Question

11) (6 points) Given the velocity field V =101 +(x2 + y2); -2xy [m/s] a) b) c) Is the flow steady or unsteady? Is this motion kinematically possible for an incompressible fluid? Do you think that velocity field can represent a potential flow at specific positions of (x, y)? What is the acceleration of a particle at position (x, y, z) = (3, 1, 0) m? d)

Answer 1

Given V = 10 i + (x² + y² ) j - 2xy Ř (mis) Solutions. PARTla) If flow is Steady then its time derivative must be zero O do dt flow is steady STEADY Flow PART(B) for Incompressible How v. V av du 2x + ay dw az 음을 = 0 +2% - to since 2 to So frow is Not Incompressible.

PARTC Potential flow -> flow Together Incompressible and Irrotational known as Potential How. is Check for Rotation Rotation oxu <.. Ts 5.3 cold - 2x อ x 2 + 2y + 2x ax az V hty) - 2xy Rotation - 2x + 2y ý + 2 x to vector for flow to be Potential flow u x must be Zero x = 0, y = 0 the Jx kenen so At (x,y) = (ogo) flow is Potential.

PART D accelerotion At (X,Y,Z) - (3, 1,0 m an = u au + vdu + w du dx JY dz 22 1o xo + (x+y?) - 2xy (0) a, 11 m/s? n ay - a av + v du + W au ax ay de 1062x) + 2 + y²) (24) - 2xy xo ay=(20x + 2x²y + 2y3) an u do do + + dz axx ay a = 10l-2y) + (x + y 2 (-2x) to 07 1 - 204 223 – 27 ) 11

2. + (23= »m's=3jazza, put n = 3, Y = 1 Z - O 1 ä = oit 80 9 - 80 80k il o i +80) – 8o a) Wie (0) 4 (803 + 1-803 802. m/s² or la I 11 113. 137 m/s²