a) Given the stream function = x² + y³, provide an equation for the streamline that passes through the point (x, y) = (1,2). What is the speed and direction of the fluid at this point? Is the speed zero at any point on this streamline? b) A velocity field has the following form in cylindrical coordinates: v=Kr. Compute the angular velocity, the vorticity vector, and the rate of strain tensor for this flow. For what value of n is the flow irrotational? For what value of n is the strain identically zero? c) For 0≤ y ≤8, the velocity of a fluid near a horizontal wall has the form У uuo 2 v = 0 Determine the vorticity vector, the rate of strain tensor, and the stream function. Show that the velocity potential does not exist. For an initially-square fluid element at y = 8/2, sketch its shape a short time later.

1. Stagnation Points A steady incompressible three dimensional velocity field is given by: V = (2 – 3x + x²) î + (y² – 8y + 5)j + (5z² + 20z + 32)k Where the x-, y- and z- coordinates are in [m] and the magnitude of velocity is in [m/s]. a) Determine coordinates of possible stagnation points in the flow. b) Specify a region in the velocity flied containing at least one stagnation point. c) Find the magnitude and direction of the local velocity field at 4- different points that located at equal- distance from your specified stagnation point.

How would I calculate the fluid acceleration along the nozzle centerline. Here, there is steady flow of water through an axisymmetric garden hose nozzle and alongthe centerline the water speed increases from uentrance to uexit . The centerline water speed increases parabolically through the nozzle. What would be an equation for centerline speed u(x), based on the parameters given in the drawing from x = 0 to  x = L ?

THREE DIMENSIONAL ( NEED NEAT HANDWRITTEN SOLUTION ONLY OTHERWISE DOWNVOTE).