Tangential Component of Velocity

The tangential component of velocity in a two-dimensional flow of incompressible fluid is given by vθ = -Csinθ/r^2, where C is a constant. This equation represents the velocity component in the circumferential direction (θ) at any point in the flow field.

Derivation of the Resultant Velocity

To find the magnitude of the resultant velocity, we need to consider both the radial and tangential components of velocity. The radial component of velocity (vr) is given as the derivative of the stream function (Ψ) with respect to the radial distance (r). However, the stream function is not provided in the given equation, so we will solely focus on the tangential component.

Vector Addition of Velocity Components

To find the resultant velocity, we can use vector addition. The magnitude of the resultant velocity can be obtained by calculating the magnitude of the vector sum of the tangential and radial components of velocity. The resultant velocity vector (v) can be expressed as:

v = √((vr)^2 + (vθ)^2)

Since the radial component of velocity is not provided, we will only consider the tangential component for this analysis.

Calculating the Magnitude of Resultant Velocity

To find the magnitude of resultant velocity, we substitute the given equation for the tangential component of velocity into the equation for the resultant velocity:

v = √(0 + (-Csinθ/r^2)^2)
v = √(C^2sin^2θ/r^4)
v = Csinθ/r^2

Therefore, the magnitude of the resultant velocity is given by v = Csinθ/r^2.

Conclusion

In conclusion, the magnitude of the resultant velocity in a two-dimensional flow of incompressible fluid, when only the tangential component of velocity is provided, is given by v = Csinθ/r^2. This equation represents the velocity magnitude at any point in the flow field, considering the tangential direction. It is important to note that the radial component of velocity is not considered in this analysis, as it is not provided in the given equation.