Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, 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 = 81, z ≥ 0.

Flow (Q) = 0 kg/s      

Step-by-step explanation:

Given:-

- The density of seawater, ρ = 1025 kg/m^3

- The velocity field of flow, v ( x,y,z ) = y i + x j

Find:-

The rate of flow outward through the hemisphere x2 + y2 + z2 = 81, z ≥ 0.

Solution:-

For some flow, if we know the point density at (x,y,z) is ρ ( x , y ,z ).

And the velocity of a unit element at point (x,y,z) is v ( x , y ,z ).

Then the rate of outflow of substance through surface "S" is given by:

Where,                    F = ρ ( x , y ,z )*v ( x , y ,z )

- Therefore for the given data:

                               F = ( 1025 kg/m^3)*(y i + x j)

- We will now parametrize the surface "S" given as:

                              x2 + y2 + z2 = 81, z ≥ 0

           S: r(u) = 3 sin (v) cos (u) i + 3 sin (v) sin (u) j + 3 cos (v) k  

Where,         u ε [ 0 , 2π) and since z ≥ 0, v ε [ 0 , π/2 ]

Therefore,

                   F [ r(u) ] = ( 1025 kg/m^3)*( 3 sin (v) sin (u) i + 3 sin (v) cos (u) j).

Note:

Where,           n: Unit vector normal to surface "S"

- Such that,          

The Flux through a surface is defined only if the surface is orientable.

Note that if the substance is orientable, there will be two normal unit vectors at every point (x,y,z):

- We can write:

- For finding the flux we always use the positive orientation of unit normal vector (n).

- The parameterisation for sphere "S":

              r ( u , v ) = 3 sin (v) cos (u) i + 3 sin (v) sin (u) j + 3 cos (v) k          

Where,         u ε ( 0 , 2π) and since z ≥ 0, v ε ( 0 , π/2 ),

- Compute ( r_u x r_v ) :

- The vector ( r_u x r_v) points towards the origin, therefore the positive unit normal vector ( r_u x r_v) would be:

Therefore,

So the flow through the surface "S" is Flow (Q) = 0 kg/s            

step-by-step explanation:

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step-by-step explanation: