Kinematics of Moving Fluid and Some Terms

Kinematics of moving fluid

Newton's law of viscosity

Viscosity is the property of fluids due to which an internal resistance or friction comes into play, when a liquid is in motion. This viscous drag between any two adjacent liquid layers is,\begin{align*} F=-\eta A \frac{dv}{dx} \end{align*} where, \(\eta\) is the coefficient of viscosity of the liquid, \(A\) is the area of contact between the liquid layers and \(\frac{dv}{dx}\) is the velocity gradient perpendicular to the liquid flow.

Stoke's law

The viscous force acting on a sphere of radius \(r\) flowing through aliquid having coefficient of viscosity \(\eta\) with terminal velocity \(v\) is,\begin{align*} F=6\pi \eta rv \end{align*}

Equation of continuity

For an incompressible liquid, volume of liquid entering per second in a tube is equal to that emerging per second from next end.

Volume of liquid entering per second in the tube A is \(V_{1}=\frac{a_{1}d_{1}}{t}=a_{1}v_{1}\) and volume of liquid emerging out from the end B in the same time is, \(V_{2}=\frac{a_{2}d_{2}}{t}=a_{2}v_{2}\). Thus \(a_{1}v_{1}=a_{2}v_{2}\) is the continuity equation.

Energy of fluid flow

The kinetic energy per unit mass is called velocity head and given by \(\frac{1}{2}v^2\). The potential energy per unit mass is given by \(gh\) and is called gravitational head. The pressure energy per unit mass is called pressure head is given by \(\frac{P}{\rho}\).

Streamline flow, critical velocity and Turbulent flow

When a liquid flows steadily, such that each particle passing a certain point flows exactly the same path and has the same velocity as the preceding particle, the flow is said to be streamline. It is also called laminar flow. But this flow holds up to a certain limiting value of velocity, called critical velocity.When the velocity exceeds this critical velocity, the velocity of liquid particle changes haphazardly. This kind of flow is called turbulent flow.

Reynold's number

Critical velocity of a liquid flowing through a tube depends on the coefficient of viscosity of the fluid, radius of the tube and density of the liquid. From dimensional analysis, we can obtain,\begin{align*} v=\frac{R\eta}{\rho r} \end{align*} where, \(R\) is the proportionality constant called Reynold's number. If the value of \(R\) is less than 2000, the flow is streamline and if it is more than 3000, the flow is turbulent. If \(R\) lies between 2000 and 3000, hen the flow is called unsteady flow.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.