Consider an incompressible 2D flow with stream function Y = xy + y³. Determine (a) the velocity at point (3,5). (b) the flow rate per depth for a curved segment y + x² = 4 in the second quadrant.

Parallel laminar Flow Between Infinite Planes with a Manometer A Newtonian fluid with viscosity μf and density pf is contained between two infinite horizontal parallel planes (separated by a distance d as shown below). The fluid flows laminarly under the action of a pressure gradient and the velocity U of the upper plane (the bottom plane is fixed). A manometer (with fluid density pm and viscosity μm) is connected between two points L apart along the bottom plane and indicates a differential reading of Ah. Using the coordinate system depicted in the figure below, calculate the velocity distribution in terms of known variables by performing a shell balance. What value must U have so that the frictional drag force on the upper plate is zero? Sketch the velocity profile for this last case (i.e., zero frictional drag force on the upper plate). d Pf ༑ Ah Pm L X

In planar flow of an incompressible fluid, the x component of the velocity is given by u = ½ x² + 2x – 4y. Use the equation of continuity to derive the expression for v. %3D

1. Water (p= 1000 kg m-³ and μ = 0.0011 kg m-¹ s-1) flows at 5 m s-1 over a 0.3 m long flat plat. Determine if the flow in the boundary layer is laminar over the entire length of the plat or it becomes turbulent at a downstream location. If latter is the case, determine the location of transition from laminar to turbulent. [Ans. BL flow is Laminar & turbulent, x = 0.11 m]