Question

A fluid motion is given by V =(y +z) i + (z + x) j +(x+ y) k. Show that themotion is irrotational and hence find the velocity potential also show that motion inpossible for an incompressible fluid.

Answer 1

For steady incompressible flow, the co ntinuity equation is

[$\partial u / \partial x + \partial v/\partial y+\partial w/\partial z$ ]=0

u=y+z , v=z+x, w=x+y

\partial(y+z)/\partial x+\partial(z+x)/\partial y+\partial(x+y)/\partial z=0\\ [0+0+0]=0\\ Therefore,the given flow field is a possible case of steady incompressible fluid flow\\ --show that the motion is irrotational

V=(y+z)i + (z+x)j +(x+y)k

V is irrotational if curl V=0

$I$

curl V=$\nabla×V$

$\begin{vmatrix} i &&& j&&& k\\hi \partial/ \partial x &&& \partial/\partial y&&& \partial/ \partial z \\ (y+z)&&&(z+x)&&&(x+y) \end{vmatrix}\\ i[\partial/\partial y(x+y)-\partial/\partial z (z+x)]-j[\partial/\partial x(x+y)-\partial/\partial z(y+z)]+k[\partial/\partial x(z+x)-\partial/ \partial y(y+z)]\\ =0\\ Curl V=\nabla×V=0\\ V \> is \>irrotational\\ 3) Velocity \> potential=\phi =? \\ V=\nabla\phi------(1)\\ d\phi={(\partial\phi/\partial x)dx + (\partial\phi/\partial y)dy + (\partial\phi/\partial z)dz}.(idx +jdy+kdz)\\ d\phi=(i\partial /\partial x)+(j\partial/\partial y)+(k\partial/\partial z)\phi.dr \\ d\phi = \nabla \phi.dr------(2) \\ dr=idx+jdy+kdz \\ V=\nabla\phi \> put \> in \> eqn(2) \\ d\phi = V.dr \> equation (3)\\ V=(y+z)i+(z+x)j+(x+y)k \\ put\>in \> eqn(3) \\ d\phi={(y+z)i +(z+x)j + (x+y)k}.dr\\ d\phi={(y+z)i +(z+x)j + (x+y)k}.(idx+jdy+dz) \\ d\phi=(y+z)dx + (z+x)dy + (x+y)dz \>(4)\\ d\phi = ydx+zdx+zdy+xdy+xdz+ydz\\ d\phi=(xdy+ydx)+(xdz+zdx)+(ydz+zdy)\\ Product \> of \> two \> function\\ d\phi=dxy+dxz+dyz\\ On \> integration \\ \phi=xy+xz+yz+ C \\ Where\>C\> is \> integration \> constant\\ \phi \> is\> the\>velocity\>potential.$