Question

A velocity field is given by u = Ax, v = -Ay; the units of velocity are m/s; x and y are given in meters; A = 0.3 s-¹. a) Obtain an equation for the streamlines in the xy plane. b) Plot the streamline passing through the point (xo, yo) = (2,8). c) Determine the velocity of a fluid particle at the point (2,8). d) If the fluid particle passing through the point (xo, yo) is marked at time t = 0, determine the location of the particle at time t = 6 s. e) What is the velocity of this fluid particle at time t = 6 s? f) Show that the equation of the fluid particle path (i.e., the pathline) is the same as the equation of the streamline.​

Answer 1

Step-by-step explanation:

(A)The equation for the streamlines can be obtained by equating the differential element of the stream function to zero:

ψ = constant

dψ = 0

We know that u = ∂ψ/∂y and v = -∂ψ/∂x, so

dψ = udy - vdx

0 = Ady(x-x0) - Adx(y-y0)

0 = Ady x - Adx y - Ady x0 + Adx y0

0 = x(A dy) - y(A dx) + (dx y0 - dy x0)A

Thus, the equation for the streamlines is:

ˣ²-ʸ² = C

where C is a constant.

(B) To plot the streamline passing through point (2,8), we need to substitute these values into the equation for the streamline:

2² - 8² = C

C = -60

So the equation for the streamline passing through the point (2,8) is:

ˣ²-ʸ² = -60

(C) To determine the velocity of a fluid particle at the point (2,8), we substitute x = 2 and y = 8 into the given velocity field:

u = Ax = (0.3 ˢ⁻¹)(2 m) = 0.6 m/s

v = -Ay = -(0.3 ˢ⁻¹)(8 m) = -2.4 m/s

Therefore, the velocity of the fluid particle at points (2,8) is (0.6 m/s, -2.4 m/s).

(D) If the fluid particle passing through the point (2,8) is marked at time t = 0, we can determine its location at time t = 6 s by using the following equations for the path of the fluid particle:

dx/dt = A x

dy/dt = -A y

We can separate the variables and integrate them to obtain the following:

x = x0 ᵉ⁽ᴬ ᵗ⁾

y = y0ᵉ⁽⁻ᴬ ᵗ⁾

Substituting x0 = 2, y0 = 8, and A = 0.3 ˢ⁻¹, we get:

x = 2 ᵉ(⁰.³ ᵗ)

y = 8 ᵉ(-⁰.³ ᵗ)

So the location of the fluid particle at time t = 6 s is:

x = 2 ᵉ(⁰.³⁶) = 7.13 m

y = 8 ᵉ(-⁰.³⁶) = 3.19 m

Therefore, the fluid particle is located at (7.13 m, 3.19 m) at time t = 6 s.

(E) To find the velocity of the fluid particle at time t = 6 s, we differentiate the equations for x and y with respect to time:

dx/dt = A x = 0.3(7.13) = 2.14 m/s

dy/dt = -A y = -0.3(3.19) = -0.96 m/s

Therefore, the velocity of the fluid particle at time t = 6 s is (2.14 m/s, -0.96 m/s).