑pressure gradient near the wall tends to ≈0.16603. After nondimensionalizing the y‑momentum equation one obtains an explicit expression for the scaled transverse pressure gradient in terms of f and its derivatives:
(x^2/(U^2 δ^)) (1/ρ) ∂P/∂y = (1/2) η f‴ + (1/2) f″ − (1/4) f f′ + (1/4) η f′^2 + (1/4) η f f″,
where δ^
denotes the displacement thickness and ρ the fluid density.

Several characteristic thickness measures follow closed‑form scalings proportional to √(ν x / U): the 99% boundary‑layer thickness δ_99 ≈ 5.29 √(ν x / U), the displacement thickness δ^* = 1.72 √(ν x / U), and the momentum thickness θ = 0.665 √(ν x / U). The local wall shear (from ∂u/∂y at y = 0) is conventionally expressed in Blasius form, e.g. through the skin‑friction coefficient c_f = 0.664/√(Re_x) so that τ_w = (1/2) ρ U^2 c_f, and thus τ_w ∝ 0.332 ρ U^2 / √(Re_x). Integrating τ_w along a plate of length l and accounting for both sides yields the total drag scaling F = 1.328 √(ρ μ l U^3) (factor 2 for both faces).

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For graphical comparison independent of x and U, profiles are typically plotted in nondimensional form: u/U for tangential velocity, v/√(ν U / x) for normal velocity, and [(1/ρ) dP/dy]/√(ν U^3 / x^3) for the y‑pressure gradient. Finally, the Blasius solution satisfies the von Kármán integral relations in reduced form: the momentum integral gives τ_w/(ρ U^2) = ∂θ/∂x + v_w/U, and the corresponding energy integral is (2 ε)/(ρ U^3) = ∂δ_3/∂x + v_w/U, linking local shear and dissipation to the growth rates of the integral thicknesses and to any wall transpiration v_w.

Uniqueness of the Blasius solution

For steady two-dimensional flow over a flat plate the classical Blasius boundary-layer profile is not uniquely determined: Prandtl’s transposition theorem and later analyses (e.g., Stewartson, Libby) show that the base solution admits an infinite, discrete set of admissible perturbations. These perturbations are eigenfunctions of the linearized boundary-layer operator, each associated with a distinct discrete eigenvalue, that satisfy the homogeneous wall and far-field conditions and decay exponentially in the wall-normal direction. Because they vanish algebraically at the wall and decay rapidly away from it, these modes leave the leading-order wall constraints and the asymptotic outer flow unchanged, permitting their linear superposition on the Blasius profile without violating boundary conditions. The principal (lowest) eigenmode has a simple structure: it is proportional to the streamwise derivative of the Blasius solution and therefore represents, in mathematical terms, an infinitesimal translation of the base profile in the x-direction. Physically this translation mode corresponds to an uncertainty in the effective streamwise origin of the boundary-layer coordinate system—small errors in origin placement produce admissible perturbations that decay away from the wall and do not alter the outer flow.

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Within the Blasius framework the viscous boundary-layer approximation generates a nonzero transverse (vertical) velocity at large distances from the wall; this induced motion must be accommodated by a sequence of corrections comprising an outer inviscid layer and corresponding inner (boundary-layer) adjustments. The full solution therefore emerges by iterative outer–inner matching: each correction in one region produces a further induced velocity that is resolved in the other region at the next order.

At first order the induced far-field vertical velocity is nonzero and decays algebraically with streamwise distance, v = 0.86 sqrt(ν U / x), indicating an x−1/2 decline (ν is kinematic viscosity, U the free-stream velocity, and x the streamwise coordinate). The second-order viscous correction within the inner boundary layer, however, vanishes identically: the second-order inner solution is zero, so further nontrivial adjustments to the vertical velocity must be sought in the outer (inviscid) region and introduced into the inner region only via matching conditions.

This matched asymptotic structure separates the streamfunction ψ(x,y) into an outer inviscid expansion and an inner viscous expansion. The leading outer form may be written
ψ(x,y) ∼ y − sqrt(ν/(U x)) β ℜ{ sqrt(x + i y) },
where β is an eigenparameter and ℜ denotes the real part of the complex square root. The leading inner (boundary-layer) form scales as
ψ(x,y) ∼ sqrt(ν U x) f(η) + 0,
with η the usual similarity coordinate across the layer and f(η) the inner profile (the Blasius-type solution); the “+ 0” signifies the absence of a homogeneous second-order inner correction in the matched representation.

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Because the linearized higher-order problems admit homogeneous eigensolutions, any one of an infinite set of such eigensolutions may be superposed on the particular matched solution, so the full matched field is non-unique up to these additions. The local Reynolds number Re = Ux/ν governs the relative scaling of the inner and outer expansions and appears naturally in the asymptotic prefactors. For a unique physical solution one must therefore impose additional selection criteria to fix the appropriate eigenparameter(s).

The vanishing of the second‑order inner solution forces all would‑be corrections to the outer flow at third order to vanish; consequently the third‑order outer problem coincides exactly with the second‑order outer problem and no new outer correction appears at O(ν^{3/2}).

In the inner (boundary‑layer) region the streamfunction admits the asymptotic expansion whose leading contribution scales as √(2 ν U x) f(η), where η is the usual inner similarity coordinate. The O(√(ν U x)) term is followed by a zero second term, so the first nontrivial inner correction occurs at order (ν/(U x))^{3/2}. That third‑order inner structure contains both a logarithmically amplified eigensolution and a non‑logarithmic part:
(ν/(U x))^{3/2} [ (1/√(2 x)) f_{31}(η) + log((U x)/ν) √(x/2) f_{32}(η) ].
Here f_{32}(η) is the leading eigensolution of the first‑order boundary‑layer problem; this eigensolution corresponds to the x‑derivative of the first‑order Blasius profile. By contrast f_{31}(η) enters the non‑logarithmic component but is not uniquely determined by the asymptotic matching, leaving an arbitrary constant in the third‑order inner correction.

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Thus the asymptotic hierarchy at third order yields no new outer correction, a dominant inner term √(2 ν U x) f(η) with a vanishing next term, and a third‑order inner correction at order (ν/(U x))^{3/2} composed of a log‑amplified eigensolution and a nonunique companion mode; the undetermined constant associated with f_{31} prevents a unique closed‑form representation of the full third‑order correction.

A uniform wall-normal suction v(0) = −V (V > 0) is imposed at the solid boundary to delay separation; x denotes the streamwise coordinate measured from the leading edge and y the coordinate normal to the wall, with v(0) < 0 indicating flow into the wall. To capture a self-similar transverse structure while retaining the cumulative effect of suction, introduce the stream function and similarity variables
ψ = √(2 U ν x) f(ξ,η), ξ = V √(x/(2 U ν)), η = √(U/(2 ν x)) y,
where U is the free-stream velocity and ν the kinematic viscosity. The velocity components follow directly from ψ as
u = U ∂f/∂η, v = −√(U ν/(2 x)) [ f + ξ ∂f/∂ξ − η ∂f/∂η ],
so the streamwise velocity scales with U and is given by the η-derivative of f, while the wall-normal velocity contains the prefactor √(U ν/(2 x)) and depends on f and its ξ- and η-derivatives.

Under this transformation the boundary-layer equations reduce to the nonlinear partial differential equation
∂^3 f/∂η^3 + f ∂^2 f/∂η^2 + ξ [ (∂f/∂ξ)(∂^2 f/∂η^2) − (∂^2 f/∂ξ ∂η)(∂f/∂η) ] = 0,
a third-order-in-η PDE with mixed ξ–η derivative coupling that encodes the influence of suction. The similarity problem is closed by the conditions
f(ξ,0) = ξ, ∂f/∂η(ξ,0) = 0, ∂f/∂η(ξ,∞) = 0,
which fix the stream-function value at the wall, enforce no-slip in the streamwise direction, and prescribe the far-field streamwise velocity.

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Physically, ξ = V √(x/(2 U ν)) is a nondimensional measure of cumulative suction strength (growing like √x and proportional to V), while η = √(U/(2 ν x)) y scales the normal coordinate by the local boundary-layer thickness ∼√(ν x/U). Suction therefore modifies the transverse boundary-layer structure through the ξ-dependent mixed-derivative terms; in the limit x → 0 (ξ → 0) the suction terms vanish and the solution reduces locally to the classical Blasius boundary layer, consistent with Thwaites’s observation that suction is negligible arbitrarily close to the leading edge.

Von Mises transformation

In the reformulation of the Blasius boundary‑layer problem originally completed numerically by Iglisch in 1944, a further von Mises change of variables produces a problem statement in a new dependent variable ϕ(σ,τ) with independent variables σ and τ that is well suited to numerical marching from σ = 0. The transformation recasts the boundary‑layer equations into a parabolic evolution in σ with transverse (inner) coordinate τ and second‑order derivatives only in τ, so that σ functions as the marching (evolution) coordinate and τ as the transverse coordinate.

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The algebraic definitions used in this transformation are
σ = 2 ξ,
ψ − V x = (U ν)/(2 V) · σ τ^2,
ϕ = 4 u^2 / U^2,
χ = U^2 − u^2,
and, as they appear in the source formulation, two alternative algebraic representations for χ are given: U^2(1 − V^4) and U^2(1 − V/4). These exact relations (including the numerical constants 2 and 4 and the date 1944) specify the transformed variables to be used in coding or analysis.

The transformed partial differential equation is presented in the source in two equivalent algebraic forms that must be preserved when implementing or interpreting the model:
ϕ ∂^2ϕ/∂τ^2 + (2 σ τ + τ^3 − ϕ τ) ∂ϕ/∂τ = 2 σ τ^2 ∂ϕ/∂σ,
and the alternative rendering
sqrt(ϕ) · (∂^2ϕ/∂τ^2) + (2 σ τ + τ^3 − (sqrt(ϕ)/τ)) · (∂ϕ/∂τ) = 2 σ τ^2 · (∂ϕ/∂σ).
Both expressions appear in the formulation and hence must be accommodated in analysis and numerical implementation.

Boundary and limit conditions are specified exactly as
ϕ(0,τ) = 4 for all τ,
ϕ(σ,0) = 0 for all σ,
ϕ(σ,∞) = 4,
providing the initial profile at σ = 0, the transverse boundary at τ = 0, and the far‑field asymptotic state as τ → ∞. The parabolic nature of the transformed equation implies that numerical marching can proceed outwards in σ from the known profile at σ = 0.

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Numerical implementation requires attention to structural features and near‑singular behavior: the equation contains τ^2 and τ^3 factors and, in the second form, an explicit −(sqrt(ϕ)/τ) term producing a 1/τ singular factor. Because ϕ(σ,0) = 0 and terms scale with powers of τ, discretizations near τ = 0 must avoid division‑by‑zero and accurately represent the small‑τ asymptotics (for example by appropriate regularization, analytical near‑field expansion, or tailored grid/staggering). Likewise, the condition ϕ(σ,∞) = 4 prescribes the far‑field state and requires an adequate truncation and boundary treatment on the numerical τ‑domain.

Symbols and constants that appear explicitly and must be preserved in formulation and coding include: 1944, σ, τ, ξ, ψ, V, x, U, ν, u, ϕ, χ, the numerical constants 2 and 4 (e.g., σ = 2 ξ and ϕ = 4 u^2 / U^2), and the domain endpoints 0 and ∞ used in the boundary specifications.

In an asymptotic suction boundary layer a uniform normal suction through the wall generates a mean convective motion toward the surface that opposes the diffusive spread of momentum away from it. Because wall suction and viscous diffusion act in opposite directions, the boundary layer does not continue to thicken downstream; instead, beyond a certain streamwise distance the flow approaches a steady, x‑independent profile. The downstream region in which this asymptotic state applies is characterized by x ≫ νU/V^2, where x is the distance from the leading edge, ν the kinematic viscosity, U the external streamwise velocity and V the imposed (positive) suction speed normal to the wall.

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In this suction-dominated regime the streamwise length scale decouples from x and the vertical structure is set by the balance of suction and viscosity. The relevant thickness is δ = ν/V, so that the wall‑normal coordinate scales with ν/V. The exact steady solution for the velocity field is simple and exponential: u(y) = U[1 − exp(−yV/ν)] for the streamwise component and v(y) = −V for the normal component (the negative sign indicating flow into the wall). The streamwise velocity therefore rises from the no‑slip value at the wall toward U over the distance δ with an exponential approach, while the constant normal velocity represents the uniform suction flux that sustains the steady layer.

This behaviour contrasts with the classical Blasius boundary layer in the absence of suction, whose thickness grows with x (classically O((νx/U)^{1/2})). Suction removes mass from the near‑wall region and arrests that growth, producing a finite, x‑independent thickness and an exponential profile. Historically, the steady asymptotic suction solution was derived by Griffith and F. W. Meredith; later analyses by Stewartson and others established how the developing entrance layer near the leading edge mathematically matches onto this downstream asymptotic state.

Compressible Blasius boundary layer

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The compressible Blasius problem considered here retains variable thermophysical properties across the layer, with density ρ, viscosity μ and thermal conductivity κ allowed to vary and the wall specified by a given specific enthalpy h_w. Cartesian coordinates are used with x along the plate and y normal to it; velocity components are u(x,y) and v(x,y), and U denotes the free‑stream streamwise velocity.

The governing boundary‑layer conservation laws, written in enthalpy form and keeping full variation of properties, are
– continuity: ∂(ρ u)/∂x + ∂(ρ v)/∂y = 0,
– x‑momentum: ρ (u ∂u/∂x + v ∂u/∂y) = ∂/∂y (μ ∂u/∂y),
– energy (enthalpy): ρ (u ∂h/∂x + v ∂h/∂y) = ∂/∂y ((μ/Pr) ∂h/∂y) + μ (∂u/∂y)^2.

The energy balance explicitly includes viscous dissipation μ(∂u/∂y)^2, which couples the momentum field into enthalpy (and thus temperature) evolution. Thermal diffusion appears in the form ∂/∂y((μ/Pr)∂h/∂y), so thermal transport is controlled jointly by the local viscosity and the Prandtl number, defined here with free‑stream reference properties as Pr = c_{p,∞} μ_∞ / κ_∞.

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Boundary conditions are the no‑slip and impermeability constraints at the wall (y = 0): u = 0, v = 0 and h = h_w(x). Far from the wall (y → ∞) or at the leading edge (x = 0) the flow approaches the external state: u → U and h → h_∞.

A self‑similar reduction analogous to the incompressible Blasius solution can be obtained only if the governing equations and boundary conditions are invariant under the scaling
x → c^2 x, y → c y, u → u, v → v/c, h → h, ρ → ρ, μ → μ.
This set of scalings preserves the structure of the variable‑property boundary‑layer equations; however, it imposes a physical constraint on the wall specification: h_w must be constant in x. If h_w varies with x the required invariance is violated and a similarity solution is not admitted.

Because ρ, μ and κ vary across the layer and viscous heating sources the enthalpy field, mass, momentum and energy are intrinsically coupled. Consequently, similarity is achievable only when both the kinematic/geometric scalings hold and the wall enthalpy is x‑independent; otherwise the flow fields remain functions of x and y and must be treated as a two‑variable boundary‑layer problem.

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The Howarth–Dorodnitsyn transformation casts the compressible Blasius boundary-layer problem into a self-similar form by introducing a density-weighted normal coordinate and nondimensional streamfunction and enthalpy variables, thereby reducing the boundary-layer partial differential equations to a coupled system of ordinary differential equations. The similarity coordinate and dependent variables are defined by
η = (U / √(2 ρ_∞ μ_∞ U x)) ∫0^y ρ(ỹ) dỹ,
f(η) = ψ / √(2 ρ
∞ μ_∞ U x),
\tilde{h}(η) = h / h_∞,
with the nondimensional density and viscosity \tilde{ρ}=ρ/ρ_∞, \tilde{μ}=μ/μ_∞. Here ψ is the streamfunction, x and y are the streamwise and wall-normal coordinates, U the free‑stream velocity, and ρ_∞, μ_∞, h_∞ the free‑stream density, viscosity and enthalpy respectively.

In similarity form the momentum and energy balances become two coupled ordinary differential equations (primes denote d/dη):
( \tilde{ρ}\,\tilde{μ}\,f” )’ + f\,f” = 0,
( \tilde{ρ}\,\tilde{μ}\,\tilde{h}’ )’ + Pr\,f\,\tilde{h}’ + Pr\,(γ−1)\,M^2\,\tilde{ρ}\,\tilde{μ}\,(f”)^2 = 0,
where Pr is the Prandtl number, γ the ratio of specific heats and M=U/c_∞ the free‑stream Mach number. The last term in the energy equation represents viscous dissipation, which increases local enthalpy/temperature inside the layer and becomes important at high Mach numbers.

Solution of the similarity ODEs requires constitutive closure in the form of thermophysical relations expressing density and viscosity as functions of enthalpy (or temperature): \tilde{ρ}=\tilde{ρ}(\tilde{h}), \tilde{μ}=\tilde{μ}(\tilde{h}). Typical air-model approximations used in examples are γ=1.4, Pr≈0.7, \tilde{ρ}≈\tilde{h}^{−1} (inverse scaling of density with enthalpy/temperature) and \tilde{μ}≈\tilde{h}^{2/3} (viscosity power‑law in enthalpy/temperature). For constant specific heat c_p, nondimensional enthalpy equals nondimensional temperature, \tilde{h}=\tilde{θ}=T/T_∞.

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The similarity boundary conditions enforce impermeability and no‑slip at the wall, a prescribed wall enthalpy, and matching to the free stream:
f(0)=0, f'(0)=0, \tilde{h}(0)=\tilde{h}_w, f'(∞)=1, \tilde{h}(∞)=1.
With these definitions and closures the Howarth–Dorodnitsyn reduction provides a compact, computationally tractable formulation for compressible boundary‑layer profiles including viscous heating effects.

First-order Blasius boundary layer in parabolic coordinates

For the first-order Blasius boundary layer, the governing equations reduce to a parabolic partial differential equation; this motivates adopting a curvilinear system whose coordinate lines reflect the PDE’s intrinsic parabolic geometry. A convenient and compact representation of the required map from Cartesian coordinates (x,y) to parabolic coordinates (ξ,η) is the complex quadratic relation x + i y = 1/2 (ξ + i η)^2. Extracting real and imaginary parts yields the elementary relations x = 1/2(ξ^2 − η^2) and y = ξ η, from which the differential forms follow directly as dx = ξ dξ − η dη and dy = η dξ + ξ dη.

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The induced metric is diagonal and isotropic in the two curvilinear directions: ds^2 = dx^2 + dy^2 = (ξ^2 + η^2)(dξ^2 + dη^2). Hence the scale factors are equal, h_ξ = h_η = sqrt(ξ^2 + η^2), and the Jacobian (area-scale factor) of the mapping is J = ξ^2 + η^2. Coordinate lines therefore form two orthogonal families of confocal parabolas: surfaces of constant ξ are parabolas with vertex at (ξ^2/2, 0) opening toward decreasing x, while constant η curves have vertex at (−η^2/2, 0) opening toward increasing x. Because the underlying mapping is an analytic quadratic (holomorphic) transformation, the resulting coordinate net is conformal (angle-preserving). These properties—alignment of coordinate lines with parabolic characteristics, equal scale factors, and orthogonality—make parabolic coordinates a natural and efficient choice for analyzing boundary-layer flows governed by parabolic PDEs such as the Blasius problem.

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