Streamline Flow

So far we have studied fluids at rest. The study of the fluids in motion is known as fluid dynamics. When a water tap is turned on slowly, the water flow is smooth initially, but loses its smoothness when the speed of the outflow is increased. In studying the motion of fluids, we focus our attention on what is happening to various fluid particles at a particular point in space at a particular time. The flow of the fluid is said to be steady if at any given point, the velocity of each passing fluid particle remains constant in time. This does not mean that the velocity at different points in space is same. 

The velocity of a particular particle may change as it moves from one point to another. That is, at some other point the particle may have a different velocity, but every other particle which passes the second point behaves exactly as the previous particle that has just passed that point. Each particle follows a smooth path, and the paths of the particles do not cross each other.

Fig. 1 The meaning of streamlines. (a) A typical trajectory of a fluid particle. (b) A region of streamline flow.

The path taken by a fluid particle under a steady flow is a streamline. It is defined as a curve whose tangent at any point is in the direction of the fluid velocity at that point. Consider the path of a particle as shown in Fig.1 (a), the curve describes how a fluid particle moves with time. The curve PQ is like a permanent map of fluid flow, indicating how the fluid streams. No two streamlines can cross, for if they do, an oncoming fluid particle can go either one way or the other and the flow would not be steady. Hence, in steady flow, the map of flow is stationary in time. 

How do we draw closely spaced streamlines ? If we intend to show streamline of every flowing particle, we would end up with a continuum of lines. Consider planes perpendicular to the direction of fluid flow e.g., at three points P, R and Q in Fig.1 (b). The plane pieces are so chosen that their boundaries be determined by the same set of streamlines. This means that number of fluid particles crossing the surfaces as indicated at P, R and Q is the same.

If area of cross-sections at these points are A$_P$, A$_R$  and A$_Q$  and speeds of fluid particles are v$_P$, v$_R$  and v$_Q$ , then mass of fluid ∆m$_P$  crossing at A$_P$  in a small interval of time ∆t is $\rho_PA_Pv_P \Delta vt$. Similarly mass of fluid ∆m$_R$  flowing or crossing at AR in a small interval of time ∆t is $\rho_RA_Rv_R \Delta vt$ and mass of fluid ∆m$_Q$  is $\rho_QA_Qv_Q \Delta vt$ crossing at A$_Q$. The mass of liquid flowing out equals the mass flowing in, holds in all cases. Therefore,

$\rho_PA_Pv_P \Delta vt=\rho_QA_Qv_Q \Delta vt=\rho_RA_Rv_R \Delta vt$     (1)

For flow of incompressible fluids 

ρ$_P$  = ρ$_R$  = ρ$_Q$ 

Equation (1) reduces to 

A$_P$ v$_P$  = A$_R$ v$_R$  = A$_Q$ v$_Q$ (2) 

which is called the equation of continuity and it is a statement of conservation of mass in flow of incompressible fluids. In general

Av = constant    (3)

Av gives the volume flux or flow rate and remains constant throughout the pipe of flow. Thus, at narrower portions where the streamlines are closely spaced, velocity increases and its vice versa. From (Fig 1b) it is clear that A$_R$ > A$_Q$ or v$_R$  < v$_Q$ , the fluid is accelerated while passing from R to Q. This is associated with a change in pressure in fluid flow in horizontal pipes.

Steady flow is achieved at low flow speeds. Beyond a limiting value, called critical speed, this flow loses steadiness and becomes turbulent. One sees this when a fast flowing stream encounters rocks, small foamy whirlpool-like regions called ‘white water rapids are formed. 

Figure 2 displays streamlines for some typical flows. For example, Fig. 2(a) describes a laminar flow where the velocities at different points in the fluid may have different magnitudes but their directions are parallel. Figure 2(b) gives a sketch of turbulent flow.

Fig. 2 (a) Some streamlines for fluid flow. (b) A jet of air striking a flat plate placed perpendicular to it. This is an example of turbulent flow.