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MMLU-Pro / 12065 / At low velocities (laminar flow), the volume flux Q through a small-boretube is…

Problem

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question

At low velocities (laminar flow), the volume flux Q through a small-boretube is a function only of the pipe radius r, the fluidviscosity \mu, and the pressure drop per unit pipe length (dp/dx). Using the power-product method, rewrite the suggestedrelationship Q = f{r, \mu, (dp/dx)} in dimensionless form. Table 1 DIMENSIONS OP FLUID-MECHANICS QUANTITIES Dimensions Quantity Symbol {MLT\Theta} {FLT\Theta} Length L L L Area A L^2 L^2 Volume U L^3 L^3 Velocity V LT^-1 LT^-1 Speed of sound a LT^-1 LT^-1 Volume flux Q L^3T^-1 L^3T^-1 Mass flux m MT^-1 FTL^-1 Pressure, stress p, \sigma ML^-1T^-2 FL^-2 Strain rate \epsilon T^-1 T^-1 Angle \texttheta None None Angular velocity \omega T^-1 T^-1 Viscosity \mu ML^-1T^-1 FTL^-2 Kinematic viscosity v L^2T^-1 L^2T^-1 Surface tension \gamma MT^-2 FL^-1 Force F MLT^-2 F Moment, torque M ML^2T^-2 FL Power P ML^2T^-3 FLT^-1 Density \rho ML^-3 FT^2L^-4 Temperature T \Theta \Theta Specific heat c_p,c_v L^2T^-2 \Theta^-1 L^2T^-2 \Theta^-1 Thermal conductivity k MLT^-3 \Theta^-1 FT^-1 \Theta^-1 Expansion coefficient \beta \Theta^-1 \Theta^-1

category

engineering

options

  1. Q = (const) (μ/r^4) (dp/dx)
  2. Q = (const) (r/\mu) (dp/dx)
  3. Q = (const) (r^3/\mu) (dp/dx)
  4. Q = (const) (r^5/μ^2) (dp/dx)
  5. Q = (const) (r^2/\mu) (dp/dx)
  6. Q = (const) (r^2μ^2) (dp/dx)
  7. Q = (const) (r^4/μ^2) (dp/dx)
  8. Q = (const) (r^4/\mu) (dp/dx)
  9. Q = (const) (dp/dx)/(rμ)

src

stemez-FluidMechanics

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Source and history

upstream_id / Original problem ID
12065
task_id
94382d0c-7e33-574e-b19b-39181a784953
task_key
default--test--12065
task_revision_id
1
config / split
default / test

Official source

initial import