The Euler's equation is ρ(d v)/(d t)=f⃗-∇⃗ p in the space fixed frame where f⃗=-ρg k⃗ downward.

step 1 Hi friends, the Euler equation is row d b upon d t is equal to f minus d d d p in the space fixe

The Euler's equation is ρ(d v)/(d t)=f⃗-∇⃗ p in the space...

The Euler's equation is $\rho \frac{d v}{d t}=\vec{f}-\vec{\nabla} p$ in the space fixed frame where $\vec{f}=-\rho g \vec{k}$ downward. We assume incompressible fluid so $\rho$ is constant. Then $\vec{f}=-\vec{\nabla}(\rho g z)$ where $z$ is the height vertically upwards from some fixed origin. We go to rotating frame where the equation becomes $$ \rho \frac{d \vec{v}}{d t}=-\vec{\nabla}(p+\rho g z)+\rho \omega^{2} \vec{r}+2 \rho(\vec{v} \times \vec{\omega}) $$ the additional terms on the right are the well known coriolis and centrifugal forces. In the frame rotating with the liquid $\vec{v}=0$ so or $$ \begin{gathered} \vec{r}\left(p+\rho g z-\frac{1}{2} \rho \omega^{2} r^{2}\right)=0 \\ p+\rho g z-\frac{1}{2} \rho \omega^{2} r^{2}=\text { constant } \end{gathered} $$ On the free surface $p=$ constant, thus $$ z=\frac{\omega^{2}}{2 g} r^{2}+\text { constant } $$ If we choose the origin at point $r=0$ (i.e. the axis) of the free surface then "cosntant" $=0$ and $z=\frac{\omega^{2}}{2 g} r^{2}$ (The paraboloid of revolution) At the bottom $z=$ constant So $$ p=\frac{1}{2} \rho \omega^{2} r^{2}+\text { constant } $$ If $p=p_{0}$ on the axis at the bottom, then $$ p=p_{0}+\frac{1}{2} \rho \omega^{2} r^{2} $$

The Euler's equation is $\rho \frac{d v}{d t}=\vec{f}-\vec{\nabla} p$ in the space fixed frame where $\vec{f}=-\rho g \vec{k}$ downward. We assume incompressible fluid so $\rho$ is constant. Then $\vec{f}=-\vec{\nabla}(\rho g z)$ where $z$ is the height vertically upwards from some fixed origin. We go to rotating frame where the equation becomes
$$
\rho \frac{d \vec{v}}{d t}=-\vec{\nabla}(p+\rho g z)+\rho \omega^{2} \vec{r}+2 \rho(\vec{v} \times \vec{\omega})
$$
the additional terms on the right are the well known coriolis and centrifugal forces. In the frame rotating with the liquid $\vec{v}=0$ so
or
$$
\begin{gathered}
\vec{r}\left(p+\rho g z-\frac{1}{2} \rho \omega^{2} r^{2}\right)=0 \\
p+\rho g z-\frac{1}{2} \rho \omega^{2} r^{2}=\text { constant }
\end{gathered}
$$
On the free surface $p=$ constant, thus
$$
z=\frac{\omega^{2}}{2 g} r^{2}+\text { constant }
$$
If we choose the origin at point $r=0$ (i.e. the axis) of the free surface then "cosntant" $=0$ and $z=\frac{\omega^{2}}{2 g} r^{2}$ (The paraboloid of revolution) At the bottom $z=$ constant
So
$$
p=\frac{1}{2} \rho \omega^{2} r^{2}+\text { constant }
$$
If $p=p_{0}$ on the axis at the bottom, then
$$
p=p_{0}+\frac{1}{2} \rho \omega^{2} r^{2}
$$

Key Concepts

Free Surface Equilibrium

Free surface equilibrium refers to the condition where the surface of a liquid adjusts itself such that the pressure is constant along the surface, typically equal to the atmospheric pressure. When a fluid rotates, the free surface adopts a particular shape—in this case, a paraboloid—to balance the centrifugal effects with gravitational forces, ensuring that the net pressure is uniform along the surface.

Hydrostatic Pressure Distribution in Rotating Fluids

Hydrostatic pressure distribution in a rotating fluid describes how the pressure within the fluid varies spatially in equilibrium due to both gravity and rotational effects. In the absence of motion relative to the rotating frame, the pressure gradient must balance gravitational force and the centrifugal force, leading to pressure increasing with radial distance and resulting in a curved free surface. This concept is crucial in understanding how rotation influences fluid pressure fields.

Incompressible Flow

The assumption of incompressible flow implies that the fluid density remains constant throughout the fluid motion. This simplification allows the conservation of mass to be described by a divergence-free velocity field and simplifies the momentum equation by effectively removing variations in density. It is particularly relevant in problems dealing with liquids where density changes are negligible.

Rotating Frame of Reference

A rotating frame of reference is a non-inertial coordinate system that rotates relative to an inertial frame. When analyzing fluid motion from such a frame, additional fictitious forces must be introduced in order to use Newton's laws of motion correctly. These fictitious forces, which do not arise from physical interactions but from the acceleration of the reference frame, include the Coriolis and centrifugal forces.

Fictitious Forces (Coriolis and Centrifugal Forces)

In a rotating frame, the Coriolis force and centrifugal force appear as additional terms in the equations of motion. The Coriolis force acts perpendicular to the velocity of the moving fluid and the axis of rotation, modifying the trajectory of fluid elements, while the centrifugal force appears as a radially outward force that depends on the square of the rotation rate. These forces are key to understanding the equilibrium and dynamics of fluids in rotating systems.

Euler's Equation for Inviscid Flow

Euler's equation is a fundamental relation in fluid dynamics that expresses the conservation of momentum for an inviscid (non-viscous) fluid. It establishes a balance between the inertial forces (represented by the fluid's acceleration) and the forces due to pressure gradients and external body forces such as gravity. In problems involving rotating fluids, Euler's equation forms the starting point to account for additional apparent forces in non-inertial reference frames.

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