derive continuity equation

p1a1v1=p2a2v2 or A1V1=A2V2

Let us consider a tube AB of varing cross-section with water is flowing through that tube. At side A, area of cross-section be `A1` velocity of fluid be `v1` At side B, area of cross-section be `A2` velocity of fluid be `v2` and density of water be `d` and let `A1`>`A2` Now v=distance travelled/time=l/t --->l=vt ---> Mass of fluid entered the tube at A in `t`seconds, m1=dV1=dAl=dA1v1t ----->(1) ---> Mass of fluid leaved the tube at B in `t`seconds, m2=dV2=dAl=dA2v2t ----->(2) But mass of fluid which entered into the tube is equal to the mass of fluid leaved the fluid. Therefore from equations (1) and (2) m1=m2 dA1v1t=dA2v2t Therefore A1v1t=A2v2t Therefore Av=constant

Consider the mass of a fluid entering a at A in `t` seconds M1=density*A1*V1*t Mass of the fluid leaving the tube at B in `t` seconds M2=A2*density*V2*t We know that M1=M2, density*A1*V1*t=density*A2*V2*t A1*V1=A2*V2 Therefore A1*V1=constant