For the problem of low Reynolds number flow past a sphere described in question 1, the stream function in spherical coordinates is given by 1020 [2² +2²³ 2 2r y = Up = 3 Rr 2 where R is the radius of the sphere and V is the liquid velocity far from the sphere. a. Find the velocity components in spherical coordinates as a function of r and using the following definitions 1 dy r2 sin Ꮎ ᎧᎾ ve sin²0 1 de r sin 0 dr b. Show that the velocity components found in part (a) satisfy the simplified form of the continuity equation that you found in question 1. c. Now assume that the sphere radius is R = 0.48 mm and the settling velocity is Vo = 14 cm/s. From the answer to part (a), find the ve component of velocity (in cm/s) at the position 0 = π/2 and r = 0.5 mm.
For the problem of low Reynolds number flow past a sphere described in question 1, the stream function in spherical coordinates is given by 1020 [2² +2²³ 2 2r y = Up = 3 Rr 2 where R is the radius of the sphere and V is the liquid velocity far from the sphere. a. Find the velocity components in spherical coordinates as a function of r and using the following definitions 1 dy r2 sin Ꮎ ᎧᎾ ve sin²0 1 de r sin 0 dr b. Show that the velocity components found in part (a) satisfy the simplified form of the continuity equation that you found in question 1. c. Now assume that the sphere radius is R = 0.48 mm and the settling velocity is Vo = 14 cm/s. From the answer to part (a), find the ve component of velocity (in cm/s) at the position 0 = π/2 and r = 0.5 mm.
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Parts a-c please.
Transcribed Image Text:For the problem of low Reynolds number flow past a sphere described in question 1, the stream function in
spherical coordinates is given by
R³
1020 [12² + 2/²0
3 Rr
2r 2
2
where R is the radius of the sphere and V is the liquid velocity far from the sphere.
a. Find the velocity components in spherical coordinates as a function of r and using the following definitions
1 dy
r2 sin Ꮎ ᎧᎾ
y =
Up =
ve
1 de
r sin 0 dr
sin²0
b. Show that the velocity components found in part (a) satisfy the simplified form of the continuity equation that
you found in question 1.
c. Now assume that the sphere radius is R = 0.48 mm and the settling velocity is U = 14 cm/s. From the
answer to part (a), find the ve component of velocity (in cm/s) at the position = π/2 and r = 0.5 mm.
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