Weber number
The Weber number (We) is a dimensionless quantity in fluid mechanics that characterizes the ratio of deforming inertial forces to stabilizing surface tension forces in multiphase fluid flows, particularly those involving an interface between two immiscible fluids.[1][2] It is mathematically defined as We = \frac{\rho v^2 L}{\sigma}, where \rho is the density of the fluid, v is the characteristic velocity, L is the characteristic length scale, and \sigma is the surface tension.[2] This parameter plays a crucial role in determining the dominance of kinetic energy over surface energy in such systems, helping to predict behaviors like droplet breakup, bubble formation, and interface stability.[2][1] When the Weber number is low, surface tension forces prevail, maintaining coherent structures such as intact droplets or films; conversely, high values indicate that inertial forces overcome surface tension, leading to fragmentation or dispersion.[2] The Weber number finds extensive applications in engineering contexts, including the analysis of spray atomization for fuel injection in combustion systems, emulsion production in chemical processing, and the dynamics of thin liquid films in coating or wetting processes.[1][2] It is often used alongside other dimensionless numbers, such as the Reynolds number for viscous effects and the Bond number for gravitational influences, to provide a comprehensive scaling of multiphase phenomena.[2]Fundamentals
Definition
The Weber number (We) is a dimensionless quantity in fluid mechanics that characterizes the relative importance of inertial forces to surface tension forces in flows involving interfaces between fluids. It is defined mathematically as \We = \frac{\rho v^2 L}{\sigma}, where \rho is the density of the phase exerting the inertial forces (e.g., the continuous phase in multiphase flows, in kg/m³), v is the characteristic velocity relative to the interface (in m/s), L is the characteristic length scale (in m, such as the diameter of a droplet or the radius of a liquid jet), and \sigma is the surface tension coefficient (in N/m).[2] This formulation arises as the ratio of inertial forces, approximated as \rho v^2 L^2, to surface tension forces, approximated as \sigma L. Equivalently, from an energy perspective, the Weber number represents the ratio of kinetic energy, on the order of \rho v^2 L^3, to surface energy, on the order of \sigma L^2.[3][2] The dimensionless nature of We is confirmed by dimensional analysis: the units of the numerator \rho v^2 L yield kg/(m·s²), matching those of the denominator \sigma (also kg/s²), resulting in a unitless quantity.[2]Physical Significance
The Weber number characterizes the relative dominance of inertial forces over surface tension forces in fluid flows involving interfaces between phases, such as liquid-gas or liquid-liquid systems.[1] When the Weber number exceeds unity (We > 1), inertial effects prevail, promoting deformation, disruption, and breakup of fluid structures like droplets or interfaces.[4] Conversely, when We < 1, surface tension forces dominate, favoring stable, minimally deformed configurations such as spherical droplets or bubbles.[4] In low-Weber-number regimes, flows are capillary-dominated, where surface tension maintains coherent shapes, as observed in small bubbles rising in liquids that remain nearly spherical due to negligible inertial deformation.[4] High-Weber-number regimes, on the other hand, are inertia-dominated, leading to phenomena like splashing upon drop impact on surfaces, where kinetic energy overcomes interfacial resistance to eject secondary droplets.[5] The Weber number plays a critical role in multiphase flows by predicting interface stability, the onset of wave formation on free surfaces, and thresholds for structural breakup.[6] For instance, a critical Weber number of approximately 12 marks the transition to bag breakup of droplets in air streams, beyond which inertial forces cause fragmentation.[7]Historical Development
Origin and Introduction
The Weber number emerged in the early 20th century amid advancements in fluid mechanics, particularly within the fields of naval architecture and wave theory spanning the 1910s to 1930s, as researchers sought to apply principles of dynamic similitude for scaling model experiments to full-scale phenomena.[8] These efforts were driven by the need to predict behaviors in ship hydrodynamics and surface wave propagation, where forces like inertia, viscosity, gravity, and surface tension interact across different scales.[9] Building on foundational ideas from dimensional analysis, the number addressed the challenge of maintaining similarity in model testing for interfacial flows.[8] Moritz Weber, a professor of naval mechanics at the Technische Hochschule Charlottenburg in Berlin, formalized the Weber number in his 1919 work on similitude mechanics, Die Grundlagen der Ähnlichkeitsmechanik und ihre Verwertung bei Modellversuchen.[10] In this treatise, Weber introduced the dimensionless group to quantify the balance between inertial forces and surface tension in fluid flows, particularly in the context of capillary waves and ship hydrodynamics.[8] His formulation arose from applying Buckingham's pi theorem to problems involving free surfaces, enabling predictions of wave stability and droplet formation without scale-dependent discrepancies.[9] Weber's initial applications focused on analyzing surface waves and interfacial stability in naval mechanics, where the number helped correlate model-scale experiments with prototype behaviors in water wave resistance and spray dynamics.[8] Preceding Weber's formalization, related dimensionless considerations appeared in Lord Rayleigh's 1879 analysis of liquid jet instability, where the ratio of inertial to surface tension effects governed breakup wavelengths, though not explicitly as a named parameter.[11] Weber's contribution built upon this by integrating it into a broader similitude framework for practical engineering applications.[8]Naming and Legacy
The Weber number is named after Moritz Weber (1871–1951), a German professor of naval mechanics at the Technical University of Berlin, who formalized its use as a dimensionless parameter in similitude analysis and model studies during the early 20th century.[12] In his seminal 1919 paper, Weber developed the parameter as a ratio characterizing capillary effects relative to inertial forces, though it was later designated in his honor. Weber also introduced the term "similitude" to denote the principle of dynamic similarity achieved through such dimensionless groups in physical modeling. Post-World War II, the Weber number saw widespread adoption in engineering literature as computational and experimental techniques advanced, solidifying its role in analyzing interfacial phenomena. It appeared prominently in standard references, including the CRC Handbook of Chemistry and Physics from the 1980s onward, where it is listed among key dimensionless quantities for fluid systems. This recognition stemmed from its utility in scaling complex flows, building on Weber's foundational contributions to similarity principles outlined in his 1930 work on the general similitude principle in physics.[13] The legacy of the Weber number endures in its profound influence on multiphase flow modeling, where it provides a critical balance between inertial and surface tension forces to predict interface stability and breakup. It remains indispensable in computational fluid dynamics (CFD) simulations and experimental setups for deriving scaling laws in diverse systems, from macroscale industrial processes to nanoscale interactions. Notable milestones highlight its evolution: it achieved first widespread application in the 1950s within aerospace engineering, notably for simulating rocket fuel atomization and spray dynamics in propulsion systems.[14] In more recent decades, since the 2000s, the parameter has been extended to microscale flows, such as in microfluidics for controlling droplet formation and transport in lab-on-a-chip devices.Mathematical Formulation
General Expression
The Weber number We is expressed in its standard form as We = \frac{\rho v^2 L}{\sigma}, where \rho is the density of the fluid, v is the characteristic velocity of the flow, L is the characteristic length scale associated with the fluid interface (such as a droplet diameter or jet radius), and \sigma is the surface tension at the interface.[2] This formulation quantifies the ratio of inertial forces to surface tension forces within multiphase flows. The nondimensional Weber number emerges from scaling analysis of the relevant pressures: the inertial pressure scales as \rho v^2, while the capillary pressure scales as \sigma / L, yielding We as their ratio to compare the dominance of each in deforming interfaces. In scenarios involving two immiscible fluids, the expression accounts for the relative motion across the interface by replacing the velocity with the difference between the phase velocities: We = \frac{\rho (v_1 - v_2)^2 L}{\sigma}, where v_1 and v_2 are the velocities of the two fluids. For gas bubbles rising in a liquid, the characteristic length L is conventionally the bubble diameter, emphasizing deformation under ambient flow.[15] Similarly, for liquid jets issuing into a surrounding medium, L is taken as the jet radius to capture surface wave instabilities leading to breakup.[16] Critical values of the Weber number delineate regimes of stability and instability at interfaces; for instance, in inviscid liquid bridge pinch-off, We \approx 4 signals the onset of capillary-driven rupture.[17]Derivation via Buckingham Pi Theorem
The derivation of the Weber number employs the Buckingham Pi theorem, a method of dimensional analysis that identifies dimensionless groups from the variables governing a physical phenomenon. This approach is particularly useful for problems involving free-surface flows where surface tension plays a role, such as the breakup of liquid jets or droplet formation. For the Weber number, the analysis assumes an inviscid flow (neglecting viscosity to focus on inertia and surface tension) with a free surface, and gravity effects are excluded (as they lead to a separate dimensionless group, the Froude number).[18][19] The relevant variables are the fluid density \rho, a characteristic velocity v, a characteristic length scale L, and the surface tension \sigma. These have the following dimensions in terms of mass (M), length (L), and time (T): [\rho] = \mathrm{M L^{-3}}, = \mathrm{L T^{-1}}, [L] = \mathrm{L}, and [\sigma] = \mathrm{M T^{-2}}. There are n = 4 variables and j = 3 fundamental dimensions, so the Buckingham Pi theorem predicts k = n - j = 1 dimensionless \pi group.[20][18] To form the \pi group, select the repeating variables \rho, v, and L (which collectively span the three dimensions without forming a dimensionless combination on their own). The non-repeating variable \sigma is combined as \pi = \sigma \rho^a v^b L^c, where the exponents a, b, and c are chosen to render \pi dimensionless: [\pi] = [\sigma] [\rho]^a ^b [L]^c = \mathrm{M T^{-2}} \cdot (\mathrm{M L^{-3}})^a \cdot (\mathrm{L T^{-1}})^b \cdot \mathrm{L}^c = \mathrm{M}^{1+a} \mathrm{L}^{-3a + b + c} \mathrm{T}^{-2 - b}. Setting the exponents to zero yields the system of equations:- For M: $1 + a = 0 \implies a = -1,
- For T: -2 - b = 0 \implies b = -2,
- For L: -3a + b + c = 0 \implies -3(-1) + (-2) + c = 0 \implies 3 - 2 + c = 0 \implies c = -1.