Answer and Explanation:
To determine the values of b and c that make the velocity field a non-rotational flow, we need to find the conditions under which the curl of the velocity field is equal to zero.
The curl of a vector field can be calculated using the following formula:
curl(F) = (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k
In this case, we have the velocity field F = (1.35 + 2.78x + 0.754y + 4.21z) + (3.45 + cx - 2.78y + z) + (-4.21x - 1.89y).
Let's calculate the curl of this velocity field:
∂F₁/∂x = 2.78
∂F₁/∂y = 0.754 - 2.78c
∂F₁/∂z = 4.21
∂F₂/∂x = -4.21
∂F₂/∂y = -2.78 - 1.89c
∂F₂/∂z = 1
∂F₃/∂x = 0
∂F₃/∂y = -3.45
∂F₃/∂z = 1
Using these partial derivatives, we can calculate the curl of the velocity field:
curl(F) = (0 - (1))i + ((-4.21) - (-3.45))j + ((-2.78 - 1.89c) - 0)k
= -i - 0.76j - (2.78 + 1.89c)k
For the velocity field to be non-rotational, the curl must be equal to zero. Thus, we have the following conditions:
-1 = 0 (from the i-component)
-0.76 = 0 (from the j-component)
2.78 + 1.89c = 0 (from the k-component)
From these conditions, we can see that there is no value of c that satisfies the equation 2.78 + 1.89c = 0. Therefore, there are no values of b and c that make the velocity field a non-rotational flow.