7.2 Euler's Equation and Bernoulli's Equation

Euler's Equation and Bernoulli's Equation

Euler's equation describes how an inviscid (frictionless) fluid accelerates in response to pressure gradients and gravity. Bernoulli's equation is a special case derived from Euler's equation under additional assumptions, and it provides a direct energy-like relationship between pressure, velocity, and elevation along a streamline. Together, these two results form the analytical backbone for a huge range of fluid flow problems.

Euler's Equation

Derivation of Euler's Equation

The derivation starts from the Navier-Stokes equations and strips away viscous effects.

1. Start with the Navier-Stokes equations for an incompressible fluid:

$\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{g} + \mu \nabla^2 \vec{V}$

1. Assume inviscid flow ($\mu = 0$). This eliminates the viscous diffusion term entirely:

$\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{g}$

1. Expand the material derivative. The material derivative $\frac{D\vec{V}}{Dt}$ captures how velocity changes for a fluid particle as it moves through the flow field. It splits into a local (time) part and a convective (spatial) part:

$\rho \left(\frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V}\right) = -\nabla p + \rho \vec{g}$

This is Euler's equation. It governs the motion of any inviscid fluid, whether the flow is steady or unsteady, compressible or incompressible.

Physical Interpretation of Euler's Equation

Each term in Euler's equation has a distinct physical role. Reading the equation as a force balance per unit volume:

• Local acceleration $\rho \frac{\partial \vec{V}}{\partial t}$: How velocity changes with time at a fixed point. This term is zero in steady flow.
• Convective acceleration $\rho (\vec{V} \cdot \nabla)\vec{V}$: How velocity changes because the fluid particle moves to a new location where the velocity field is different. Even in steady flow, a particle can accelerate if the velocity varies in space (e.g., flow through a converging nozzle).
• Pressure gradient force $-\nabla p$: The net pressure force per unit volume. Fluid accelerates from high-pressure regions toward low-pressure regions.
• Body force $\rho \vec{g}$: Gravity (or any other body force) acting per unit volume. For most problems, $\vec{g}$ points downward with magnitude $g$.

The key takeaway: in an inviscid fluid, the only forces driving the flow are pressure differences and gravity. There's no viscous shear to slow things down or redistribute momentum.

Bernoulli's Equation

Deriving Bernoulli's Equation from Euler's Equation

Bernoulli's equation follows from Euler's equation with three additional assumptions: steady flow, incompressible flow, and irrotational flow (or integration along a streamline). Here's the step-by-step path:

1. Assume steady flow ($\frac{\partial \vec{V}}{\partial t} = 0$). Euler's equation reduces to:

$\rho (\vec{V} \cdot \nabla)\vec{V} = -\nabla p + \rho \vec{g}$

1. Assume incompressible flow ($\rho = \text{constant}$). Divide through by $\rho$:

$(\vec{V} \cdot \nabla)\vec{V} = -\frac{1}{\rho}\nabla p + \vec{g}$

1. Apply the vector identity for the convective term:

$(\vec{V} \cdot \nabla)\vec{V} = \nabla\left(\frac{V^2}{2}\right) - \vec{V} \times (\nabla \times \vec{V})$

Substituting:

$\nabla\left(\frac{V^2}{2}\right) - \vec{V} \times (\nabla \times \vec{V}) = -\frac{1}{\rho}\nabla p + \vec{g}$

1. Assume irrotational flow ($\nabla \times \vec{V} = \vec{0}$). The cross-product term vanishes:

$\nabla\left(\frac{V^2}{2}\right) = -\frac{1}{\rho}\nabla p + \vec{g}$

Note: if the flow is rotational but you integrate along a streamline (where $d\vec{s}$ is parallel to $\vec{V}$), the cross-product term also drops out because $\vec{V} \times (\nabla \times \vec{V})$ is perpendicular to $\vec{V}$. So Bernoulli's equation holds along a streamline even in rotational flow, but for irrotational flow it holds between any two points in the field.

1. Integrate between two points (along a streamline, or anywhere if irrotational). With gravity acting in the $-z$ direction ($\vec{g} = -g\hat{k}$, so $\vec{g} \cdot d\vec{s} = -g\,dz$):

$\frac{V_1^2}{2} + \frac{p_1}{\rho} + gz_1 = \frac{V_2^2}{2} + \frac{p_2}{\rho} + gz_2$

This is Bernoulli's equation. Each group of terms has units of energy per unit mass (J/kg), so the equation is really a statement of mechanical energy conservation for an inviscid, incompressible fluid.

Applications of Bernoulli's Equation

The compact form is:

$\frac{V^2}{2} + \frac{p}{\rho} + gz = \text{constant along a streamline}$

The three terms are often called the dynamic pressure head ($\frac{V^2}{2}$), the pressure head ($\frac{p}{\rho}$), and the elevation head ($gz$). Their sum stays constant, so if velocity increases, pressure must drop (and vice versa).

Solving problems with Bernoulli's equation:

• Pick two points along the same streamline where you know (or want to find) $V$, $p$, or $z$.
• Write Bernoulli's equation between those two points.
• Use the continuity equation for incompressible flow, $A_1 V_1 = A_2 V_2$, to relate velocities at different cross-sections.
• Solve the resulting system for the unknowns.

Common applications:

• Flow measurement devices: Venturi meters and orifice plates use the pressure drop caused by a velocity increase in a constriction to measure flow rate.
• Pipe and duct analysis: Determining pressure changes or required pipe diameters for a given flow rate.
• Open-channel flows: Relating water depth and velocity in rivers or canals.

Limitations and modifications:

Bernoulli's equation assumes no viscous losses, no shaft work, and no heat transfer. For real engineering systems, you'll often use the extended Bernoulli equation (also called the energy equation):

$\frac{V_1^2}{2} + \frac{p_1}{\rho} + gz_1 + w_{\text{pump}} = \frac{V_2^2}{2} + \frac{p_2}{\rho} + gz_2 + h_L + w_{\text{turbine}}$

Here $h_L$ accounts for friction (major losses) and fittings/valves (minor losses), while $w_{\text{pump}}$ and $w_{\text{turbine}}$ represent energy added or removed by machinery. Always check whether the assumptions behind the standard Bernoulli equation are reasonable before applying it; if viscous effects, compressibility, or unsteadiness matter, you need the more general forms.