A steady, two-dimensional velocity field is given by V = (u, v) = (1.35 + 2.78x - 0.896y)i + (3.45x + cx - 2.78y)j
Calculate constant c such that the flow field is irrotational.
To determine if a velocity field is irrotational, we need to calculate its curl (or rotational). In two dimensions, the curl of a vector field V = (u, v) is given by:
curl(V) = ∂v/∂x - ∂u/∂y
In this case, we are given that the velocity field is steady and two-dimensional, given by V = (1.35 + 2.78x - 0.896y)i + (3.45x + cx - 2.78y)j.
Calculating the partial derivatives, we have:
∂v/∂x = c
∂u/∂y = -0.896
Substituting these values into the curl equation, we get:
curl(V) = c - (-0.896) = c + 0.896
For the velocity field to be irrotational, the curl must be equal to zero. Therefore, we set c + 0.896 = 0 and solve for c:
c + 0.896 = 0
c = -0.896
Hence, the constant c that makes the flow field irrotational is c = -0.896.