   Chapter 16.7, Problem 43E

Chapter
Section
Textbook Problem

A fluid has density 870 kg/m3 and flows with velocity v = z i + y2 j + x2 k, 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 cylinder x2 + y2 = 4, 0 ≤ z ≤ 1.

To determine

To find: The rate of flow outward through the cylinder x2+y2=4, 0z1.

Explanation

Given:

The density of the fluid is 870kgm3 and the velocity is v=zi+y2j+x2kms.

The equation of the cylinder is given by x2+y2=4, 0z1.

Formula used:

SρvndS=SρvdS (1)

ru=xui+yuj+zuk (2)

rv=xvi+yvj+zvk (3)

Calculation:

By using the parametric representation of the cylinder, consider x=2cosu, y=2sinu, z=v and limits for u and v are 0 to 2π and 0 to 1.

r(u,v)=2cosui+2sinuj+vk

Substitute 2cosu for x, 2sinu for y and v for z in equation (2) and find ru as,

ru=u(2cosu)i+u(2sinu)j+u(v)k=2sinui+2cosuj+0k=2sinui+2cosuj

Substitute 2cosu for x, 2sinu for y and v for z in equation (3) and find rv.

rv=v(2cosu)i+v(2sinu)j+v(v)k=(2cosu)v(1)i+(2sinu)v(1)j+v(v)k=(2cosu)(0)i+(2sinu)(0)j+(1)k=(0)i+(0)j+(1)k

rv=k

Find ru×rv as,

ru×rv=(2sinui+2cosuj)×(k)=|ijk2sinu2cosu0001|=(2cosu0)i+(0+2sinu)j+(00)k=2cosui+2sinuj

To find SρvdS, apply limits and substitute zi+y2j+x2k for v, 2cosui+2sinuj for

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