Answer:
To determine the values of b and c such that the velocity field represents a non-rotational flow, we need to ensure that the curl of the velocity field is zero. The curl of a vector field can be calculated using the cross product of the del operator (∇) with the vector field.
Given the velocity field:
V = (1.35 + 2.78x + 0.754y + 4.21z) + (3.45 + cx - 2.78y + z) + (-4.21x - 1.89y)
Let's calculate the curl of V:
∇ × V = (∂/∂y)(-4.21x - 1.89y) - (∂/∂z)(3.45 + cx - 2.78y + z)
+ (∂/∂z)(1.35 + 2.78x + 0.754y + 4.21z) - (∂/∂x)(-4.21x - 1.89y)
+ (∂/∂x)(3.45 + cx - 2.78y + z) - (∂/∂y)(1.35 + 2.78x + 0.754y + 4.21z)
Simplifying the above expression, we have:
∇ × V = (-∂(4.21x)/∂y - ∂(1.89y)/∂y) - (∂(3.45)/∂z + ∂(cx)/∂z - ∂(2.78y)/∂z + ∂(z)/∂z)
+ (∂(1.35)/∂z + ∂(2.78x)/∂z + ∂(0.754y)/∂z + ∂(4.21z)/∂z) - (-∂(4.21x)/∂x - ∂(1.89y)/∂x)
+ (∂(3.45)/∂x + ∂(cx)/∂x - ∂(2.78y)/∂x + ∂(z)/∂x) - (∂(1.35)/∂y - ∂(2.78x)/∂y - ∂(0.754y)/∂y - ∂(4.21z)/∂y)
Now, we can compute each partial derivative term:
∂(4.21x)/∂y = 0
∂(1.89y)/∂y = 1.89
∂(3.45)/∂z = 0
∂(cx)/∂z = 0
∂(2.78y)/∂z = 0
∂(z)/∂z = 1
∂(1.35)/∂z = 0
∂(2.78x)/∂z = 0
∂(0.754y)/∂z = 0.754
∂(4.21z)/∂z = 4.21
∂(4.21x)/∂x = 4.21
∂(1.89y)/∂x = 0
∂(3.45)/∂x = 0
∂(cx)/∂x = c
∂(2.78y)/∂x = 2.78
∂(z)/∂x = 0