Gaussian divergence theorem
The velocity field of a fluid is given by F (?, ?, ?) = 5?k and let ? be the sphere ?2 + ?2 + ?2 = 16. Find the flow from ? through ?.
Homework Answers
Use the divergence theorem:
p = 5 = density
Mass flux = p ???r div(v) dVol
=p ???r div(-yi + xj + 2zk) dVol
= p ???r (-1 + 1+ 2 ) dVol
= 2p ?1 dVol
r^2 = x^2 + y^2 + z^2 = 16
r = 4 units
==>Vol = [0,r=4] ? 1 dVol = (4/3)?43
Mass flux = 2*p*(4/3)?43
Mass flux = 853.33? units/second
If a fluid with density 1000 flows with a velocity field v=3yi−3xj−5zkv=3yi−3xj−5zk, find the fluid flow...
If a fluid with density 1000 flows with a velocity field
v=3yi−3xj−5zkv=3yi−3xj−5zk, find the fluid flow rate outward
through the sphere x2+y2+z2=1x2+y2+z2=1.
flow rate =
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In a bathtub, the velocity of water near2 the drain is given by the vector field (22 +1)2222 + 1 ...
Question 5
In a bathtub, the velocity of water near2 the drain is given by the vector field (22 +1)2222 + 1 where r, y, and z are measured in centimeters and (0, 0,0) is at the center of the drain. 1. Rewriting F as follows, describe in words how the water is moving: Consider each of the three terms in equation (4). (Look at some plots.) For fixed z, what is the flow like? How does the flow change...
If a fluid with density 1000 flows with a velocity field
v=3yi−3xj−5zkv=3yi−3xj−5zk, find the fluid flow rate outward
through the sphere x2+y2+z2=1x2+y2+z2=1.
flow rate =
5z k find the fluid flow rate outward through the sphere z2 + y2 + z2-1 Sy-3zj lf a fluid with density 1000 flows with a velocity field v flow rate =
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x> 0, y> 0 and z> 0 )j+ (3z + 7)k be the velocity field of a fluid. Let B be the Determine the flux of F through B in the direction of the outward unit normal
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x>...
Use a computer algebra system to find the rate of mass flow of a fluid of density ρ through the surface S oriented upward if the velocity field is given by F(x, y, z) 0.52k
Use a computer algebra system to find the rate of mass flow of a fluid of density ρ through the surface S oriented upward if the velocity field is given by F(x, y, z) 0.52k
please help me.
Z=x+ly Ideal potential flow field with velocity vector in 2- dimensional virtual (x, y) plane Complex potential function of a fluid flowing through a range with a volumetric flow m and F(2) = aperture range is given with. Where In; natural logarithmic function and sinh; sinus is hyperbolic function abbreviation, m and b are constant. Find the velocity components in Cartesian coordinates for this flow area in (sinh (5)
Use the Divergence Theorem to find the total flux out of the unit sphere of the field F = 2*, -y, z). Write the triple integral in shperical coordinates, but do not evaluate.
Please answer without using previously posted answers.
Thanks
Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a gradient field, and φ s called a potential function. Ideal Fluid Flow Let F represent the two-dimensional velocity field of an inviscid fluid that is incompressible, ie. . F-0 (or divergence-free). F can be represented by (1), where ф is called the velocity potential-show that o is harmonic....
Question Help 15.8.4 The picture to the right shows the flow of fluid in a long cylindrical pipe. The vectors v= (a2-P) k inside the cylinder that have their bases in the xy-plane have their tips on the paraboloid z = a2-f. Find the divergence of the velocity field when a = 7 2 +y a The divergence is 1. (Simplify your answer.)
Question Help 15.8.4 The picture to the right shows the flow of fluid in a long cylindrical...
21 Problem 20. Let S be the surface bounded by the graph of f(x,y)-2+y2 . the plane z 5; Os1; and .0sys1. In addition, let F be the vector field defined by F(x, y,z):i+ k. (1) By converting the resulting triple integral into cylindrical coordinates, find the exact value of the flux integral F.n do, assuming that S is oriented in the positive z-direction. (Recall that since the surface is oriented upwardly, you should use the vector -fx, -fy, 1)...
Just question 5
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In a bathtub, the velocity of water near2 the drain is given by the vector field (22 +1)2222 + 1 where r, y, and z are measured in centimeters and (0, 0,0) is at the center of the drain. 1. Rewriting F as follows, describe in words how the water is moving: Consider each of the three terms in equation (4). (Look at some plots.) For fixed z, what is the flow like? How...
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In a bathtub, the velocity of water near2 the drain is given by the vector field (22 +1)2222 + 1 where r, y, and z are measured in centimeters and (0, 0,0) is at the center of the drain. 1. Rewriting F as follows, describe in words how the water is moving: Consider each of the three terms in equation (4). (Look at some plots.) For fixed z, what is the flow like? How does the flow change...
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