Pressure gradient \( \Large{\frac{ dp }{ dz }} \)

-0.65

Pa/m

Pipe radius \( r \)

2.50

cm

Dynamic viscosity \( \mu \)

0.0011

Pa·s

Directions

Details

About

Volumetric flow rate

=

89

cm^{3}/s

Average velocity

=

4.5

cm/s

Maximum velocity

=

9.1

cm/s

Reynold's number

=

2024

This simulation demonstrates laminar flow of viscous fluid in a pipe. The plot shows fluid velocity as a function of radial distance from the pipe center. Velocity is proportional to the length of the arrows. The pressure gradient, pipe diameter, and fluid viscosity are set with sliders.

Fully-developed laminar flow in an horizontal pipe means the fluid: (1) only moves in the axial (z) direction. (2) flow is at steady state. (3) velocity is zero at the pipe wall. (4) is constant velocity along a streamline. (5) does not have axial mixing. (6) velocity is maximum at the pipe center. (7) has a Reynolds number \( Re < 2100. \) Gravitational force is assumed negligible.

In terms of cylindrical polar coordinates, the z-direction Navier-Stokes equation can be written as:

\(
\rho (\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} +
\frac{u_\theta}{r}\frac{\partial u_r}{\partial_\theta}
- \frac{u^2_\theta}{r} + u_z \frac{\partial u_r}{\partial z}) \\
= - \frac{\partial P}{\partial z} + \rho g_z + \mu (
\frac{1}{r} \frac{\partial}{\partial r}(r\frac{\partial u_z}{\partial r}) + \frac{1}{r^2}\frac{\partial ^2
u_z}{\partial \theta ^2}
+ \frac{\partial ^2 u_z}{\partial z^2}
)
\)

Where \(u_r, u_\theta , u_z \) is the fluid velocity in \(r, \theta, z\) directions; \(\rho\) is the fluid density; \(\mu\) is the fluid viscosity; \(P\) is pressure; \(g\) is the gravitational force; \(t\) is time.

This equation simplifies to: $$ 0 = - \frac{\partial P}{\partial z} + \mu \frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial u_z}{\partial r}) $$ Where the pressure gradient in the z direction \(\frac{\partial p}{\partial z}\) is constant and equal to \(\frac{\Delta p}{l}\). The boundary conditions are: $$ u_z(R) = 0 $$ $$ \left. \frac{\partial u_z}{\partial r} \right|_{r=0} = 0 $$ Where \(R\) is the pipe radius. The solution for \(u_z\) is: $$ u_z = \frac{R^2}{4 \mu} \frac{\Delta p}{l} (1-\frac{r^2}{R^2}) $$ Average velocity is half the maximum velocity: $$ u_{avg} = \frac{ R^{2} \Delta p }{ 8 \mu l } $$

References:

- Gerhart, P. M., Gerhart, A. L., & Hochstein, J. I. (2016). Munson, Young and Okiishi's Fundamentals of Fluid Mechanics. John Wiley & Sons. P.322.

This simulation was created in the Department of Chemical and Biological Engineering, at University of Colorado Boulder for LearnChemE.com by Mingyuan Lu and Neil Hendren under the direction of Professor John L. Falconer. Address any questions or comments to learncheme@gmail.com. All of our simulations are open source, and are available on our LearnChemE Github repository.

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