Viscous Flows, Homework Sets 1 Due on Oct. 18, 2018 Q1: Prove the following vector calculus identities using the index notation (i) ∇ × ( ∇ φ ) = 0, where φ is a scalar function. (ii) ∇ · ( ∇ × u ) = 0, where u is a vector function. (iii) u × ( ∇ × u ) = 1 2 ∇ ( u · u ) - ( u · ∇ ) u , where u is a vector function. (iv) ∇ × ( ∇ × u ) = ∇ ( ∇ · u ) - ∇ 2 u , where u is a vector function and ∇ 2 is the Laplacian operator. Q2: A steady, two-dimensional velocity field is given by u ( x, y ) =( ay ) ˆ i + ( bx ) ˆ j , where a and b are positive constants. ˆ i and ˆ j are unit vectors in the x and y direction, respectively. (a) Find the angular velocity of the fluid elements in this flow field. (b) Compute the two-dimensional strain rate tensor ˙ of this flow field. (c) Find the principal strain rates and the directions of principal axes of the strain rate tensor. (d) Compute the total derivate of the velocity field in a frame of reference moving with a velocity, V = c ˆ j , where c is a positive constant.