The Euler's equation is ρ(d v⃗)/(d t)=f⃗-∇⃗ p=-∇⃗(p+ρg z), where z is vertically upwards. Now (d

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The Euler's equation is ρ(d v⃗)/(d t)=f⃗-∇⃗ p=-∇⃗(p+ρg z),...

The Euler's equation is $\rho \frac{d \vec{v}}{d t}=\vec{f}-\vec{\nabla} p=-\vec{\nabla}(p+\rho g z)$, where $z$ is vertically upwards. Now $\frac{d \vec{v}}{d t}=\frac{\partial \vec{v}}{\partial t}+(\vec{v} \cdot \vec{\nabla}) \vec{v}$ $$ (\vec{v} \cdot \vec{\nabla}) \vec{v}=\vec{\nabla}\left(\frac{1}{2} v^{2}\right)-\vec{v} \times \operatorname{Curl} \vec{v} $$ But we consider the steady (i.e. $\partial \vec{v}$ ) $\partial t=0$ ) flow of an incompressible fluid then $\rho=$ constant. and as the motion is irrotational Curl $\vec{v}=0$ So from (1) and (2) $\rho \vec{\nabla}\left(\frac{1}{2} v^{2}\right)=-\vec{\nabla}(p+\rho g z)$ or, $\vec{\nabla}\left(p+\frac{1}{2} \rho v^{2}+\rho g z\right)=0$ Hence $$ p+\frac{1}{2} \rho v^{2}+\rho g z=\text { constant. } $$

The Euler's equation is $\rho \frac{d \vec{v}}{d t}=\vec{f}-\vec{\nabla} p=-\vec{\nabla}(p+\rho g z)$, where $z$ is vertically upwards.
Now $\frac{d \vec{v}}{d t}=\frac{\partial \vec{v}}{\partial t}+(\vec{v} \cdot \vec{\nabla}) \vec{v}$
$$
(\vec{v} \cdot \vec{\nabla}) \vec{v}=\vec{\nabla}\left(\frac{1}{2} v^{2}\right)-\vec{v} \times \operatorname{Curl} \vec{v}
$$
But we consider the steady (i.e. $\partial \vec{v}$ ) $\partial t=0$ ) flow of an incompressible fluid then $\rho=$ constant. and as the motion is irrotational Curl $\vec{v}=0$
So from
(1) and
(2) $\rho \vec{\nabla}\left(\frac{1}{2} v^{2}\right)=-\vec{\nabla}(p+\rho g z)$
or, $\vec{\nabla}\left(p+\frac{1}{2} \rho v^{2}+\rho g z\right)=0$
Hence
$$
p+\frac{1}{2} \rho v^{2}+\rho g z=\text { constant. }
$$

Key Concepts

Bernoulli's Equation

Bernoulli's equation is an expression of energy conservation in fluid flow. For steady, incompressible, inviscid, and irrotational flows, it relates the pressure, kinetic energy per unit volume, and potential energy per unit volume along a streamline, indicating that their sum remains constant.

Irrotational Flow

Irrotational flow is characterized by the absence of rotation or vorticity, allowing the velocity field to be represented as the gradient of a scalar potential. This simplifies the equations and is a key assumption in deriving potential flow solutions.

Incompressible Flow

This concept refers to flows where the fluid density remains constant. Incompressibility simplifies the continuity equation and impacts the pressure and velocity relationships in the analysis of fluid motion.

Material Derivative

The material derivative accounts for the change in a fluid property as one moves with the flow. It combines the local time derivative with the convective derivative, reflecting both temporal and spatial changes experienced by a moving fluid particle.

Euler's Equation

This equation represents the conservation of momentum for an inviscid fluid. It balances the inertial forces, represented by the acceleration term, against the body forces and pressure gradients in the fluid, forming the basis for many analyses in fluid dynamics.

Steady Flow

In steady flow, fluid properties do not change with time at any fixed point in space. This assumption significantly simplifies the equations of motion by eliminating the local time derivative components.

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