(Solved): fluid flows question for aerothermodynamics Question 1 a) In a steady incompressible two-dimensional ...

Question 1 a) In a steady incompressible two-dimensional flow the velocity vector in \( \mathrm{m} / \mathrm{s} \) is given as: \( \vec{V}=(6 x-6 y) \vec{i}+(5 x-6 y) \vec{j} \). With the aid of sketches give a clear account of the distortion of a rectangular fluid element (Figure Q1-1). Figure Q1-1 With clear explanation, determine whether this flow is irrotational. [7.5 marks] b) Give a brief definition of the boundary layer shape factor including a clear explanation of how the magnitude of the boundary layer shape factor is related to the boundary layer profile. Air is moving along one side of a flat plate at a constant speed of \( 40 \mathrm{~m} / \mathrm{s} \). The flat plate is \( 3 \mathrm{~m} \) long and its width is \( 1 \mathrm{~m} \). Measurement shows the momentum thicknesses of the laminar boundary layer at \( \mathrm{x}_{1}=1.0 \mathrm{~m} \) and \( \mathrm{x}_{2}=4.0 \) \( \mathrm{m} \) are about \( 0.401 \) and \( 0.802 \mathrm{~mm} \) respectively. Estimate the frictional drag acting on the plate between \( x_{1} \) and \( x_{2} \). [7.5 marks] c) Two flat plates (A and B) are orientated to a uniform flow as indicated in Figure Q1-2. If the boundary layer flow on these plates is laminar, determine the ratio of the total friction drag for plate \( A \) to that for plate \( B \quad\left(i . e\right. \). Drag Ratio \( =\frac{D^{D}}{\text { Drag}_{A}} \) ). Explain your answer. The wall shear stress acting on the plates has been given as \( \tau_{W}=K x^{-0.5} \) where \( \mathrm{k} \) is a known constant. Ignore any three dimensionality effects in the flow. Question 1 continued Figure Q1-2