(x0 x = + 1)e(t-t0) - t 1 – (x0 x = + t0 + 1)e(t-t0) - t 1 – (x0 x = + t0 + 1)e(t-t0) - t SOLUTION (c) – GIVEN One-Dimensional flow with field defined by u = x + t FIND Equation for path followed by particle. x SOLUTION Since u = dx/dt Flow equation, can be written as u = x + t dx/dt = x + t. Solving the above first order ordinary differential equation. dx/dt - x = t where operator (D 1)x = t, D = d/dt – Solution is x = Aet - t 1 – The constant A is found from initial conditions, x0 = Aet0 - t0 1, – Hence (x0 A = + t0 + 1)/ et0 Substituting for gives, A The Lagrangian representation of fluid particle having the identity (x0 ,t0) (x0 x = + t0 + 1)e(t-t0) - t 1. – Illustrates Flow field description using Eulerian and Lagrangian approach Q10) The particle positions in a two-dimensional flow is given as below and x = x0 e-kt + y0 (1-e-2kt) y = y0 ekt Find the equation of the trajectory of a fluid particle in the flow field. Find the flow velocity components in the Eulerian system. (i) x = x0 (y0/ y) + y0 (1-(y0/ y)2)
x = (x0 + 1)e(t-t0) - t – 1 x = (x0 + t0 + 1)e(t-t0) - t – 1 x = (x0 + t0 + 1)e(t-t0) - t SOLUTION – (c) GIVEN One-Dimensional flow with field defined by u = x + t FIND Equation for path x followed by particle. SOLUTION Since u = dx/dt Flow equation, u = x + t can be written as dx/dt = x + t. Solving the above first order ordinary differential equation. dx/dt - x = t (D – 1)x = t, where operator D = d/dt Solution is x = Aet - t – 1 The constant A is found from initial conditions, x0 = Aet0 - t0 – 1, Hence A = (x0 + t0 + 1)/ et0 Substituting for A gives, The Lagrangian representation of fluid particle having the identity (x0 ,t0) x = (x0 + t0 + 1)e(t-t0) - t – 1. Illustrates Flow field description using Eulerian and Lagrangian approach Q10) The particle positions in a two-dimensional flow is given as below x = x0 e-kt + y0 (1-e-2kt) and y = y0 ekt Find the equation of the trajectory of a fluid particle in the flow field. Find the flow velocity components in the Eulerian system. (i) x = x0 (y0/ y) + y0 (1-(y0/ y)2)