1.

Derive the Bernoulli equation for an incompressible liquid flowing in an inclined tube of variable cross section in a gravitational field.

Answer»


Solution :Apply the law of conservatiou of energy in conservative SYSTEMS. In our case the work of the FORCES of pressure is accompaniod by the change in the total MECHANICAL energy of the system : `W = W_2 - W_1 = (K_2 + U_2) - (K_1 + U_1)`. Consider separately a volume of liquid `V = l_1 S_1 = l_2S_2` (Fig.), the mass of the this volume is `m = rho V`. The work of the forces of pressure is
`W = F_1l_1 - F_2 l_2 = p_1 S_1 l_1 - p_2 S_2 l_2 = (p_2 - p_1) V`
Subtituting the result obtained into the expression for the change in energy , we obtain
`(p_1 - p_2) V = (mv_2^2)/(2) + mgh_2 - (mv_1^2)/(2) - mgh_1`
WHENCE
`p_1 + (rho v_1^2)/(2) + rho gh_1 = p_2 = (rho v_2^2)/(2) + rho gh_2`


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