91Ó°ÊÓ

A cylinder containing a fluid of mass density \(\rho\) is placed on the center of a phonograph turntable and rotated with constant angular velocity \(\omega\). After some initial sloshing of the fluid, everything calms down to a steady state. a. The pressure is a function of height \(\mathrm{h}\) and distance from the axis \(\mathrm{r}\). Show that the variation of pressure with radial distance is $$\frac{\partial p(r, h)}{\partial r}=\rho \omega^{2} r$$ while the variation with vertical distance is $$\frac{\partial p(r, h)}{\partial h}=-\rho g$$ (Where \(g\) is the acceleration of gravity.) b. The pressure at the surface of the fluid at the center of the cylinder \(\left(\mathrm{r}=0, h=h_{c}\right.\) ) is of course atmospheric pressure pa. Integrate the differential equations of part (a.) to show that, at any point in the fluid, $$p(r, h)=p_{a}+\frac{1}{2} \rho \omega^{2} r^{2}-\rho g\left(h-h_{c}\right)$$ c. Show that the profile of the fluid surface is given by $$y(r)=\frac{\omega^{2}}{2 g} r^{2}$$ \({ }^{2}\) As such, the aim of this section is quite modest. If you want to learn more about the interesting subject of fluid flow, see the "91Ó°ÊÓ" section of this chapter.

Step by step solution

01

Understanding Pressure Variation with Radial Distance

In a rotating frame of reference, the fluid experiences a centripetal acceleration. This acceleration causes a force per unit volume that acts radially outward. The pressure gradient balances this effective force. The equation for this is given by Newton's second law in the radial direction, \( \frac{d^2r}{dt^2} = \omega^2r \), thus leading to \( \frac{\partial p(r, h)}{\partial r} = \rho \omega^2 r \).

02

Understanding Pressure Variation with Vertical Distance

For a fluid at rest (in the rotating frame), the pressure must balance the force due to gravity. Therefore, the pressure change with height is given by the hydrostatic pressure equation, resulting in \( \frac{\partial p(r, h)}{\partial h} = -\rho g \).

03

Integrating with Respect to Radial Distance

Integrate \( \frac{\partial p(r, h)}{\partial r} = \rho \omega^2 r \) with respect to \( r \), which results in: \( p(r, h) = \frac{1}{2} \rho \omega^2 r^2 + f(h) \), where \( f(h) \) is an integration function of \( h \).

04

Integrating with Respect to Vertical Distance

Integrate \( \frac{\partial p(r, h)}{\partial h} = -\rho g \) with respect to \( h \), giving \( p(r, h) = -\rho g (h - h_c) + g(r) \). Here, \( g(r) \) is an integration function of \( r \).

05

Combining the Integration Results

Combine the results from both integrations: \( p(r, h) = \frac{1}{2} \rho \omega^2 r^2 - \rho g(h - h_c) + p_a \). Here, \( p_a \) is chosen as the integration constant because at \( r = 0 \) and \( h = h_c \), pressure is atmospheric \( p = p_a \).

06

Determining the Fluid Surface Profile

At the fluid surface, pressure is constant and equal to atmospheric pressure. Therefore, set \( p(r, h) = p_a \) and solve for \( h \) as a function of \( r \). Setting \( h = y(r) \), we find \( y(r) = h_c + \frac{\omega^2}{2g} r^2 \). Since \( y(r) \) represents the deviation from \( h_c \), the fluid surface is \( y(r) = \frac{\omega^2}{2g} r^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotating Fluids

Rotating fluids are a fascinating topic in fluid mechanics, often seen when dealing with scenarios like liquids in a spinning container. When a fluid is spun with constant angular velocity, each particle within the fluid is subjected to centripetal acceleration.
This setup is commonly described in a rotating frame of reference, where the fluid experiences certain unique forces. These forces can influence the shape and pressure distribution within the fluid.

• Angular Velocity: This is the rate at which the fluid is rotating. It is denoted by \( \omega \) and defines how fast the fluid spins around the axis.
• Centripetal Acceleration: As the fluid rotates, its particles are forced to move in a circular path, creating an acceleration directed towards the center.
• Steady State: After brief initial disturbances or 'sloshing', the fluid settles into a steady state where the forces acting on it are balanced.

This concept helps us understand how the distribution of forces affects the pressure and shape of the fluid surface over time.

Pressure Variation

Pressure variation in fluids is crucial to understand how different forces act within a rotating system.
In the scenario of rotating fluids, pressure changes both radially and vertically. Let's delve into these aspects:

• Radial Pressure Variation: As you move outward from the rotation axis, the pressure in the fluid changes due to the powerful influence of centripetal forces. The expression \( \frac{\partial p(r, h)}{\partial r} = \rho \omega^2 r \) illustrates how the pressure increases with distance from the axis. This relation stems from Newton’s law in radial direction.
• Vertical Pressure Variation: Gravity continues to play a vital role, creating a pressure difference with vertical height. Given by \( \frac{\partial p(r, h)}{\partial h} = -\rho g \), this equation reflects the reduction in pressure as height increases, following the typical hydrostatic condition.

Understanding pressure variation helps predict how the fluid behaves both on the surface and deep within, significantly impacting the entire system dynamics.

Hydrostatic Equation

The hydrostatic equation is one of the core principles governing static fluids. It explains the pressure variation in fluids at rest.
In our rotating fluid system, this equation applies vertically and showcases how gravity affects pressure.

• Basic Formulation: The hydrostatic equation \( \frac{\partial p}{\partial h} = -\rho g \) reveals that pressure decreases with height due to the weight of the fluid above.
• Gravity's Role: Gravity always pulls the fluid downward, and this force's effect is counteracted by internal fluid pressure, ensuring equilibrium in a steady state.
• Vertical Balance: This equation helps determine how fluid pressure varies precisely with height in a vertically oriented fluid, even when part of a larger, more complex system like our rotating cylinder.

Grasping the hydrostatic equation provides the foundation for understanding the majestic equilibrium in fluid systems, especially when subjected to rotational forces.

Centrifugal Force in Fluids

Centrifugal force is a pseudo-force that appears when analyzing motion in a rotating frame. It's crucial in explaining the behavior of rotating fluids.
Here’s how it impacts the system:

• Pseudo-force Effect: Though not a real force in non-rotating frames, centrifugal force must be considered in rotating ones. It acts outward from the axis of rotation, affecting fluid particles.
• Radial Force Influence: Expressed as \( F_c = m\omega^2 r \), this force increases with distance from the rotation axis. It contributes significantly to pressure variation in fluids.
• Surface Profile Change: Due to the centrifugal force's effects, the fluid surface adopts a parabolic shape. This is mathematically represented by \( y(r) = \frac{\omega^2}{2g} r^2 \), indicating the surface's curved profile.

By understanding centrifugal force, we can predict how rotating fluid surfaces form and behave, giving insight into complex fluid dynamics and applications beyond simple theoretical models.