A hollow tube (with inner radius Ro) is inside another concentric tube which rot

Question

A hollow tube (with inner radius Ro) is inside another concentric tube which rotates with an angular frequency w. The gap between the two tubes is 6 and is filled with lubrication oil with constant density p, dynamic viscosity u, and thermal conductivity k. The inner radius of the outer tube is R, and R?> ?. The outer tube is thermally isolated outside. Coolant continuously flows through the inner tube to remove heat generated during the operation of the system. The axial heat conduction can always be neglected for coolant, oil, and tubes. Also neglected is the temperature difference across the thickness of tube walls. Please consider the following questions: Assume the flow is laminar. Drive the velocity profile inside the gap.

Explanation / Answer

When u = u(r), the governing equation becomes

d / dr (r * mu * du/dr) = (dp/dx)*r

Writing dp/dx = K and integrating it twice,

u = (r2 / 4) (K / mu) + C1 ln r + C2

Boundary conditions:

At inner tube where r = Ro, u = 0 (no slip)

At outer tube where r = R1, u = R1 * w

Thus,

(Ro2 / 4) (K / mu) + C1 ln Ro + C2 = 0....................eqn1

(R12 / 4) (K / mu) + C1 ln R1 + C2 = R1 * w...........eqn2

Subtracting them we get

((R12 - Ro2) / 4) (K / mu) + C1 ln (R1 / Ro) = R1 * w

C1 = [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] / ln (R1 / Ro)

Putting it back in eqn1 we get

(Ro2 / 4) (K / mu) + [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] (ln Ro) / ln (R1 / Ro) + C2 = 0

C2 = - [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] (ln Ro) / ln (R1 / Ro) - (Ro2 / 4) (K / mu)

Therefore,

u = (r2 / 4) (K / mu) + [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] (ln r) / ln (R1 / Ro) - [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] (ln Ro) / ln (R1 / Ro) - (Ro2 / 4) (K / mu)

u = ((r2 - Ro2) / 4) (K / mu) + [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] (ln r) / ln (R1 / Ro) - [(R1 * w) - ((R12 - Ro2) / 4) (K / mu)] (ln Ro) / ln (R1 / Ro)

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.