A dual-mode flip-flop has two inputs, M and X. M is the mode input which causes the deviceto operate either as a D flip-flop or T flip-flop. When M is 0, the device behaves like a Dflip-flop with X serving as the D input. When M is 1, the device behaves like a T flip-flopwith X serving as the T input. Design the flip-flop using NAND cell.

a. 0

b. -½(U/b)k

c. (-U/b)k

d. U/b

Explanation:

Given

u = U(y/b)

a.

The rate of volumetric dilation is calculated by using the general formula;

u = U(y/b)

v = 0 and w = 0

The volumetric dilation rate is then given as

∆V = ∂u/∂x + ∂v/∂y + ∂w/∂z

∆V = ∂/∂x(U(y/b)) + ∂/∂y(0) + ∂/∂z(0)

∆V = 0 + 0 + 0

∆V = 0;

The volumetric dilation rate is zero than flow is an incompressible fluid.

b.

Calculating Rotation Vector

The rotation Vector of an element about one axis is the average of the Angular Velocity and the two perpendicular lines.

This is calculated as follows;

W = ½((∂w/∂x - ∂v/∂z)i + (∂u/∂z - ∂w/∂x)j + (∂v/∂x - ∂u/∂y)k)

W = (∂v/∂x - ∂u/∂y)k

Since vector W is not a 0 and everywhere the flow field is not irrational

W ≠ -½(U/b)k

c. The vorticity

The vorticity is the Vector that is twice the rotation Vector.

V = 2W

V = 2( -½(U/b)k)

V = (-U/b)k

d. The rate of angular deformation.

This is defined as

Y = ∂v/∂x + ∂u/∂y where ∂v/∂x = 0

Y = ∂u/∂y

Y = ∂/∂y(U(y/b))

Y = U/b