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Streamline Flow

Figure (1)shows streamlines traced out by dye injected into a moving fluid. A streamline is the path traced out by a tiny fluid element, which we may call a fluid "particle". As the fluid particle moves, its velocity may change, both in magnitude and in direction. However, its velocity vector at any point Figure (2). Streamlines never cross because, if they did, a fluid particle arriving at the intersection would have two velocities simultaneously, an impossibility.

In flows like that of Figure (1), we can isolate a tube of flow whose boundary is made up of streamlines. Such a tube acts like a pipe because any fluid particle that enters it cannot escape through its walls; if a particle did escape, we would have a case of streamlines crossing each other.

Figure (3)  shows two cross sections, of areas A1 and A2 along a thin tube of low. Let us station ourselves at B and monitor the fluid, moving with speed v1 , for a short time interval Δt. During this interval, a fluid particle moves a small distance v1Δ t and a volume Δ V of luid, given by

     Δ V = A1v1Δ t
passes through area A1.

Assume that the fluid is incompressible and cannot be created or destroyed. Thus in this same time interval, the same volume of fluid must pass point C, farther down the tube of flow. If the speed there is v2 this means that

Δ V = A1v1Δ t = A2c2Δ t
A1v1 = A2v2

   Fig1: Streamline Flow



Figure 2: A fluid particle P traces out a streamline as it moves. The velocity vector of the particle is tangent to the streamline at every point.

Figure 3: A tube of flow is defined by the streamlines that form its boundary. The flow rate of fluid must be the same for all cross sections of the tube of flow.
When a channel, such as a pipe, constricts, the streamlines draw closer together, signaling an increase in the fluid velocity. The arrow shows the direction of the flow.

Thus, along the tube of flow,

R = Av = a constant

In which R, whose SI unit is the cubic meter per second, is the volume flow rate of the fluid. The above equation is called the equation of continuity for fluid flow. It tells us that the flow is faster in the narrower parts of the tube of flow, where the streamlines are closer together.

The above equation is actually an expression of the law of the law of conservation of mass in a form useful in fluid mechanics. In fact, if we multiply R by the (assumed uniform) density of the fluid, we get the quantity Avp, which is the mass flow rate, whose SI unit is the kilogram per second. Then this equation effectively tells us that the mass that flows through point B in Figure (3) each second must be equal to the mass that flows through point C each second.

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