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Fluid Kinematics

Key Concepts: The important kinematical variables in fluid mechanics include fluid velocity,

fluid acceleration, the rotation vector, and the vorticity vector. Each of these variables may depend

on location and time (x,y,z,t) for 3˗D flow or (x,y,t) for 2˗D flow within the velocity field. If the

flow is independent of time, t, then the flow is said to be steady.

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: a_x = u ∂u/∂x + v ∂u/∂y + w ∂u/∂z + ∂u/∂t

a_y = u ∂v/∂x + v ∂v/∂y + w ∂v/∂z + ∂v/∂t

a_z = 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.