Ax Consider the steady Cartesian velocity field 12MAB 2MAB0

+Ax Consider the steady Cartesian velocity field 1-(2MAB, 2MAB,0 (x2+y2)2 , (x2+y2) , a) Determine the streamline that passes through x = (x,, yo,0) b) Compute Rij for this velocity field. c) For A> 0, explain the sense of rotation (i.e. clockwise or counter clockwise) for fluid elements for 1.

Solution

let the stream line function be x² +y² = Constant, the stream line function of that passing through (x0,y0) is

x0² + y0² = Constant with U= [ given function but x is changed to x0 and y to Y0}

For the given velocity function ,

Modulus of U = A where A is a constant and positive. The flow velocity is same all over equal to A

x² + y² = xo² + y0² = constant

The stream lines are circles

For x>0 and y>0 ; u_x < 0 and u_y >0 ( A is positive), so that the flow is moving in a counter clockwise direction.

This is for beta =1, for beta <1 or beta>1 the direction do not changeonly the magnitude changes.