Kinematic Boundary Condition Fluids

[member 428835]

Hi PF!

Given a fluid/fluid interface $F(t,x,y,z) = 0$ the kinematic boundary condition states $DF/Dt = 0$. Given $y=F$ the text states $v = \partial_tF+u\partial_xF+w\partial_zF$. How is this possible? I thought $DF/Dt = 0 = \partial_tF+u\partial_xF+v\partial_yF+w\partial_zF$ and $F=y \implies \partial_yF = 1$, thus implying $-v = \partial_tF+u\partial_xF+w\partial_zF$. What happened to the negative sign?

 

[eljose79]

Hello,

Thank you for your question. The kinematic boundary condition, $DF/Dt = 0$, is a statement of mass conservation at the fluid interface. This means that the rate of change of the fluid interface with respect to time is equal to the sum of the velocities in the x, y, and z directions (i.e. $\partial_tF+u\partial_xF+v\partial_yF+w\partial_zF = 0$).

In the given text, the velocity, $v$, is defined as the velocity of the fluid interface in the y direction. This is why there is no negative sign in the equation $v = \partial_tF+u\partial_xF+w\partial_zF$. The negative sign is only present in the equation when the velocity is defined as the rate of change of the fluid interface with respect to time (i.e. $-v = \partial_tF+u\partial_xF+w\partial_zF$). When using the kinematic boundary condition, the negative sign is not needed because the equation is already stating that the rate of change is equal to zero.

I hope this helps clarify the confusion. Let me know if you have any further questions. Best of luck in your studies!