Pressure, Resistance, Flow and Velocity

- Flow rate (Q) is directly proportional to the pressure gradient (?P) and inversely proportional to the resistance: Q = ?P/R

- Resistance is inversely proportional to the 4thpower of the radius (r) of a tube: R = 1/r⁴

- ?P is directly proportional to resistance: ?P = Flow x R

- velocity = Flow rate (Q) ÷ cross-sectional area (A): v = Q/A

1. Refer to Figure 1 at the end of this document and answer the questions that follow.

1.1 The upper part of the diagram shows a pipe of constant diameter, presumably containing a fluid (liquid). Values of pressure at selected points are indicated. Will there be flow in this pipe? How do you know?

Yes there will be a flow in the pipe.

This is because fluids only flow if there is a positive pressure gradient.

1.2 If the internal diameter of this pipe is 3.6 cm, what will will be its cross- sectional area (A)? (round your answer to the nearest whole number)

Area = πr²

Radius (r) = Diameter /2
Radius (r) = 3.6 /2
Radius (r) = 3.6 /2
Radius (r) = 1.8

Area = πr²
Area =3.14 X (1.8)²
Area =3.14 X (1.8)²
Area =10.17 cm²

1.3 If flow did occur in this pipe, and the flow rate (Q) was 1 mL/min, what will be the velocity (v) of flow?

Velocity (v) = Flow Rate (Q) / Cross-sectional Area (A)

Velocity (v) = 1 / 10.17 mL/min/cm²

Velocity (v) = Flow Rate (Q) / Cross-sectional Area (A)

1.4 If the diameter of this pipe were to change to 0.36 cm, but ?P stayed the same, what would be the new flow rate? Give a brief written explanation (hint: what has happened to the resistance, R?).

2. Figure 2 at the end of this document shows a large vessel with a constriction in the middle: notice that the cross-sectional areas are the same before and after the constriction. Assume a constant flow rate (Q) of 5 L/min and answer the questions that follow. Table cells for which calculated values are required are identified by letters, (a) - (i).

2.1 Calculate the fluid velocities in the 3 parts of the vessel, (a), (b), and (c) [corresponding to cross-sectional areas of 4, 1, and 4 cm2, respectively].

2.2 Explain briefly in words the findings from 2.1.

2.3 Clearly, a flowing fluid will have a kinetic energy: an energy of motion. Calculate the kinetic energy of the fluid at different points-(d), (e), and (f)-using the formula shown in the Figure. (The ρ term, which represents the density of the fluid, can be ignored, as we are considering an homogeneous fluid in this vessel-i.e. the density is constant everywhere. Thus we will calculate a relative measure of kinetic energy, and no units are required.)

Assessed Study Guide

2.4 Comparing the kinetic energies calculated in 2.3, by what factor (multiple) does the kinetic energy at (b) differ from that seen at (a) and (c)?

2.5 A flowing fluid will have a total energy, E, which is the sum of its kinetic energy and its potential energy. The pressure, P, at a given point is a form of potential energy. On this basis, calculate the pressure, P, at each point, (g), (h), and (i). [This will be a relative measure of kinetic energy, so no units are required.]

2.6 You have learned a basic rule: fluids flow down a pressure gradient, from high pressure to low pressure. You will have found in 2.5 that P falls from (g) to (h), so we expect flow from (g) to (h). But what about flow from (h) to (i)? Between these two points, P increases! But, I assure you, there IS flow from (h) to (i). Perhaps our rule concerning flow is incomplete. Taking account of the information compiled, propose a better general rule about fluid flow.

3. Look up the term vascular flutter (NOT atrial flutter). From the information you find and from analogy with the situation arising in Question 2, briefly explain the root cause of vascular flutter.

4. Consider the following statement: In the mammalian circulatory system, the reason velocity of blood flow is lowest at the capillaries is because the very narrow capillaries offer great resistance to blood flow. Is this TRUE or FALSE? Defend your diagnosis. You should include some quantitative evidence in your account.

5. Name at least 3 distinct principles that apply equally regarding air flow in the respiratory system and blood flow in the cardiovascular system. Air and blood are both considered fluids. In what major way do the properties of these fluids differ?

Assessed Study Guide A (Autumn Term, 2015) 3

Figure 1: A pipe of constant diameter. The values of "P" indicated at intervals represent the pressure at that point along the pipe. For Question 1.

Figure 2: The Bernoulli effect exemplified. Letters (a), (b), etc. in the table locate values to be calculated. For Question 2.