Course description

3.1 Flow of liquids & Equation of continuity:

At rest the properties of fluids can be described by the concepts of pressure and density, by Archimedes’ principle of buoyancy and by Pascal’s law of transmission of pressure (Fig. 1).

   Archimedes’ principle of buoyancy                             Pascal’s law

                                            Fig. 1

In motion the phenomenon of fluids can be described by the familiar principles of mechanics. The knowledge of the behavior of fluids in motion is required to know the harnessing of water power, the building of efficient steam turbines (Fig. 2), the designing of streamlined cars, trains and airplanes.

                   Mechanics                                  Water power                                     Steam turbines

                                                                              Fig. 2

Rate of flow of a liquid: We have to consider an ideal liquid which to be perfectly mobile, practically incompressible, and non-viscous i.e., having no internal friction.

For such a liquid the amount of liquid flowing across any section of a tube in a given time is always same.

Let A and B be two sections of a tube of cross-sectional area  as shown in Fig. 3.

The volume of the liquid flowing through the section AB=αl=α×vt  ; where v is the velocity of the liquid and t is the time taken from A to B.

 =α×v  c.c/sec

Moreover, mass (m) = density (ρ) × volume (V)

 m = ρ ×α×vt

Streamline, turbulent and Laminar flow: There are two motions of a fluid (i) steady and (ii) unsteady.

Streamline: In steady or orderly motion, the velocity of a fluid at a given point is constant in time. Every particle arriving at the given point will pass on with the same speed in the same direction. If we trace out the path of the particle, we will get a curve or straight line called streamline as indicated the curve in the Fig. 4. This kind of motion is called orderly or streamlines motion.

Turbulent motion: Exceeding a particular limiting value, called the critical velocity, the steady motion of the liquid loses all its orderliness, and becomes sinuous or zigzags and the motion is then called turbulent motion (Fig. 5).

Laminar flow: Let us consider a certain amount of liquid as shown in Fig. 6. The bottom plate is stationary. The top surface of the liquid is moving. In between the top and bottom plates the velocities of the intermediate layers increases uniformly from the bottom surface to the top surface. The magnitudes of the velocities of the different layers are indicated by the lengths of the arrows in the Fig. 6. Such a liquid flow, in which the different layers or laminae glide (to descend gradually) over one another at a slow and steady velocity (not exceeding critical velocity) without intermixing is called a laminar, streamline or viscous flow.

Equation of continuity:

Let us consider a thin tube of flow as shown in Fig. 7. The line of velocity of the fluid inside is parallel to the tube at any point. Let v1 and v2 be the velocities of the fluid at sections P and Q of the tube respectively. If A1 and A2 are the respective areas of cross-sections of these sections then the

                                                                                                                                                                                                                                                                                                                                                                         Fig 7

rate of flow of fluid into the tube at the point P is A1v1 and the rate of flow at the point Q is A2v2. If ρ1 and ρ2 be the respective densities of the fluid at sections P and Q; then the mass of the fluid flowing in per second is A1v1ρ1 and that flowing out is A2v2ρ2.

Since there are no sources or sinks wherein the fluid can be created or destroyed, the mass crossing each section of the tube per unit time must be the same.

Hence,  A1v1ρ1 = A2v2ρ2

If the fluid is incompressible, then ρ1 = ρ2. Then we have A1v1 = A2v2

or, Av =Al/t=Volume/time   = rate of flow of liquid  constant -----------------------(i)

where l is the length of the tube.

This equation is known as equation of continuity and states that in steady incompressible flow the volume flux or the flow rate given by the product Av across any section is constant.

This equation predicts that the speed of flow varies inversely with the cross-sectional area, being larger in narrower parts. Therefore, in the narrower section of the tube, the streamlines must crowd together than in the wider part.

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