Question

The potential function for a flow is ϕ=(x²−y²)ft²/s, where x and y are in feet. Determine the magnitude of the velocity of a fluid particle at point A(3ft,1ft). Show that continuity is satisfied, and find the streamline that passes through point A.

Answer 1

To determine the magnitude of the velocity at point A, differentiate the potential function with respect to x and y, and use the formula v = sqrt(vx^2 + vy^2). Verify continuity by checking if the mass flow rate is conserved. To find the streamline passing through point A, solve the equation for the potential function.

Step-by-step explanation:

The magnitude of the velocity of a fluid particle at point A can be determined using the potential function. First, differentiate the potential function with respect to x and y to obtain the x and y components of the velocity respectively. The magnitude of the velocity can then be calculated using the formula v = sqrt(vx^2 + vy^2).

Continuity is satisfied when the mass flow rate is constant. In this case, the mass flow rate is given by Q = A * v, where A is the cross-sectional area and v is the velocity. By finding the cross-sectional area at point A and substituting the given values, you can verify that the mass flow rate is conserved.

To find the streamline that passes through point A, you can solve the equation for the potential function by setting it to a constant value, which is determined by the potential function at point A. Substituting the given values, you can find the equation of the streamline.