Question

Derive Bernoulli's Equation.

Answer 1

1) Calculate the work that is done on the fluid of vertical layer thickness x₁ and x₂ in a pipe:

$$dW=F_1dx_1-F_2dx_2=p_1A_1dx_1-p_2A_2dx_2=\\=p_1dV-p_2dV=(p_1-p_2)dV.$$

2) This work was done to change the kinetic and potential energy of the fluid:

$$dK=\frac12m_2v_2^2-\frac12 m_1v_1^2=\frac12\rho(v_2^2-v_1^2)dV,\\\space\\ dU=mgh_2-mgh_1=\rho g(h_2-h_1)dV,\\\space\\ dW=dK+dU,\\\space\\ (p_1-p_2)dV=\frac12\rho(v_2^2-v_1^2)dV+\rho g(h_2-h_1)dV.$$

3) Rearrange the equation so all members with index 1 are on the left and all members with index 2 are on the right:

$$p_1+\frac12\rho v_1^2+\rho gh_1=p_2+\frac12\rho v_2^2+\rho gh_2.$$