Answer:
Step-by-step explanation:
To find the component of the vorticity along the z-axis, we need to calculate the curl of the given velocity field, V = 54x î + 10y ĵ.
The curl of a vector field in three dimensions is given by the cross product of the del operator (∇) and the vector field. The formula for the curl is:
∇ × V = (∂Vᵧ/∂x - ∂Vₓ/∂y) k,
where ∇ × V represents the curl of the vector field V, ∂Vᵧ/∂x and ∂Vₓ/∂y are the partial derivatives of the y and x components of V, respectively, and k is the unit vector along the z-axis.
Let's calculate the partial derivatives:
∂Vₓ/∂y = 0 (since the derivative of 54x with respect to y is zero),
∂Vᵧ/∂x = 0 (since the derivative of 10y with respect to x is zero).
Plugging these values into the formula for the curl, we have:
∇ × V = (0 - 0) k = 0k = 0.
The computed value of the component of the vorticity along the z-axis is zero.
Comparing this result with the given options, we can see that option a. O corresponds to the correct answer since the vorticity component along the z-axis is indeed zero.