Question

10. The Navier-Stokes equation for the case of pressure driven flow along a circular pipe, with $\vec{v} = v_z(r)\hat{e}_z$, while neglecting gravity effects, is given by $\frac{dp}{dz} = \mu \left[ \frac{1}{r} \frac{d}{dr} \left( r \frac{dv_z}{dr} \right) \right]$ With the boundary conditions, $p(z = 0) = p_0$, $p(z = L) = p_L$, $v_z(r = 0) < \infty$ and $v_z(r = R) = 0$, obtain $v_z$ in terms of $\mu$, $(p_0 - p_L)$, $L$, $r$ and $R$.

          10. The Navier-Stokes equation for the case of pressure driven flow along a circular pipe,
with $\vec{v} = v_z(r)\hat{e}_z$, while neglecting gravity effects, is given by
$\frac{dp}{dz} = \mu \left[ \frac{1}{r} \frac{d}{dr} \left( r \frac{dv_z}{dr} \right) \right]$
With the boundary conditions, $p(z = 0) = p_0$, $p(z = L) = p_L$, $v_z(r = 0) < \infty$ and
$v_z(r = R) = 0$, obtain $v_z$ in terms of $\mu$, $(p_0 - p_L)$, $L$, $r$ and $R$.

10. The Navier-Stokes equation for the case of pressure driven flow along a circular pipe,
with v⃗ = vz(r)êz, while neglecting gravity effects, is given by
(dp)/(dz) = μ[ (1)/(r)(d)/(dr)( r (dvz)/(dr)) ]
With the boundary conditions, p(z = 0) = p0, p(z = L) = pL, vz(r = 0) < ∞ and
vz(r = R) = 0, obtain vz in terms of μ, (p0 - pL), L, r and R.

Added by Martin F.

Introduction to Fluid Mechanics

Philip J. Pritchard, Robert W. Fox 8th Edition

Chapter 7

Instant Answer

Solved by Expert John Nicolle

Step 1

First, let's write down the Navier-Stokes equation for pressure-driven flow along a circular pipe: ∂vz/∂t + vz∇vz = -1/ρ ∇p + ν∇^2vz where vz is the axial velocity component, t is time, ρ is the fluid density, p is the pressure, ν is the kinematic viscosity, and Show more…

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10. The Navier-Stokes equation for the case of pressure-driven flow along a circular pipe with y = vz(r)ez, while neglecting gravity effects, is given by: With the boundary conditions: p(z = 0) = Po, p(z = L) = PL, vz(r = 0) < co, and vz(r = R) = 0, obtain vz in terms of âˆ‚, (Po - p), L, r, and R.