The Four Roll Mill is a device that is used to generate a wide class of two-dimensional flows with controllable ratios of rate of strain and vorticity. It is pictured in Figure 1 and consists of four identical rollers set in a square. By rotating the relative role of rotation of the rollers different flow fields are set up, Rollers 1 and 3 always rotate with the same angular speed and in the counterclowise direction. Rollers 2 and 4 rotate with the same angular velocity with respect to each other but at different speeds and different directions compared to rollers 1 and 3. A parameter A specifies the 1 angular velocities of rollers 2 and 4 relative to rollers 1 and 3. If X = -1 they rotate with the same angular velocity as rollers 1 and 3. If =0 rollers 2 and 4 are stationary. If x= 1 rollers 2 and 4 rotate in the opposite direction as 1 and 3 but at the same speed. The parameter A can be take on any value between 1 and -1. al b) (c) --1 4 2-0 Figure 1: The Four Roll Mill. (a) = -1: All rollers rotate counterclockwise and with the same ingular speed. (b) = 0; and 4 are stationary but s 1 an rotate with same angular speed and in the counterclockwise direction. (c) = 1: All the rollers have the same angular speed. Rollers 1 and 2 are conterclockwise and rollers 2 and 4 are clockwise. Locally in the center of the four roll mill, the velocity gradient tensor of the flow field set up in a liquid residing in the mill can be approximated to be 1 0 0 dv dx Vy = 0 0 (1) 0 0 0 The parameter 1 is a velocity gradient and is proportional to the angular speed of rollers 1 and 3. Note that when I = -1 the mill produces an antisymmetric, purely rotational flow; when ) = 0 a simple shear flow is produced; when X = 1 a symmetric, purely extensional flow is created. Question 1: Derive an equation for the streamlines. Sketch the streamlines for = 1,0.5, 0, -0.5, -1). Question 2: By considering the time-dependent trajectories, place arrows indicating the direction of fluid motion on your sketches