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A steady, two-dimensional velocity field in the xy-plane is given by               V-›= (a + bx)i-›+ (c + dy)j-›+ 0k-›.  (a) What are the primary dimensions (m, L, t, T, . . .) of coefficients a, b, c, and d? (b) What relationship between the coefficients is necessary in order for this flow to be incompressible? (c) What relationship between the coefficients is necessary in order for this flow to be irrotational? (d ) Write the strain rate tensor for this flow. (e) For the simplified case of d = −b, derive an equation for the streamlines of this flow, namely, y = function(x, a, b, c)

(2) Consider the following fluid velocity fields: F(x,y) = (x,y), F(x,y)=(-x, y), F(x,y) = (y, 0). (a) Plot the three fields as glyphs. Which of these vector fields represent an expansion, a compression and a shear flow? (b) Calculate the divergence of the three fields V F. Can you relate the value of the divergence with the nature (compression, expansion or shear of the flow)? (c) Calculate the circulation V x F and relate it with the nature of the flow.

A steady, incompressible, two-dimensional velocity field is given by                 V-›= (u, ? ) = (2.5 − 1.6x) i-›+ (0.7 + 1.6y) j-› where the x- and y-coordinates are in meters and the magnitude of velocity is in m/s. The x-component of the acceleration vector ax is (a) 0.8y (b) −1.6x (c) 2.5x − 1.6 (d ) 2.56x − 4 (e) 2.56x + 0.8y