Physics, Chapter 9: Hydrodynamics (Fluids in Motion)

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Physics, Chapter 9: Hydrodynamics (Fluids in Motion) University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 9: Hydrodynamics (Fluids in Motion) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 9: Hydrodynamics (Fluids in Motion)" (1958). Robert Katz Publications. 143. https://digitalcommons.unl.edu/physicskatz/143 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 9 Hydrodynamics (Fluids in Motion) 9-1 Steady Flow of a Liquid When a liquid flows through a pipe in such a way that it completely fills the pipe, and as much liquid enters one end of the pipe as leaves the other end of the pipe in the same time, then the liquid is said to flow at a steady rate. At any point of the pipe, the flow of the liquid does not change with time. The path of any particle of liquid as it moves through the pipe is Fig. 9-1 Streamlines of a liquid flowing through a pipe at a steady rate. called a streamline. We can map the flow of liquid through the pipe by drawing a series of streamlines following the paths of the particles of liquid, as shown in Figure 9-1. The instantaneous velocity of a liquid particle is always tangent to the streamline. The rate of flow may be represented by the density of streamlines, or the number of streamlines passing through a surface of unit area perpendicular to the direction of flow. Thus the stream­ lines will be close together where the liquid is moving rapidly and farther apart in regions of the pipe where the liquid is moving slowly. Similar conventions were used to represent the direction and magnitude of the gravitational field intensity. Since the liquid is incompressible and there are no places in the pipe where the liquid can be stored, the volume of liquid which flows through any plane perpendicular to the streamlines in any interval of time must be the same everywhere in the pipe. Consider two typical planes whose inter­ sections with the pipe have areas Al and A 2 perpendicular to the stream- 165 166 HYDRODYNAMICS §9-1 lines. The volume of liquid Q passing through area Al in unit time is Q = AIvI, (9-1) where VI is the velocity of the liquid at this point. Similarly, the volume of liquid passing through A 2 in unit time is Q = A 2V2. Since these two quantities must be equal for steady flow, we have AIvI = A 2U2, (9-2) VI A 2 ,or -= -. V2 Al Thus the velocity of the liquid at any point in the pipe is inversely propor­ tional to the cross-sectional area of the pipe. The liquid will be moving slowly where the area is large and will be moving rapidly where the area is small. Illustrative Example. Water flows out of a horizontal pipe at the steady rate of 2 ft 3/min. Determine the velocity of the water at a point where the diameter of the pipe is I in. The area A of the I-in. portion of the pipe is A = 1r X 1 ft2 = 0.0055 ft2. 4 X 144 When we substitute the values for Q and A in Equation (9-1), 2 ft 3 we obtain -- = 0.0055 ftz X v, 60 sec ft from which v = 6.10-· sec Fig. 9-2 The number of streamlines enter­ ing a volume element is equal to the num- , bel' leaving; the volume element when there is steady flow. In the steady or streamline flow of a liquid, the total quantity of liquid flowing into any imaginary volume element of the pipe must be equal to the quantity of liquid leaving that volume element. If we represent the flow by streamlines, this implies that the streamlines are continuous and do not pile up anywhere within the liquid. The same number of streamlines enter §9-2 BERNOULLI'S THEOREM 167 a volume element as the number which leave it, as shown in Figure 9-2. Another characteristic of streamline flow is that it is layerlike, or lamellar. There is no circulation of the liquid about any point in the pipe. We might imagine a small paddle wheel, as shown in Figure 9-3, placed in the pipe. When the flow is lamellar, no rotation of the paddle wheel is produced by the flow of liquid. Fig. 9-3 A small paddle wheel placed in a flowing liquid will not rotate when the liquid is in steady or lamellar flow. .~ 9-2 Bernoulli's Theorem The fundamental theorem regarding the motion of fluids is due to Daniel Bernoulli (1700-1782), a Swiss physicist and mathematician. Bernoulli's theorem is essentially a formulation of the mechanical concept that the work done on a body is equal to the change in its mechanical energy, in the case that mechanical energy is conserved; that is, where there is no loss of mechanical energy due to friction. Let us consider the motion of an incompressible fluid of density p along an imaginary tube bounded by streamlines, as shown in Figure 9-4. We shall call such a tube a 8treamtube. Since each streamline represents the direction of motion of a particle of liquid in steady flow, no particle of liquid may cross a streamtube. At the left-hand end of the tube, the liquid has a velocity VI, the tube has cross-sectional area AI, the pressure is PI, and the tube is at a height hI above some reference level. At the right-hand end of the tube, the velocity is V2, the cross-sectional area is A 2 , the pressure is P 2 , and the height is h2 . When a small quantity of fluid of volume V is moved into the tube through the action of the external fluid, an equal volume of fluid must emerge from the streamtube against the force exerted by the pressure P 2 of the fluid outside the tube at the right-hand end of the stream­ tube. Let us imagine that the flow of fluid in the streamtube takes place as the result of the displacement 81 of a piston of area Al which just fits the streamtube at its left-hand end such that the volume swept out by the 168 HYDRODYNAMICS §9-2 motion of the piston is A1s1 V, and of a corresponding displacement S2 of a piston of area A 2 which just fits the streamtube at its right-hand end such that the volume swept out by the second piston is A 2 s2 = V; that is, a quantity of fluid of volume V has just passed through the streamtube. The piston at position 1 has done work on the fluid equal to the product of the force exerted by the displacement of that force, hence the work done on Fig. 9-4 the fluid is P1A1sl, while the fluid has pushed back the piston at position 2 so that the fluid has done work on the second piston of amount P2 A 2 s2• The net work done on the fluid is therefore equal to P1A1s1 - P2A 2 s2 . Since we have assumed the fluid to be incompressible, we have A1s1 = A 2 s2 = V. From the above equations, we find that the net work done on the fluid within the streamtube is P1V - P2 V. If there was no loss of energy of the fluid due to frictional forces, this work done on the fluid in the streamtube must have resulted in a change of mechanical energy of the liquid which flowed into the tube at position 1 and out of the tube at position 2. The sum of the potential and kinetic §9-3 TORRICELLI'S THEOREM 169 energies of the liquid flowing into the tube at position 1 is pVgh l + ! pVvi, while the sum of these energies of the liquid flowing out of the tube at positio~ 2 is pVgh2 + !pVv~. Equating the work done on the fluid to the difference in its energy, we find PIV - P 2 V = pVgh2 + !pVv~ - (pVgh l + !pVvi). (9-3a) Dividing the equation through by V and transposing all quantities with subscript 1 to the left-hand side and all quantities with subscript 2 to the right-hand side of the equation, we find PI + pgh l + !pvi = P 2 + pgh2 + !pv~. (9-3b) This equation expresses Bernoulli's theorem, which states that at any two points along a streamline the sum of the pressure, the potential energy of a unit volume of fluid, and the kinetic energy of a unit volume of fluid has the same value. As indicated in the derivation, Bernoulli's theorem holds rigorously only for frictionless, incompressible, streamline flow. Bernoulli's theorem is a statement of the principle of conservation of energy expressed in a form suited to the description of fluids in steady frictionless flow. Although Bernoulli's theorem holds rigorously only for an incompres­ sible fluid, experience indicates that it is valid for air in streamline flow at speeds up to about half the speed of sound. (The speed of sound is about 740 mijhr.) Actual fluids such as water and air have internal fluid friction, or viscosity, so that, to be strictly true, the equal sign of Equation (9-3a) should be replaced by a greater than or equal sign (~), meaning that the work done on the fluid in the streamtube is greater than, or at least equal to, the increase in mechanical energy.
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