Bernoulli's Theorem and its Applications

Bernoulli's theorem

For the streamline flow of a non-viscous, incompressible liquid, the sum of the energy heads rmains constant throughout the flow ie.\begin{align*} \frac{P}{\rho}+gh+\frac{1}{2}v^2=constant \end{align*}

Let, a,v,P and h denote the corss section, velocity of flow, pressure and height respectively at the end A with supscript 1 and at the end B with subscript 2. Suppose in time \(\Delta t\), the liquid at \(A\) reaches to \(A'\). So, the force acting on the liquid is \(f_{1}=a_{1}P_{1}\). \(\therefore\) Workdone on the liquid in moving it from \(A\) to \(A'\) is \(W_{1}=P_{1}a_{1}v_{1}\Delta t\). During the same interval, the liquid at \(B'\) reaches to \(B\) doing work agianst pressure \(P_{2}\). Work done by the liquid against pressure \(P_{2}\) at the end \(B\) is\(W_{2}=P_{2}a_{2}v_{2}\Delta t\). The net work done on the liquid by the pressure energy is,\begin{align*} W=W_{1}-W_{2}=P_{1}a_{1}v_{1}\Delta t-P_{2}a_{2}v_{2}\Delta t \end{align*} From equation of continuity, \(a_{1}v_{1}\Delta t=a_{2}v_{2}\Delta t=V\) \begin{align*} \therefore w=(P_{1}-P_{2})V=(P_{1}-P_{2})\frac{m}{\rho}---1 \end{align*} where, \(m\) is the mass of the liquid passed and \(\rho\) is the density of the liquid.

The work done by pressure energy increases both the kinetic and potential energy of the liquid ie.\begin{align*} W=increase \space in \space KE+increase \space in \space PE \end{align*}\begin{align*} or,(P_{1}-P_{2})\frac{m}{\rho}=(\frac{1}{2}mv_{2}^2-\frac{1}{2}mv_{1}^2)+(mgh_{2}-mgh_{1}) \end{align*}\begin{align*} or, \frac{P_{1}}{\rho}+\frac{1}{2}v_{1}^2+gh_{1}=\frac{P_{1}}{\rho}+\frac{1}{2}v_{2}^2+gh_{2} \end{align*}\begin{align*} \frac{P}{\rho}+gh+\frac{1}{2}v^2=constant \end{align*} which is Bernoulli's theorem. For the flow along a horizontal line,\begin{align*} \frac{P}{\rho}+\frac{1}{2}v^2=constant \end{align*}

Derivation of Bernoulli's equation from Euler's equation

Consider an infinitesimally small portion AB of a tube through which a fluid flows. Let, the cross-section be uniformm over \(dl\). If \(\rho\) be the density of the fluid, the mass of fluid in between A and B is \(m=da dl \rho\) and whe weight is\(w=mg=da dl \rho\) which is acting vertically downward at O and making an angle \(\theta\) with the direction of the flow. The component \(mg\cos \theta\) of the weight has a tendency to flow from A to B. Let, the fluid pressure at A be P. Therefore, pressure on the face B is \(P+\frac{\partial P}{\partial l}dl\). The net force acting on the fluid to drive it forward is,\begin{align*} F=Pda+mg\cos \theta-(P+\frac{\partial P}{\partial l}dl).da \end{align*}\begin{align*} F=mg\cos \theta-\frac{\partial P}{\partial l}dl.da---1 \end{align*} If \(V\) be the velocity of the fluid at the end A then acceleration is \(\frac{dv}{dt}\).\begin{align*} \therefore F=da.dl. P \frac{dv}{dt}---2 \end{align*} Equating 1 and 2, we get,\begin{align*} da.dl. P \frac{dv}{dt}=mg\cos \theta-\frac{\partial P}{\partial l}dl.da \end{align*}\begin{align*} or, \frac{dv}{dt}=-\frac{1}{\rho}\frac{\partial P}{\partial l}+g\cos \theta---3 \end{align*} Since, velocity \(v\) of fluid flow is the function of both the distance and time, it changes from point to point. Thus, \(v\) can be expressed as,\begin{align*} dv=\frac{\partial v}{\partial l}.dl+\frac{\partial v}{\partial t}.dt\end{align*}\begin{align*} \frac{\mathrm{d} v}{\mathrm{d} t}=v.\frac{\partial v}{\partial l}+\frac{\partial v}{\partial t}\end{align*} from figure, \(cos \theta =\frac{\partial h}{\partial l}---4\) where, h is the vertical height of the tube. From 3 and 4,\begin{align*} v\frac{\partial v}{\partial l}+\frac{\partial v}{\partial t}=-\frac{1}{\rho}\frac{\partial P}{\partial l}+g\frac{\partial h}{\partial l}---5 \end{align*} This is the general equation of fluid flow applicable fro both the steady and turbulent flow and is known as Euler's equation. For steady flow, \(\frac{\partial v}{\partial t}=0\) and all parial derivatives become total derivatives. So, equation 5 becomes,\begin{align*} \rho v dv + 0= -dP- \rho g dh \end{align*}\begin{align*} or, \rho v dv +dP+\rho g dh =0 \end{align*} Integrating, we get,\begin{align*} \frac{\rho v^2}{2}+P+\rho gh=constant \end{align*}\begin{align*} or, \frac{P}{\rho}+\frac{1}{2}v^2+gh=constant \end{align*} which is the required Bernoulli's equation.

Venturimeter

It is an arrangement to measure the rate of flow of a liquid in a pipe and is based on the Bernoulli's principle.

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