The temperature distribution in a fluid is given by T = 10x + 5y, where x and y are the horizontal and vertical coordinates in meters and T is in degrees centigrade. Determine the time rate of change of temperature of a fluid particle traveling:
(a) horizontally with u = 20 m/s, v = 0.
(b) vertically with u = 0, v = 20 m/s.
(c) diagonally with u = 20 m/s, v = 20 m/s.

The rates of change of T for each case are the following:

(a) = 200 °C/s

(b) = 100 °C/s

(c) = 300 °C/s

Explanation:

We need to find the rate of change for temperatura of the fluid, which is the derivate of the function as a function of time (t). We know that T is defined as a function of coordinates x and y, but we also know that velocity u is the derivate of x in terms of time (t) of the coordinate x and v is the derivate of y in terms of time (t).

So, to obtain the derivate of T in terms of t, we have:

And, as we said before,

    and    

Then,

And, as we know the equation that defines T in terms of x and y, we can derivate this function and obtain,

So,

And finally, we replace the value of u and v for each case,

(a) u=20 and v=0

(b) u=0 and v=20

(c) u=20 and v=20

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