Answer:
a. To determine if the flow field is steady or unsteady, we need to check if the velocity field changes with time (t). In this case, V=[4tx]i-[2t^2y]j-[4xz]k, we can see that the velocity field does depend on time since it includes the variable t. Therefore, the flow field is unsteady.
b. To determine if the flow field is two or three-dimensional, we need to check the number of spatial variables (x, y, and z) involved. In this case, the velocity field has all three spatial variables (x, y, and z) present, so the flow field is three-dimensional.
c. To find the total acceleration vector at the point (x, y, z)=(-1, +1, 0), we need to differentiate the velocity field with respect to time (t) and evaluate it at the given point.
Given: V=[4tx]i-[2t^2y]j-[4xz]k
The velocity vector is V = (4tx)i - (2t^2y)j - (4xz)k
Differentiating with respect to time (t), we have:
a = dV/dt = 4x i + (-4ty) j - 4z k
Now, plugging in the given coordinates (x, y, z) = (-1, +1, 0), we get:
a = 4(-1) i + (-4t(1)) j - 4(0) k
= -4i - 4tj
The total acceleration vector at the point (-1, +1, 0) is A = -4i - 4tj.
To find the magnitude of the acceleration vector, we use the formula:
|A| = sqrt((-4)^2 + (-4t)^2)
= sqrt(16 + 16t^2)
= sqrt(16(1 + t^2))
Therefore, the magnitude of the acceleration vector at the point (-1, +1, 0) is |A| = sqrt(16(1 + t^2)).