Prove that the velocity potential and the stream function for a uniform flow, Equations
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- 1. Consider a three-dimensional steady incompressible flow with components: u =-x(y+z), v= y, w=÷(z²-2yz), (a) Is the flow locally rotating? Justify your answer. (b) Determine the equation of vortex lines. (c) Are the vortex lines perpendicular to the streamlines?arrow_forwardThe stream function of a flow field is y = Ax3 – Bxy², where A = 1 m1s1 and B = 3 m-1s1. (a) Derive the velocity vector (b) Prove that the flow is irrotational (c) Derive the velocity potentialarrow_forwardConsider a uniform stream of magnitude V inclined at angle ?. Assuming incompressible planar irrotational flow, find the velocity potential function and the stream function. Show all your work.arrow_forward
- An incompressible stream function is given byψ = aθ +br sin θ. ( a ) Does this flow have a velocitypotential? ( b ) If so, find it.arrow_forwardOne of the corner flow patterns of Fig. 8.18 is given by thecartesian stream function ψ = A(3yx2 - y3). Which one?Can the correspondence be proved ?arrow_forwardFor the flow field described by u = 2, v = yz2t, w = −z3t/3. (a) Is this flow one-, two-, or three-dimensional? (b) Is this flow steady? (c) Is this flow incompressible? (d) Find the z-component of the acceleration vector.arrow_forward
- 6. Find the stream function associated with the two-dimensional incompressible, possible flow with velocity components given by a2 Ur = b|1- cos e and ug = -b1+- sin 0 r2 Where a, and b are known constants.arrow_forward(b) Two velocity components of a steady, incompressible flow field are given as follows; u = 2ax + bxy + cy? v = axz – byz? where a, b and c are constants. Determine an expression for w as a function of x, y, and z.arrow_forwardA fluid flows along a flat surface parallel to the x-direction. The velocity u varies linearly with y, the distance from the wall, so that u = ky. (a) Find the stream function for this flow. (b) Is this flow irrotational?arrow_forward
- Example 6.5. For a three-dimensional flow field described by V = (y? + 22) i + (x² +z?) j + (x²+y²) k find at (1, 2, 3) (i) the components of acceleration, (ii) the components of rotation.arrow_forward. Find the stream function associated with the two-dimensional incompressible, possible flow with velocity components given by a2 cos 0 and u, = -b 1+ --(1+)sino a2 b1 r2 r2 Where a, and b are known constants.arrow_forwardQ6 In an incompressible flow, we know that u and v are both nonzero but constant in magnitude. What can we infer about w from the differential continuity equation? About the density?arrow_forward
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