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Tutorial 1 Chapter 1: 1. Starting from 3D Navier-Stokes equation, derive the Euler's equation for 2D flow. State the assumptions made for Euler's equation. 2. In parallel 1D flow in the positive x-direction, the flow velocity varies from 0 m/s at y=0m to 10m/s at y = 5 m. Derive the stream function and potential function in terms of y. 3. The stream function of an inviscid and incompressible flow field is given by: $psi = 2x^2y - frac{2}{3}y^3$ Where $psi$ has the unit of $m^2/s$ and x and y are in meters. Determine the pressure at point x = 4 m, y = 5 m, if the pressure at x = 1 m and y = 1 m is 100kPa. Neglect the change in elevation, and the fluid is water. 4. Flow through a diverging nozzle can be approximated by a 1D velocity distribution u = u(x). Assume the velocity varies linearly from u = V$_o$ at the inlet to u = 0.25V$_o$ at the outlet, compute the deceleration of the fluid particles as a function of x. 5. Show that the continuity equation for steady incompressible flow is $frac{partial u}{partial x} + frac{partial v}{partial y} + frac{partial w}{partial z} = 0$

          Tutorial 1
Chapter 1:
1. Starting from 3D Navier-Stokes equation, derive the Euler's equation for 2D flow. State the
assumptions made for Euler's equation.
2. In parallel 1D flow in the positive x-direction, the flow velocity varies from 0 m/s at y=0m to
10m/s at y = 5 m. Derive the stream function and potential function in terms of y.
3. The stream function of an inviscid and incompressible flow field is given by:
$psi = 2x^2y - frac{2}{3}y^3$
Where $psi$ has the unit of $m^2/s$ and x and y are in meters. Determine the pressure at point x =
4 m, y = 5 m, if the pressure at x = 1 m and y = 1 m is 100kPa. Neglect the change in
elevation, and the fluid is water.
4. Flow through a diverging nozzle can be approximated by a 1D velocity distribution u = u(x).
Assume the velocity varies linearly from u = V$_o$ at the inlet to u = 0.25V$_o$ at the outlet,
compute the deceleration of the fluid particles as a function of x.
5. Show that the continuity equation for steady incompressible flow is
$frac{partial u}{partial x} + frac{partial v}{partial y} + frac{partial w}{partial z} = 0$

$Tutorial 1 Chapter 1: 1. Starting from 3D Navier-Stokes equation, derive the Euler's equation for 2D flow. State the assumptions made for Euler's equation. 2. In parallel 1D flow in the positive x-direction, the flow velocity varies from 0 m/s at y=0m to 10m/s at y = 5 m. Derive the stream function and potential function in terms of y. 3. The stream function of an inviscid and incompressible flow field is given by: psi = 2x^2y - frac23y^3 Where psi has the unit of m^2/s and x and y are in meters. Determine the pressure at point x = 4 m, y = 5 m, if the pressure at x = 1 m and y = 1 m is 100kPa. Neglect the change in elevation, and the fluid is water. 4. Flow through a diverging nozzle can be approximated by a 1D velocity distribution u = u(x). Assume the velocity varies linearly from u = Vo at the inlet to u = 0.25Vo at the outlet, compute the deceleration of the fluid particles as a function of x. 5. Show that the continuity equation for steady incompressible flow is fracpartial upartial x + fracpartial vpartial y + fracpartial wpartial z = 0$

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