Equation of Continuity
We will cover following topics
PYQs
Equation Of Continuity
1) For an incompressible fluid flow, two components of velocity (u,v,w) are given by u=x2+2y2+3z2, v=x2y−y2z+zx. Determine the third component w so that they satisfy the equation of continuity. Also, find the z−component of acceleration.
[2018, 10M]
Using the continuity equation in cartesian coordinates:
∂u∂x+∂v∂y+∂w∂z=0
⟹2x+x2−2yz+∂w∂z=0
⟹∂w∂z=2yz−2x−x2
Integrating wrt z, we get:
w=yz2−2xz−x2z+f(x,y)
⟹ The z component of acceleration:
az=(q⋅∇)w+∂ω∂t=u∂w∂x+v∂w∂y+w∂ω∂z
Therefore,
az=(x2+2y2+3z2)(∂f∂x−2z−2xz)+(x2y−y2z+xz)(∂f∂y+z2)+(yz2−2xz−x2z+f(x,y))(2yz−2z−x2)2) Show that (x2a2)cos2t+(y2b2)sec2t=1 is a possible form for the boundary surface of a liquid.
[2007, 12M]
3) Show that: u=−2xyz(x2+y2)2, v=(x2−y2)z(x2+y2)2, w=yx2+y2 are the velocity components of a possible liquid motion. Is this motion irrotational?
[2002, 15M]
Here, we have:
∂u∂x=−2yz1⋅(x2+y2)2−x⋅2(x2+y2)⋅2x(x2+y2)4=−2yzx2+y2−4x2(x2+y2)3=−2yzy2−3x2(x2+y2)3 ∂v∂y=z−2y(x2+y2)2−2(x2+y2)⋅2y(x2−y2)(x2+y2)4=−2yzx2+y2+2(x2−y2)(x2+y2)3=−2yz3x2−y2(x2+y2)3and
∂w/∂z=0Hence, the equation of continuity ∂u/∂x+∂v/∂y+∂w/∂z=0 is satisfied and so the liquid motion is possible.
Furthermore, we have
Ωx=∂w∂y−∂v∂z=x2−y2(x2+y2)2−x2−y2(x2+y2)2=0Ωy=∂u∂z−∂w∂x=−2xy(x2+y2)2+2xy(x2+y2)2=0Ωz=∂v∂x−∂u∂y=2xz(3y2−x2)(x2+y2)3−2xz(3y2−x2)(x2+y2)3=0 ∴∂w/∂y=∂v/∂z,∂u/∂z=∂w/∂x,∂v/∂x=∂u/∂yand hence the motion is irrotational.