流体动力学代做 MT4509代写 流动运动学代写 数学专业代写

MT4509 — Fluid Dynamics

Example sheet 1

Flow kinematics I

流体动力学代做 1. In a two dimensional flow, the velocity is given by (u, v) = (x + t, −y + t). Find: (i) the general equation of the streamlines

1. 流体动力学代做

In a two dimensional flow, the velocity is given by (u, v) = (x + t, −y + t). Find:

(i) the general equation of the streamlines

(ii) the streamline at t = 0 passing through (1, 1) and sketch it

(iii) the time T (> 0) at which the streamline through (1, 1) also passes through (0, 2)

(iv) the general equation of the particle paths

(v) the (x, y) equation of the particle path originating at (0, 0) at t = 0.

2. 流体动力学代做

In a two-dimensional flow, the velocity is given by (u, v) = (x/(1 + t), 1) for t > −1. Find:

(i) the particle path originating at (X, Y ) at t = 0

(ii) the streamline at time t through a general point (X₀, Y₀)

(iii) the (x, y) equation (at time t) of the streakline formed by all fluid particles which at some previous time have passed through the points (X′ , Y′).

3.

Find the streamlines of the uniform irrotational strain flow u = (αx, βy, γz), with α+β +γ = 0.

4. (May 2004) 流体动力学代做

(a) State the continuity equation and Euler’s equation and state the boundary conditions for these equations at a stationary impermeable surface.

(b) Define the terms streamline at an instant in time, pathline and streakline.

(c) Consider the two-dimensional flow, expressed in rectangular Cartesian coordinates, with the velocity u = (α²y, −β²x) where α and β are two constants.

(i) Determine the time dependent position of a fluid particle initially at x = x₀ = (x₀, y₀)

(ii) Determine the equation for the streamlines in this flow

(iii) Sketch an example of a streamline, a pathline and a streakline for this flow.

5.

Repeat 4(c) for the two-dimensional flow u = (2tαx, −2tαy) where α is a positive constant and t is time.

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