The given velocity field is V = (3x + 2y)i + (2z + 3x²)j + (2t - 3z)k.
Velocity Components:
The velocity field is given in vector form as (Vx)i + (Vy)j + (Vz)k, where Vx, Vy, and Vz are the velocity components along the x, y, and z axes respectively.
So, from the given velocity field:
Vx = 3x + 2y
Vy = 2z + 3x²
Vz = 2t - 3z
Speed at Point (1, 1, 1):
To find the speed (magnitude of velocity) at a given point, we use the formula:
Speed = √(Vx² + Vy² + Vz²)
At point (1, 1, 1):
Vx = 3(1) + 2(1) = 5
Vy = 2(1) + 3(1²) = 5
Vz = 2(0) - 3(1) = -3
Speed = √(5² + 5² + (-3)²) = √59 ≈ 7.68
Speed at t = 2 seconds, Point (0, 0, 2):
Substituting t = 2 into the velocity field:
Vx = 3(0) + 2(0) = 0
Vy = 2(2) + 3(0²) = 4
Vz = 2(2) - 3(2) = -4
Speed = √(0² + 4² + (-4)²) = √32 ≈ 5.66
Classification of the Velocity Field:
Steady/Unsteady: The velocity field is unsteady because it contains the variable 't', indicating that the velocity changes with time.
Uniform/Non-uniform: The velocity field is non-uniform because the velocity components are functions of position (x, y, z) and time (t), which means the velocity changes both spatially and temporally.
1D/2D/3D: The velocity field is 3-dimensional since it has components along all three spatial dimensions (x, y, z).
The velocity components are Vx = 3x + 2y, Vy = 2z + 3x², and Vz = 2t - 3z. The speed at point (1, 1, 1) is approximately 7.68, and the speed at t = 2 seconds and point (0, 0, 2) is approximately 5.66. The velocity field is unsteady, non-uniform, and 3-dimensional.