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Fluid Dynamics

Steady Flow (Stream Line Flow)

It is the flow in which the velocity of fluid particles crossing a particular point is the same at all the times. Thus, each particle takes the same path as taken by a previous particle through the point.

Enquiry: Can we study fluids in motion?

             Motions of smoke from chimney, flow of water in river, flow of any gas in a tube are examples of flow of fluids. This is generally a complex stream of mechanics. But, if we restrict ourselves to a simpler type of motion, i.e. steady, streamline or laminar flow and not turbulent flow then we can study the behavior of such fluids.

Enquiry: What is steady, streamline or laminar flow?

                The flow of fluid is said to be streamline if the velocity at any point in the fluid remains constant with time (in magnitude as well as in direction) in this case energy needed to drive the fluid is used up in overcoming the viscous force between its layers. All particles passing through a point in a steady flow follow the same path. The paths are known as streamlines> An example is water slowly flowing through a pipe.

Enquiry: What is turbulent flow?

             When the motion of a particle varies rapidly in magnitude and direction, the flow is said to be turbulent. In other words, when the velocity exceeds beyond the critical velocity, the paths and velocities of liquid change continuously and haphazardly. This flow is called turbulent flow. An example is water coming out of a fountain.

Enquiry: What is critical velocity?

             If in the case of a steady flow of fluids the velocity of flow is gradually increased it is found that the motion remains steady (streamline or laminar) upto a certain limit. If the velocity of flow crosses a certain limit the fluid particles do not follow the path of the preceding one and the motion becomes turbulent. The maximum velocity upto which fluid motion remains steady is called critical velocity. According to Reynold, in case of motion of fluids in narrow tubes, critical velocity depends on the density r and coefficient of viscosity η of the fluid as well as radius of the tube. i.e.

                V_c  = η/rρ     or   V_c = R η/ρr

             Here R is a dimensionless constant called Reynold's number. For steady flow R < 2000. For 2000<R<3000, flow is transitional and for R>3000 flow is turbulent.

Pause:   We shall be studying about viscosity later on in the course of this chapter.

Note:      In further discussions, we shall be considering only the streamline motion of a non-viscous incompressible fluid.

Enquiry: Will the velocity of a fluid remain constant even if the tube through which it flows has constrictions?

             In case of a fluid flowing through a tube of non-uniform cross-section, the product of the area of cross-section, density and the velocity of flow is same at every point in the tube. This is known as the principle of continuity, which actually means that amount of mass remains constant. For a fluid, mass flowing in through end B=mass flowing out from C. If the fluid is compressible

                                        A₁V₁ρ₁ = A₂V₂ρ₂.

If the fluid is incompressible then ρ₁ = ρ₂.

Therefore, A₁V₁ = A₂V₂ = Constant.

             Therefore, we can infer that if the cross-sectional area of a tube changes, the velocity of flow will also change.