Solving for the Constants in Fluid Flow through a Pipe

In summary, the conversation is about solving for constants in an equation for fluid flow through a rigid pipe. The notes end up with Aln(a) = 0 due to a boundary condition that requires rejecting the ln(r) complimentary function. The center of the cylinder is defined as r = 0 and the edge as r = a.
  • #1
enc08
42
0
Hi,

I'm looking at the solution to a question on fluid flow through a rigid pipe.

Original equation: [tex]\mu u = 0.25r^{2} dp/dx + Aln(r) + B[/tex]
After applying boundary conditions: [tex]\mu u = 0.25dp/dx (r^{2} - a^{2})[/tex]

I don't understand how the constants have been solved for. Below is as far as I get:
Starting with
[tex]\mu u = 0.25r^{2} dp/dx + Aln(r) + B[/tex]
Assume a no-slip boundary condition, so
[tex]u(r = a) = 0: 0 = 0.25a^{2} dp/dx + Aln(a) + B[/tex]

The notes somehow end up with [tex]Aln(a) = 0[/tex].

Thanks for any input.
 
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  • #2
A must be zero, or else [itex]u(0) = A \ln(0) + B[/itex], which is mathematical and physical nonsense.
 
  • #3
I see. So the edge of the cylinder is defined as r = 0, and the centre as r = a?

Thanks for clearing it up.
 
  • #4
enc08 said:
I see. So the edge of the cylinder is defined as r = 0, and the centre as r = a?

Thanks for clearing it up.

No, the center of the cylinder is r = 0.

You have a 2nd order ODE for u(r); it has two boundary conditions. One is that u(a) = 0, the other is that u(0) is finite. It is this condition that requires you to reject the ln(r) complimentary function.
 
  • #5
I see, thanks.
 
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FAQ: Solving for the Constants in Fluid Flow through a Pipe

What is fluid flow through pipe?

Fluid flow through pipe is the movement of a fluid, such as water or oil, through a closed pipe system. It is an important concept in the study of fluid mechanics and is used in a variety of industries, including plumbing, chemical engineering, and oil and gas production.

What factors affect fluid flow through pipe?

The rate of fluid flow through a pipe is affected by several factors including the pressure difference between the two ends of the pipe, the viscosity of the fluid, the diameter and length of the pipe, and any obstacles or changes in the pipe's geometry.

What is the equation for calculating fluid flow through pipe?

The most commonly used equation for calculating fluid flow through pipe is the Bernoulli's equation, which takes into account the fluid's velocity, pressure, and potential energy. Other equations, such as the Darcy-Weisbach equation and the Hazen-Williams equation, are also used depending on the specific application.

What are the different types of fluid flow through pipe?

The three main types of fluid flow through pipe are laminar flow, turbulent flow, and transitional flow. Laminar flow occurs when the fluid moves in smooth, parallel layers, while turbulent flow is characterized by chaotic, irregular movement. Transitional flow is a mix of both laminar and turbulent flow and occurs when the flow rate is between a certain range.

How is fluid flow through pipe measured?

Fluid flow through pipe is typically measured using flow meters, which can be either mechanical or electronic. These devices measure the volume or mass of fluid passing through the pipe and can provide information such as flow rate, total flow, and velocity. Other methods, such as pressure drop and velocity profile measurements, can also be used to determine fluid flow through pipe.

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