The velocity components u and v in a two-dimensional flow field are given by: u = 4yt ft./s, v = 4xt ft./s, where t is time. What is the time rate of change of the velocity vector V (i. e., the acceleration vector) for a fluid particle at x = 1 ft. and y = 1 ft. at time t = 1 second?

vec (a) = 16 i + 16 j

mag (a) = 22.63 ft/s^2

Explanation:

Given,

- The two components of velocity are given for fluid flow:

u = 4*y ft/s

v = 4*x ft/s

Find:

What is the time rate of change of the velocity vector V (i. e., the acceleration vector) for a fluid particle at x = 1 ft. and y = 1 ft. at time t = 1 second?

Solution:

- The rate of change of velocity is given to be acceleration. We will take derivative of each components of velocity with respect to time t:

a_x = du / dt

a_x = 4*dy/dt

a_y = dv/dt

a_y = 4*dx/dt

- The expressions dx/dt is the velocity component u and dy/dt is the velocity component v:

a_x = 4 * (4*y) = 16y

a_y = 4 * (4*x) = 16x

- The acceleration vector can be expressed by:

vec (a) = 16y i + 16x j

- Evaluate vector (a) at x = 1 and y = 1:

vec (a) = 16*1 i + 16*1 j = 16 i + 16 j

- The magnitude of acceleration is given by:

mag (a) = sqrt (a^2_x + a^2_y)

mag (a) = sqrt (16^2 + 16^2)

mag (a) = 22.63 ft/s^2