# Rate of flow outward through hemisphere

• aw90
In summary, the conversation discusses the rate of flow of seawater through a hemisphere with a given velocity field and the use of a surface integral to calculate the flow. The surface is parameterized using standard spherical coordinates and the velocity field is given in terms of these coordinates. The formula for the surface integral is mentioned and a shortcut is suggested for the bottom of the hemisphere.
aw90
1. Seawater has density 1025 kg/m3 and flows in a velocity field v=yi+xj, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x2+y2+z2=9, z≥0
2. surface integral of F over S is ∫∫ F • dS
3. I parameterized the surface by making r(∅,∂)=<3sin∂cos∅, 3 sin∂sin∅, 3cos∂>. My vector v is <y, x, 0>. Aside from that, i have no idea where to begin. Any help is appreciated.

To avoid confusion I would suggest you use the standard letters for the spherical coordinate parameterization:

$$\vec r(\theta,\phi) = \langle 3\sin\phi\cos\theta,3\sin\phi\sin\theta,3\cos\phi\rangle$$

$$\vec v = \langle 3\sin\phi\sin\theta,3\sin\phi\cos\theta,0\rangle$$

Use the formula

$$\int\int_S \delta\vec v \cdot d\vec S = \int\int_{(\phi,\theta)} \delta\vec v \cdot \vec r_\phi \times \vec r_\theta\ d\phi d\theta$$

for the spherical surface. For the bottom of the hemisphere in the xy plane you may see a shortcut.

## 1. What is the "rate of flow outward through hemisphere"?

The "rate of flow outward through hemisphere" refers to the volume of fluid flowing out of a hemisphere per unit of time. It is a measure of the outward flow of fluid from a particular point in the hemisphere's surface.

## 2. How is the rate of flow outward through hemisphere calculated?

The rate of flow outward through hemisphere is calculated by dividing the volume of fluid flowing out of the hemisphere by the time it takes for the fluid to flow out. This calculation can be expressed as Q = V/t, where Q is the rate of flow, V is the volume of fluid, and t is the time.

## 3. What factors affect the rate of flow outward through hemisphere?

The rate of flow outward through hemisphere can be affected by various factors, such as the viscosity of the fluid, the diameter of the hemisphere, and the pressure difference between the inside and outside of the hemisphere. Other factors that can influence the rate of flow include the shape and surface roughness of the hemisphere, as well as any obstacles or obstructions in the fluid's path.

## 4. Why is the rate of flow outward through hemisphere important?

The rate of flow outward through hemisphere is important in various fields of science and engineering. It is a crucial factor in understanding fluid dynamics and can be used in the design and analysis of various systems, such as pipelines, pumps, and filters. It is also relevant in fields such as meteorology, oceanography, and biology, where the flow of fluids plays a significant role in natural phenomena and processes.

## 5. How is the rate of flow outward through hemisphere measured?

The rate of flow outward through hemisphere can be measured using various techniques, such as flow meters, pressure sensors, and visual observation. The specific method used will depend on the fluid being studied and the accuracy and precision required for the measurement. In some cases, mathematical models and simulations may also be used to estimate the rate of flow outward through hemisphere.

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