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Demostrate that is the case of a steady flow of an ideal fluid turns into Bernoulli equation. |
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Answer» Solution :The Euler's equation is `rho(dvecv)/(dt)=vecf-vecnablap=-vecnabla(p+rhogz)`, where z is vertically upwards. Now `(dvecv)/(dt)=(delvecv)/(delt)+(vecv*vecnabla)vecv`. But `(vecv*vecnabla)vecv=vecnabla(1/2v^2)-vecvxxCurlvecv` (2) we consider the steady (i.e. `delvecv//delt=0`) flow of an incompressible fluid then `rho`= CONSTANT. and as the MOTION is irrotational CURL `vecv=0` So from (1) and (2) `rhovecnabla(1/2v^2)=-vecnabla(p+rhogz)` or, `vecnabla(p+1/2rhov^2+rhogz)=0` Hence `p+1/2rhov^2+rhogz=const ant`. |
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