00:01
In the first question, the temperature profile is given as theta equals to e power minus 3t divided by x square, where x square can be also written as x1 square plus x2 square plus x3 square.
00:17
We have to find the complete derivative, that is, capital d theta by d t.
00:26
Also at the same time, it has given that v1 equals to x2 plus 2x2.
00:35
V2 equals to x3 minus x1 and v3 equals to x1 plus 3 x2.
00:42
Now the formula for complete derivative of d theta by d t will be partial derivative of d theta with respect to time plus u into del theta by del x1 plus v into del theta by del x1 plus v into delta theta by del x2 plus sorry let's make let me call this as a v1 this is v2 and plus v3 delta theta by del x3 the concept is here let's take this one now v1 can be this is velocity so v1 can be written as del x1 by del t so v1 into del t so v1 into del delta by del x1 that is this particular term if we now see and analyze here now simply put v1 equals to del x1 by del t sorry into del theta by del x1 so this this will cancel out and we will get del theta by del t so in order to incorporate the partial derivative of theta with respect to 1 sorry with respect to x1 because theta is a function of x1 x2 x3 and time t because if we expand here will be e power minus 3 t divide by x1 square plus x2 square plus x3 square so that's the concept so now let's proceed and find del theta by del t so here when we differentiate with respect to time we will get minus 3 into e power minus 3 t divided by x square plus v1 is given as x2 plus 2x3 so here it will be x2 plus 2x3 into del theta by del x1 so now again if we go here that is basically equals to e power minus 3 t divided by x1 square plus x2 square plus x3 square and if we take the partial derivative of that we will have into let me write clearly d theta by d t will be d tita by d t will be e power minus 3 t divided by x squared plus x2 plus 2x3 into del theta by del x1.
04:15
So del theta by del x1 will be e power minus 3 t into 2x1 divided by x squared whole square.
04:36
So that will be the case.
04:37
Similarly for this term that is v2.
04:41
Into del theta by del x2 we will have x3 minus x1 that is the v2 here in equal into e power minus 3 t into 2 x2 divided by x square whole square one need to keep in mind here that x square is basically x1 square plus x2 square plus x3 square so i am just taking little shortcut similarly for the final term, v3 into del theta by del x3, we have x1 plus 3x2 into v .e power minus 3 t into 2x2, sorry 2x3 divided by x square whole square.
05:42
So after rearranging, you will get this result.
05:49
So this is the basic result.
05:51
Result or one can say that after differentiating this will be the temperature field obtained.
06:01
Now let's come to next question that is question number two.
06:06
We have to find this streamlined pathline in xy plane with velocity field given u equals to x into t plus 2 and v equals to y divided by t plus 1.
06:21
So let's first find this.
06:25
The pathline equation.
06:29
So simply let's try to eliminate t.
06:32
So we can write u y x minus 2 equals to t.
06:37
And in this equation we can write t equals to y y by v minus 1.
06:45
Now simply equate this two equations.
06:48
So we have u y x minus 2 equals to y y by v minus 1.
06:56
So we can write u minus 2 x.
07:02
Into v equals to y minus v into x so for the rearranging we will have xy minus v x minus uv plus 2 xb so this term and this term will get add on and we will have plus vx equals to zero so this is the pathline equation now let's come to find the streamline equation.
07:51
Now we know that the formula for the stream line is del phi by del y is equal to u and del phi by del x is equals to minus v.
08:09
So let's take del phi by del y.
08:13
Now in spite of you, we can write x into t plus 2.
08:18
And here we can write del phi by del x equals to minus y divided by t plus 1.
08:27
So when we will do with the integration we will have phi equals to xy into t plus 2 plus c...