(Problem 2.4) Consider the following velocity field: Uo(x²-y²+x)ex - U₁(2xy + y)ey (a) Compute the acceleration at y = 1 and x = 0 and x = 2. U₁ = 2 m s¹. (b) Compute the volumetric flow rate through the plane at x = 5 and extending from y = 0 to y= 5 and width 3 in the z direction.

Velocity field of an incompressible flow is given by V = 6xi − 6yj (m/s) a) Find the pathlines in x-y plane. Make a sketch of pathlines for x ≥ 0 and y ≥ 0. b) Find the streamlines. Make a sketch of streamlines for x ≥ 0 and y ≥ 0. c) At time t = 0 s, the position of a rectangular fluid element ABCD is described by the corner points A(1,3), B(2,3), C(1,2) and D(2,2). Determine the new position of the fluid element at time t = 1/6 s

a. Derive an equation for the material acceleration vector.b. Obtain the vorticity vector for the velocity field.c. Is the flow rotational or irrotational? Show through your derivation.d. Is the flow incompressible or compressible? Show through your derivation.

two-dimensional velocity field u =xt + 2y and v =xt^2- yt x=1 meter y= 1 meter and t= 1 second Find the acceleration where it is.?

3.4 Consider a steady, incompressible, 2D velocity field for motion parallel to the X-axis with constant shear. The shear rate is du/dy Ay. Obtain an expression for the velocity field V. Calculate the rate of rotation. Evaluate the stream function %3D for this flow field. Ay Ay + В і, о, Ay + By+ C 6. Ans: V= 2

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Converging duct flow is modeled by the steady, two- dimensional velocity field V = (u, v) = (U₁ + bx) i-by. For the case in which Ug = 3.56 ft/s and b = 7.66 s¯¹, plot several streamlines from x = 0 ft to 5 ft and y=-2 ft to 2 ft. Be sure to show the direction of the streamlines. (Please upload you response/solution using the controls provided below.)

Consider the velocity field represented by V = K (yĩ + xk) Rotation about z-axis is

Problem 1 Given a steady flow, where the velocity is described by: u = 3 cos(x) + 2ry v = 3 sin(y) + 2?y !! !! a) Find the stream function if it exists. b) Find the potential function if it exists. c) For a square with opposite diagonal corners at (0,0) and (47, 27), evaluate the circu- lation I = - f V.ds where c is a closed path around the square. d) Calculate the substantial derivative of velocity at the center of the same box.