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Mathematics assignment I Fluid Mechanics Math

University of Jazan College of Science Department of Mathematics Surname: Ju.Id: Section: G - MATH 2022 - Fall HOME WORK- I Fluid Mcchanics - Math - 464 MaximumValues = 5 Marks Dr. Salahuddin Mohammad Hanif Office No. 1505 Answer all Questions. Question - 1: If the velocity distribution over a place is given by 𝟐 𝒖 = 𝒚 − 𝒚𝟐 . 𝟑 In which u is the velocity in meters per second at a distance 𝒚 meter above the plane. Determine the shear stress at 𝒚 = 𝟎 and 𝒚 = 𝟎. 𝟏𝟓𝒎. Solution:- Question - 2: A Velocity field is given by ̂ . ⃗ = 𝟓𝒙𝟐 𝒊̂ − 𝟏𝟎𝒙𝒚𝒋̂ + 𝟐𝟎𝒕𝒌 𝑽 Find the velocity and acceleration at point (𝟏, 𝟐, 𝟑) at time 𝒕 = 𝟎. 𝟏 second. Solution:- Question - 3: Show that the velocity field ⃗𝑽 = 𝒙𝒚𝒛𝟐 𝒊̂ − 𝒚𝟐 𝒛𝟐 𝒛𝟑 ̂ 𝒛𝒋̂ + 𝒚( − )𝒌 𝟐 𝟐 𝟑 satisfies the continuity equation for an incompressible flow and finds the velocity at the point (𝟏, 𝟏, 𝟏). Solution:- Question - 4: For the velocity field ̂ ⃗𝑽 = 𝟐𝒙𝒚𝒊̂ + 𝟒𝒕𝒛𝟐 𝒋̂ − 𝒚𝒛𝒌. Find the acceleration, the angular velocity about the 𝒛-axis and the vorticity vector at the point (𝟐, −𝟏, 𝟏) at 𝒕 = 𝟐. Solution:- Question - 5: A velocity field in a plane flow is given by ⃗𝑽 = 𝟐𝒚𝒕𝒊̂ + 𝒙𝒋. ̂ Find the equation of the streamline passing through (𝟒, 𝟐) at 𝒕 = 𝟐. Solution:- Question - 6: Consider the steady, two-dimensional flow field ⃗ = 𝑽 𝑽𝟎 (−𝒙𝒊̂ + 𝒚𝒋̂) 𝓵 where 𝑽𝟎 and 𝓵 a constants. Determine the acceleration field for this flow. Solution:- Question - 6: The velocity components in a steady, two-dimensional incompressible flow field are 𝒖 = 𝟐𝒚 𝒗 = 𝟒𝒙 a) Determine the corresponding stream function and b) Show on a sketch several streamlines. Indicate the direction of flow along the streamlines Solution:- Question - 7: The 2D, steady, incompressible velocity field ⃗𝑽 = (𝒖, 𝒗) = (𝟎. 𝟓 + 𝟎. 𝟖𝒙)𝒊̂ + (𝟏. 𝟓 − 𝟎. 𝟖𝒚)𝒋̂. Given that the stagnation point at (−𝟎. 𝟔𝟐𝟓, 𝟏. 𝟖𝟕𝟓), and units take place in m, s and m/s. Determine a) Rate of translation. b) Rate of rotation. c) Linear strain rate. d) Shear strain rate. e) Volumetric strain rate. f) Verify the flow is incompressible. Solution:- Question - 8: Solution:- Question - 9: Solution:- Question - 10: Solution:- Question - 11: Solution:- Question - 12: