Fluid Kinematics
In a Nutshell: The velocity
field of a fluid element is an important topic in fluid mechanics. As
in calculus, the velocity field, V
, (also called velocity vector) of a fluid element depends on
location and time, V
= V(x,y,z,t). For general three-dimensional fluid motion: V = u(x,y,z,t) i + v(x,y,z,t) j + w(x,y,z,t) k. where V is the velocity vector u(x,y,z,t) is the x-component of fluid velocity v(x,y,z,t) is the y-component of fluid velocity w(x,y,z,t) is the z-component of fluid velocity i, j, k are the unit vectors in the x, y, and z
directions respectively The
derivative of fluid velocity (called the material derivative) is: (use the chain rule) a = dV/dt =
∂V/∂x /dx/∂t +
∂V/∂y /dy/∂t +
∂V/∂z /dz/∂t +
∂V/∂t a =
u ∂V/∂x + v
∂V/∂y + w
∂V/∂z + ∂V/∂t or
in component form: ax = u
∂u/∂x + v ∂u/∂y + w
∂u/∂z + ∂u/∂t ay = u
∂v/∂x + v ∂v/∂y + w
∂v/∂z + ∂v/∂t az = u
∂w/∂x + v ∂w/∂y + w
∂w/∂z + ∂w/∂t If
the flow is independent of time, t, then
∂u/∂t, ∂v/∂t, and ∂w/∂t are all
zero and the flow
is said to be steady. Click
here to continue discussion. |
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