Question

Consider a velocity field where the radial and tangential components of velocity are Vr=0 and Vθ=cr, respectively, where c is a constant. Obtain the equations of the streamlines. Is the flow field totationa or irrotational? Prove your answer.

Answer 1

The equations of the streamlines can be obtained by considering the tangential velocity and equating it to a constant. The flow field described by the given velocity is rotational, as it has a tangential component of velocity indicating rotation in the field.

Step-by-step explanation:

To obtain the equations of the streamlines, we need to use the definition of streamlines, which states that the velocity is always tangential to the streamline. In this case, we have the radial component of velocity, Vr, equal to 0 and the tangential component, Vθ, equal to cr. This implies that the velocity is entirely in the tangential direction, and the streamlines will be concentric circles centered at the origin. The equation of a circle can be written as x^2 + y^2 = r^2, where x and y represent the x and y coordinates of a point on the circle, and r is the radius. In this case, the radius is proportional to the tangential velocity, so we can write the equation as x^2 + y^2 = (c*r)^2. This represents the equations of the streamlines.

The flow field described by the given velocity field is rotational. A flow field is considered rotational if it has non-zero vorticity, which measures the local rotation of fluid elements. In this case, the presence of a tangential component of velocity, Vθ, indicates that there is rotation in the flow field. The vorticity can be calculated as the curl of the velocity field, which in this case would be non-zero.