Website owner: James Miller
Streamline fluid flow and varying tube diameter. Continuity equation.
Fluid flow. Suppose fluid is flowing in a tube in which the cross-sectional area is greater in one section than in another section. Then the velocity of the fluid will be higher in the section of the tube with the smaller cross-section. See Fig. 1. Why is this so and what is the mathematical law relating the velocities and the cross-sections?
Examples of this idea
- fluid flow in a tube where the diameter reduces down
- fluid flow in a river that is wide and deep in one place and narrow and shallow in another
- vehicles moving on a highway in which several lanes are funneled into a single lane (perhaps due to road construction)
- marbles flowing in a channel that reduces down in diameter
If a pipe has a cross-sectional area of 1 sq. ft. in one section and 1/3 sq. ft. in another section, then water flowing in the 1/3 sq. ft. section will flow at three times the speed of the water in the 1 sq. ft. section. Give an intuitive explanation of why this must be true.
Consider the example in Fig. 2 in which marbles in three channels (or lanes) are fed into a single channel. Let “d” be the diameter of a marble. Then it is intuitively obvious that when a marble in the single channel portion has moved a distance of 3d the marbles in the 3-channel section will have moved a distance of only 1d. Thus if the marbles in the 1 channel section are moving at a rate of 3d per second, then the marbles in the 3 channel section will be flowing at a rate of only 1d per second.
Consider now the fluid flow shown in Fig. 3 where the cross-sectional area at point 1 is A1 and at point 2 is A2. In time t, the fluid at point 1 will advance a distance of d1 = V1t and a volume of liquid equal to A1d1 = A1V1t will pass point 1. Similarly, the volume of fluid passing point 2 in time t will be A2V2t. The volume of fluid passing points 1 and 2 in time t must be equal so
A1V1t = A2V2t
and
A1V1 = A2V2
which is the equation of continuity for the steady flow of an incompressible liquid.
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life