The Venturi Tube

Scientists often investigate the manner in which quantities vary and whether they approach specific values under certain conditions. The apparatus illustrated in Figure [i] is used to study the flow of liquids or gases. It consists of a Venturi tube (a cylindrical pipe with a narrow constriction) and two meters that measure the pressure of a flowing liquid or gas in the nonconstricted and the constricted parts of the tube. (Other types of meters may measure the speed at which the liquid or gas flows through the tube.)

Let us suppose that a liquid enters the tube from the left, with a certain velocity. (The notion of velocity is defined precisely later.) The inward pressure meter displays a measurement x of the pressure in the nonconstricted part of the tube. As the liquid passes through the constriction it speeds up, and the pressure decreases to a value y, as indicated by the constricted pressure meter. Let us fix our attention on the two meters, as illustrated in Figure [ii]. We will use these meters to give a precise meaning to the statement y approaches L as x approaches a, or, symbolically, L = limit of y as x approaches a

When using this laboratory equipment, we would not expect the pressure y to remain exactly at L over a long period of time. Instead, our goal might be to force y to remain very close to L by restricting x to values near a. In particular, if e (epsilon) denotes a small positive real number, let us suppose it is sufficient that L - e < y < L + e, as indicated on the constricted pressure meter in Figure [ii]. An equivalent statement using absolute value is | y - L | < e

If these inequalities are true, we say that y has e-tolerance at L.   For example, the statement y has 0.01-tolerance at L means | y - L | < 0.01; that is, y is within 0.01 units of L.   This tolerance may be sufficiently accurate for experimental purposes.

Similarly, let us consider a small positive number d (delta) and define d-tolerance at a on the inward pressure meter in Figure [ii]. In our later work with functions it will be important that x ¹ a. Anticipating this restriction, we say that x has d-tolerance at a if   0 < | x - a | < d    or, equivalently, if   a - d < x < a + d   and    x ¹ a.

Let us now consider the following question:  Given any e > 0, is there a d > 0 such that if x has d-tolerance at a, then y has e-tolerance at L?   If the answer to this question is yes, we write L = limit of y as x approaches a

It is important to note that if L = limit of y as x approaches a , then no matter how small the number e, we can always find a  d > 0 such that if x is restricted to the interval ( a - d , a + d ) on the inward pressure meter (and x ¹ a ), then y will lie in the interval ( L - e , L + e ) on the constricted pressure meter.

 

© 1991, Swokowski, Calculus 5e, p 52