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Bernoulli's principle

Bernoulli's principle is a fundamental concept in that describes the relationship between the , , and in a moving , stating that an increase in the speed of a element results in a simultaneous decrease in its or potential energy. Formulated by the Swiss mathematician and physicist Daniel Bernoulli, the principle was first published in his seminal work Hydrodynamica in 1738, where it emerged from studies of motion and energy conservation. This principle applies specifically to steady, inviscid (frictionless), and incompressible flows along a streamline, providing a cornerstone for understanding phenomena in both liquids and gases. The mathematical foundation of Bernoulli's principle is expressed through Bernoulli's equation, which equates the total per unit volume along a streamline: P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}, where P is the , \rho is the fluid , v is the , g is , and h is the above a reference level. For horizontal flows where elevation changes are negligible, the equation simplifies to P + \frac{1}{2} \rho v^2 = \text{constant}, illustrating the inverse relationship between pressure and velocity. These derivations assume ideal conditions—no viscosity, steady flow without , and constant —making the principle an approximation for real-world fluids but invaluable for engineering analyses. Bernoulli's principle has wide-ranging applications in engineering and physics, most notably in , where it explains the generation of on wings: air flows faster over the curved upper surface than the flat lower surface, reducing pressure above the wing and producing an upward force. It also underpins devices like the Venturi meter for measuring fluid flow rates through pressure differences in constrictions and pitot-static tubes for determining aircraft airspeed. In , the principle informs the design of carburetors and aspirators, where high-velocity fluid draws in lower-pressure substances, and it extends to natural phenomena such as flow in vessels or patterns around structures. Despite limitations in viscous or compressible flows, Bernoulli's principle remains a key tool for predicting fluid behavior in low-speed, ideal scenarios.

Introduction and History

Statement of the Principle

Bernoulli's principle describes a fundamental relationship in where an increase in the speed of a moving , whether or gas, is accompanied by a corresponding decrease in the within that . This intuitive observation highlights how faster-moving fluids exert lower compared to slower-moving ones, a phenomenon rooted in the as the accelerates and converts potential or energy into . For instance, in airflow over an , the higher on the upper surface results in lower , contributing to generation. In its general verbal form, the principle states that along a streamline in a flowing , the total per unit volume—comprising the energy, , and gravitational —remains constant, assuming no energy losses due to or other dissipative effects. This conservation implies that an increase in (from higher velocity) must be balanced by a decrease in energy or (from elevation changes). The principle applies specifically to fluids that are inviscid (lacking ) and either incompressible or compressible under conditions where variations are negligible, and it holds for steady where conditions do not change with time along the streamline. The principle was first articulated by in his 1738 book , where he derived it from the conservation of (an early concept of ) applied to fluid motion, treating the fluid as composed of particles whose energy balance governs and relationships. Bernoulli's work emphasized in incompressible, frictionless flows, laying the groundwork for later formalizations in .

Historical Development

Daniel Bernoulli first articulated the core ideas underlying the principle in his 1738 book , where he applied Newtonian mechanics and the concept of (living force) to analyze fluid motion, particularly in , framing it as a balance of energy per unit volume between pressure, , and . This work established the principle's foundations in , emphasizing how variations in fluid speed affect pressure, though Bernoulli's formulation was initially limited to incompressible, steady flows without explicit momentum derivations. Leonhard Euler refined and generalized Bernoulli's insights in his 1757 publication "Principes généraux du mouvement des fluides," extending the principle to arbitrary fluid motions by deriving it rigorously from the momentum equations of inviscid flow, thus providing a more universal framework beyond confined pipe systems. Euler's contributions clarified the principle's applicability to broader hydrodynamic scenarios, marking a shift from empirical hydraulics to theoretical fluid dynamics. In the late 18th and 19th centuries, experimentalists like advanced practical applications through his 1797 investigations into fluid flow through constrictions, demonstrating pressure drops consistent with the principle in hydraulic setups and inspiring devices for . By the 1800s, the principle gained recognition as a manifestation of for ideal fluids, with scholars integrating it into thermodynamic contexts while distinguishing it from the more comprehensive Navier-Stokes equations, which account for and account for the principle's limitations in real, dissipative flows. The 20th century saw the principle's integration into , notably in Ludwig Prandtl's 1904 theory, which relied on Bernoulli's inviscid approximations for outer regions while addressing viscous effects near surfaces, enabling accurate predictions of and in . This evolution underscored the principle's enduring role as a foundational tool in , bridging 18th-century to modern engineering analyses.

Mathematical Formulations

Incompressible Flow

Bernoulli's principle in the context of incompressible flow applies to fluids that do not change density, such as liquids or low-speed gases, under specific conditions. The key assumptions are that the flow is steady (no time variation), inviscid (no viscosity or friction), incompressible (constant density \rho), and analyzed along a single streamline. These conditions simplify the analysis to conservation of energy without dissipative losses or density variations. The Bernoulli equation for such flow states that along a streamline, the sum of , per unit volume, and energy per unit volume remains constant: P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant} where P is the , v is the , h is the above a reference level, and g is the . This form, originally conceptualized by in his 1738 work , expresses the conservation of for the element. Each term in the equation represents : P is the energy per unit volume, \frac{1}{2} \rho v^2 is the or per unit volume, and \rho g h is the per unit volume. The units are consistent in pascals () or N/m², confirming the equation's balance as total per unit volume. For horizontal flows where elevation changes are negligible (h_1 = h_2), the equation simplifies to: P + \frac{1}{2} \rho v^2 = \text{constant} This form highlights the inverse relationship between pressure and velocity, central to many engineering applications. A representative example is the pressure drop in a narrowing pipe carrying water (\rho = 1000 kg/m³). Consider steady horizontal flow where the cross-sectional area decreases, causing velocity to increase from v_1 = 1 m/s to v_2 = 2 m/s by continuity. Assuming constant total pressure, the pressure decreases by \Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2) = \frac{1}{2} \times 1000 \times (4 - 1) = 1500 Pa, illustrating the Venturi effect.

Compressible Flow

Bernoulli's principle extends to compressible flows under specific assumptions, including inviscid conditions, steady flow, and isentropic processes, where the flow is reversible and adiabatic with constant . These assumptions allow variations with and , which are negligible in incompressible cases but critical at high speeds. For such flows, the principle takes the form \frac{v^2}{2} + \int \frac{dP}{\rho} + g z = \text{constant}, where v is the flow speed, \rho is , P is , g is , and z is ; the integral term accounts for compressible effects by relating pressure changes to density variations. Under isentropic conditions, the thermodynamic relation dh = dP / \rho (from T ds = dh - dP / \rho with ds = 0) simplifies this to the enthalpy-based equation h + \frac{v^2}{2} + g z = \text{constant}, where h is the specific , representing the total energy per unit mass along a streamline. For ideal gases undergoing isentropic compression or expansion, these equations connect directly to the Mach number M = v / a, where a is the speed of sound. The stagnation pressure P_0 (pressure at zero velocity) relates to the local pressure P by \frac{P}{P_0} = \left(1 + \frac{\gamma - 1}{2} M^2 \right)^{-\gamma / (\gamma - 1)}, with \gamma as the specific heat ratio (e.g., \gamma = 1.4 for air); this formula quantifies how pressure drops with increasing speed in compressible regimes, unlike the linear \frac{1}{2} \rho v^2 term in incompressible flow. Similar relations hold for temperature and density ratios, emphasizing the role of thermal effects in high-Mach flows. This formulation breaks down in the presence of shock waves or other non-isentropic processes, such as those involving irreversibilities like or , where increases and the constant total no longer holds uniformly.

Unsteady Potential Flow

In unsteady , the assumptions are that the fluid is inviscid and irrotational, allowing the velocity field \mathbf{v} to be expressed as the of a \phi, such that \mathbf{v} = \nabla \phi and \nabla \times \mathbf{v} = 0. These conditions apply to both compressible and incompressible flows, with as a conservative . The unsteady Bernoulli equation for such flows is derived from the Euler equations by integrating the momentum balance, yielding \frac{\partial \phi}{\partial t} + \frac{1}{2} |\nabla \phi|^2 + \int \frac{dP}{\rho} + g z = f(t), where \frac{\partial \phi}{\partial t} represents the temporal , \int \frac{dP}{\rho} accounts for pressure work, g z is the , and f(t) is an arbitrary function of time uniform throughout the domain. This form holds everywhere in the flow field, not just along streamlines, due to the irrotational nature. For incompressible flows, where density \rho is constant, the equation simplifies to \frac{\partial \phi}{\partial t} + \frac{1}{2} v^2 + \frac{P}{\rho} + g z = f(t), with v = |\mathbf{v}|. This version is particularly useful for analyzing time-varying phenomena without density variations. Physically, the \frac{\partial \phi}{\partial t} term captures the local rate of change of the velocity potential, reflecting unsteadiness such as accelerating fluid particles in dynamic environments, which modifies the balance between kinetic energy, pressure, and potential energy compared to steady flows. A representative application is in the analysis of small-amplitude water waves under theory, where the linearized unsteady Bernoulli equation at the relates to wave elevation: p_d = -\rho \frac{\partial \phi}{\partial t} = \rho g \eta, with \eta as the surface displacement, enabling computation of wave-induced pressures and forces on structures.

Derivations

From Euler's Equations

The Euler equations describe the motion of an inviscid fluid, given by \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\frac{1}{\rho} \nabla P - \nabla (g h), where \mathbf{v} is the velocity vector, \rho is the fluid density, P is the , g is the , and h is the above a reference level. This form incorporates the material acceleration on the left and the and gravitational on the right. To derive Bernoulli's principle, apply the vector identity (\mathbf{v} \cdot \nabla) \mathbf{v} = \nabla \left( \frac{v^2}{2} \right) - \mathbf{v} \times (\nabla \times \mathbf{v}), where v = |\mathbf{v}| and \nabla \times \mathbf{v} = \boldsymbol{\omega} is the . Substituting this into the Euler equations yields \frac{\partial \mathbf{v}}{\partial t} + \nabla \left( \frac{v^2}{2} \right) - \mathbf{v} \times \boldsymbol{\omega} = -\frac{1}{\rho} \nabla P - \nabla (g h). For steady flow (\partial \mathbf{v}/\partial t = 0), take the of this equation with the displacement d\mathbf{l} along a streamline, where d\mathbf{l} is parallel to \mathbf{v}. The term \mathbf{v} \times \boldsymbol{\omega} \cdot d\mathbf{l} = 0 because it is to the streamline . Integrating along the streamline from to point 2 gives \int_1^2 \nabla \left( \frac{v^2}{2} \right) \cdot d\mathbf{l} + \int_1^2 \frac{1}{\rho} \nabla P \cdot d\mathbf{l} + \int_1^2 \nabla (g h) \cdot d\mathbf{l} = 0, which simplifies to \frac{v_2^2}{2} - \frac{v_1^2}{2} + \int_1^2 \frac{dP}{\rho} + g (h_2 - h_1) = 0. Thus, \frac{v^2}{2} + \int \frac{dP}{\rho} + g h = \text{constant along the streamline}, assuming barotropic flow where \rho = \rho(P), allowing the pressure integral to depend only on the endpoints. For irrotational flow (\boldsymbol{\omega} = 0), the equation holds not just along a single streamline but between any two points in the flow field, as the term vanishes everywhere and the constant is uniform. In the unsteady case with irrotational flow, where \mathbf{v} = \nabla \phi and \phi is the , the includes the local term as \partial \mathbf{v}/\partial t = \nabla (\partial \phi / \partial t). Integrating similarly along a path yields the unsteady : \frac{\partial \phi}{\partial t} + \frac{v^2}{2} + \int \frac{dP}{\rho} + g h = \text{constant (may vary with time)}. This extends the steady form by accounting for temporal changes in the potential.

From Energy Considerations

Bernoulli's principle can be derived from the conservation of mechanical energy applied to a fluid particle or a control volume in steady flow, where the work done by pressure forces balances the changes in kinetic and potential energy of the fluid. In this approach, consider a fluid element moving along a path; the net work performed by surrounding pressure forces on the element equals the increase in its kinetic energy plus the change in gravitational potential energy, assuming no dissipative losses. This energy balance reflects the first law of thermodynamics for a system without heat transfer or shaft work. For incompressible, steady flow of an inviscid fluid, the pressure work term arises from the integral of pressure over the volume displacement, \int \frac{P \, dV}{\rho}, which simplifies under constant density \rho to \frac{P}{\rho} when evaluated between two points. Combining this with the kinetic energy change \frac{1}{2} v^2 and potential energy \rho g h, the resulting equation is: P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}, where P is pressure, v is velocity, g is gravitational acceleration, and h is elevation, all evaluated along a streamline. This form holds for fluids like liquids where density variations are negligible. The derivation assumes steady flow (no time-varying properties), inviscid conditions (no or effects), (constant ), and no or external work. These conditions ensure that mechanical energy is conserved without conversion to . For compressible flows, such as in gases, the energy balance incorporates changes in due to density variations. Under adiabatic and reversible (isentropic) conditions, leads to conservation of total , yielding: h + \frac{v^2}{2} + g h = \text{constant}, where h is the specific , which for an is c_p T with constant specific heat c_p and T. This form accounts for thermodynamic work in compressing or expanding the fluid. This energy-based derivation requires the flow to be reversible, meaning no entropy generation from shocks, friction, or heat transfer; it does not apply to dissipative or irreversible processes where energy is lost to heat. In such cases, additional terms for losses must be included, modifying the constant to account for non-conservative effects.

Applications

In Aerodynamics

Bernoulli's principle plays a central role in generating on by establishing differences arising from variations in over the wing's curved surfaces. In flight, air flowing over the upper surface of a cambered travels faster than over the lower surface due to the of the , which accelerates the flow over the curved upper surface through the effects of circulation and the Coanda effect, resulting in lower above the wing and higher below, as dictated by the principle that an increase in fluid corresponds to a decrease in . This net upward force, known as , is a direct consequence of these differentials. Within the framework of potential flow theory, the lift generated by an airfoil is quantitatively linked to circulation through the Kutta-Joukowski theorem, which states that the lift per unit span is given by L' = \rho V_\infty \Gamma, where \rho is the fluid density, V_\infty is the freestream velocity, and \Gamma is the circulation around the . The ensures smooth flow off the trailing edge, fixing the circulation value and enabling velocity fields that satisfy irrotational flow assumptions. Bernoulli's equation then relates these velocities to pressure distributions along streamlines, with higher velocities on the upper surface producing the requisite low-pressure region for . This integration of circulation theory with Bernoulli's principle provides a foundational for airfoil performance in inviscid, incompressible approximations. High-lift devices, such as trailing-edge flaps, enhance aerodynamic performance during takeoff and landing by altering the airfoil geometry to increase effective and circulation. When deployed, flaps deflect downward, accelerating airflow over the upper surface and creating steeper velocity gradients, which, per Bernoulli's principle, amplify the pressure differential across the wing and boost coefficients by up to 80-100% compared to clean configurations. This increase in circulation, as predicted by the Kutta-Joukowski , allows to operate at lower speeds without stalling, though it also elevates . In supersonic , Bernoulli's principle is adapted through its compressible form for isentropic flow regions, such as Prandtl-Meyer fans that occur at convex corners on airfoils or nozzles, where flow accelerates and decreases continuously across the fan without entropy increase. For shock waves, which are non-isentropic and involve abrupt deceleration and rise, the principle does not apply directly across the discontinuity due to total losses, but it informs the upstream and downstream conditions in conjunction with Rankine-Hugoniot relations. These adaptations enable of -velocity trades in high-speed flows around supersonic wings, where fans on the upper surface contribute to lift by lowering . The historical application of these concepts is evident in the ' 1903 glider designs, which implicitly incorporated pressure-velocity trade-offs akin to Bernoulli's principle through empirical testing of curved wing surfaces to achieve sufficient for controlled flight. Their iterative experiments with shapes demonstrated faster flow over the upper surface yielding lower and upward , enabling the first sustained powered flight later that year without explicit reference to the principle but aligning with its physical basis.

In Hydrodynamics and Devices

In hydrodynamics, Bernoulli's principle finds extensive application in devices that exploit the relationship between velocity and in incompressible liquid flows. The , a direct consequence of the principle, occurs when a fluid accelerates through a in a pipe, leading to a decrease in at the narrow section while velocity increases to conserve mass flow. This phenomenon is utilized in Venturi meters to measure liquid flow rates by detecting the pressure differential across the , which correlates with velocity via the and Bernoulli's relation. Similarly, in carburetors for systems, the Venturi creates low to draw into the airstream, facilitating and mixing, though primarily in hybrid air-liquid contexts. Pitot tubes provide a practical means to measure in liquids, such as in or channels, by capturing the between (where is brought to rest) and . The device consists of a tube aligned with the to sense total pressure P_0 and a port for P, yielding v = \sqrt{2 (P_0 - P)/\rho}, where \rho is the ; this formula assumes incompressible, steady along a streamline. In , Pitot tubes are deployed in hydroelectric systems and channels to speed accurately, aiding in control and efficiency assessments. Aspirators and atomizers leverage low-pressure zones created by high-velocity liquid jets to entrain secondary fluids or particles. In an , a fast-moving stream of through a generates reduced , drawing air or another into the for mixing or evacuation, as seen in suction devices or medical drainage tools. Atomizers operate analogously, where accelerated lowers to pull in additives, enabling fine dispersion in applications like spray nozzles for or . These devices rely on the principle's prediction that pressure drops inversely with velocity squared, ensuring effective without mechanical pumps. Natural phenomena in liquid flows also illustrate Bernoulli's principle. In river rapids, accelerates over shallow, narrow sections, increasing and thus decreasing near the surface, which can cause or influence ; observations in tidal rapids like Nakwakto confirm the principle holds until dominates. In human , blood flow through constricted vessels, such as in arterial , experiences increases that reduce local , potentially leading to vessel collapse or diagnostic indicators via Doppler , as the principle equates total energy along streamlines assuming negligible . For engineering calculations in pipe networks, Bernoulli's equation approximates flow rates by balancing pressure, velocity, and elevation heads, incorporating head loss terms for friction. The modified form \frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_f, where \gamma = \rho g and h_f is frictional head loss (often estimated via Darcy-Weisbach as h_f = f \frac{L}{D} \frac{V^2}{2g}), enables prediction of discharge in water distribution systems without full viscous solving. This approach is standard for preliminary design in civil engineering, prioritizing energy conservation over detailed turbulence modeling.

Misconceptions and Limitations

Airfoil Lift Misunderstandings

One common misunderstanding in explaining lift is the "equal transit time" theory, which posits that air particles passing over the upper surface of a wing travel the same distance as those under the lower surface in the same amount of time, resulting in higher above due to the longer path length and thus lower pressure via Bernoulli's principle. This fallacy assumes streamlines over and under the converge at the trailing edge simultaneously, but experimental visualizations and computational analyses show that upper streamlines are shorter and air over the top arrives earlier, with velocities determined by circulation rather than path length equality. The theory also underpredicts by a factor of about two compared to observations, as the implied is insufficient to account for measured forces. Bernoulli's principle alone cannot fully explain airfoil lift, as it describes the relationship between pressure and velocity but does not address the origin of the velocity differences or the net force balance required for sustained flight. Lift generation fundamentally relies on Newton's third law, where the deflects airflow downward, imparting to the air and producing an equal upward reaction force on the wing; the pressure differences observed via Bernoulli are a consequence, not the cause, of this deflection. Without this momentum transfer—achieved through the wing's —Bernoulli's equation would predict no net lift in uniform flow. Another misconception attributes the adherence of to the curved upper surface of an primarily to Bernoulli's principle creating a that "sucks" the air along the curve. In reality, the , which causes fluid to follow a nearby curved surface, arises from viscous interactions in the that entrain surrounding air toward the surface, preventing separation; Bernoulli's principle contributes to the field but does not drive the attachment without viscosity. This viscous mechanism ensures smooth flow over the , enabling the pressure differences, but pure inviscid Bernoulli flow would not produce the observed attachment. These errors persist in educational materials, with many textbooks and introductory resources incorrectly implying that stems solely from velocity-induced differences without invoking deflection or circulation, leading to confusion among students and pilots. For instance, the equal transit time concept appears in numerous technical articles and manuals despite being debunked since the mid-20th century. The accurate perspective integrates Bernoulli's principle as a descriptor of local pressure-velocity relations once is established, but the root cause of on an is the circulation induced by the , modeled as a bound vortex along the wing's span that accelerates over the upper surface relative to the lower. This circulation, enforced by the at the trailing edge, creates the asymmetric velocity field; without to initiate vortex formation, no occurs even with an shape.

Common Demonstrations and Errors

One popular classroom demonstration involves suspending a lightweight ball, such as a ping-pong ball, in a vertical of air from a blower or hairdryer, often attributed to Bernoulli's principle creating lower pressure around the ball due to faster airflow. However, this effect is primarily due to the , where the airflow adheres to the curved surface of the ball, combined with entrainment of surrounding air, which generates an asymmetric pressure distribution; plays a crucial role in maintaining the flow attachment, violating the inviscid assumption of Bernoulli's principle. Another common demonstration uses a strip of paper held loosely at one end while air is blown gently underneath it from pursed lips, causing the strip to upward toward the stream, seemingly illustrating reduced from increased per Bernoulli's principle. In reality, the arises from a pressure imbalance created by the deflection of the jet, which entrains ambient air and expands the low-pressure region below the strip; this involves unsteady flow and viscous interactions not accounted for in the simplified Bernoulli application. These demonstrations often fail to accurately represent Bernoulli's principle because they involve unsteady flows, significant viscous effects, and curved streamlines that introduce gradients beyond simple velocity-pressure trade-offs, breaching the core assumptions of steady, inviscid, along a single streamline. A more reliable alternative is the Venturi tube demonstration, where water flows through a converging-diverging connected to a manometer; the height difference in the manometer tubes directly measures the at the (higher ) compared to the inlet and outlet, validating the Bernoulli relation under controlled, steady conditions with minimal losses. To avoid overgeneralization, educators should emphasize Bernoulli's principle's strict assumptions—steady, along streamlines—and use these flawed demos as opportunities to discuss real-world deviations, encouraging students to identify when additional physics like or dominates.