Viscous Drag Forces

THIS INTERACTIVE JAVA APPLET WAS DEVELOPED AT UT AUSTIN BY
Aubrey Dugger, UNDER THE SUPERVISION OF PROFESSOR. Spyros A. Kinnas, 1997.

PERMISSION TO LINK TO THIS JAVA APPLET AND TO DOWNLOAD OR TO USE THE RELATED
JAVA SOURCE CODE, MUST BE GRANTED BY PROFESSOR Spyros A. Kinnas ( [email-masked] ).

Problem:

Attached to a cylindrical pipe is a U-tube mercury manometer, as
shown. Given the type of fluid in the pipe, the velocity (V), the diameter
of the pipe (D), and the length between sides of the U-tube manometer (L),
find the change in height (Dh) of the mercury in
the manometer.

Solution:

The first step in this problem is to determine the type of flow going
through the pipe (i.e. laminar or turbulent). To do this, we need to find the
Reynold's number (Re). This is accomplished fairly painlessly if we know the
velocity, pipe diameter, and fluid viscosity. The equation is:

where n = the kinematic viscosity of the fluid.

Once we know the Reynold's number, we can approximate the flow as:

Re<2300 means the flow is laminar
Re>2300 means the flow is
turbulent

Case 1: Laminar Flow
If the flow is laminar, our work is easy. We
can apply Bernoulli's equation and derive the formula:

where

DP = the change in pressure between ends of the
manometer tube

g = the specific weight of the fluid in the pipe

gM = the specific weight of the fluid
in the manometer

m = the dynamic viscosity of the fluid in the
pipe.

Case 2: Turbulent Flow
If the flow is turbulent, the problem is
slightly more complicated. We still use Bernoulli's equation to solve for the
change in height:

where

manometer tube

gM = the specific weight of the fluid
in the manometer

f = the friction factor

g = gravity.

All of the variables in this equation are known except for the friction
factor, f. To find this value, we can either consult the Moody diagram with our
known Reynold's number, or we can solve the equation:

Newton-Raphson Method:
Since this equation is difficult to solve
directly we must use a numerical method such as the Newton-Raphson method. In
this method we start with two initial guesses for the variable (renamed "x") and
then iteratively solve the function (g) using the following equations:

Our first initial guess (xk-1) is 0.01. Our second guess
(xk) is this number plus 1%, or 1.01(xk-1)=0.0101.

We then plug these guesses into the Newton-Raphson equation to find a new
guess, xk+1. This becomes our new xk and our previous
xk gets promoted to our new xk-1. This entire process is
repeated until the function converges, or the difference between our two guesses
is minimal. The final value of x is the friction factor (f) we are looking for.

Once we determine the friction factor for the given flow conditions, we can
plug all of our variables into the equation and solve for the change in height
of the mercury in the manometer tube.

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