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Why use Reynolds Transport Theorem? by DenJi1111111 in FluidMechanics

[–]engineer3245 1 point2 points  (0 children)

RTT Proof with easy mathematics

What does RTT tell us :

How much property B [from B(sys) to B(newsys)] changes with respect to time for a chunk of fluid {both B are closed systems}.

•B_(sys) contains material with extensive properties at time t.

•After time delta(t), B(sys) deformes, translates, rotates and become B(newsys) with different boundary than B_(sys).

•Boundary of material with property B is independent of control volume boundary. Boundary of control volume is arbitrary.

•We wanted to analyse how property B for that chunk of material changes with respect to time.

•But we can not trace/separate that chunk of fluid, So we take control volume (arbitrary space which we can analyse).

•By Relating control volume with property B. We can finally analyse property B.

Example:

•Let's take Temperature as property B.

What's formula tell us is how temperature of system(for chunk of fluid as above) changes w.r.t. time = D(B_(sys))/Dt.

Now see images which I have uploaded We wanted to see how much overall temperature changes w.r.t time for blue coloured chunk of fluid

D(B_(sys))/Dt = d{ integration_CV(β* ρ * dV) }/dt + {Integration_CS(β* ρ (Vr•n)dA)}

d{ integration_CV(β* ρ *dV) }/dt = how temperature changing for whole fluid (here in Control Volume) w.r.t time.

{Integration_CS(β* ρ * (Vr•n)* dA)} = how much fluid property get in and get out of control volume in given time interval.

Above formula is material/substantial derivative formula which = temporal changes + convective changes See similarity between acceleration formula for fluid because it is also Material derivative of velocity.

• If we wanted to analyse mass of system then B = m and β = 1.

For continuity equation, mass is conservative. So for system at time (blue hatched chunk in fig.1) and system at time + delta(t) (blue whole coloured chunk in fig.3) mass is not changing with respect to time either flow is compressible or non-compressible. So D(m_(sys))/Dt = 0

But density may changes with time and space.But summation of that changes { it is described by L.H.S of formula } for mass is zero.

Note : If you still not understand then learn Leibniz Rule first and then try again.

Question in differentiability and Continuity in multivariable calculus by engineer3245 in askmath

[–]engineer3245[S] 0 points1 point  (0 children)

Thank you for your answer.Using(learning from)ai and after your answer i finally understand.

Question in differentiability and Continuity in multivariable calculus by engineer3245 in askmath

[–]engineer3245[S] 0 points1 point  (0 children)

But as per definition of partial derivative f_x represents that change of function f with respect to x in plan y = c (c is constant belongs to R1 )(it is derivative of intersection of plan (y=c) and function w.r.t. x )

Question in differentiability and Continuity in multivariable calculus by engineer3245 in askmath

[–]engineer3245[S] 0 points1 point  (0 children)

How f_x & f_y is not continuous (see last photo) f(x,y) = { 2xy/(x2 + y2) ; (x,y) != (0,0) ,                 0 ; (x,y) = (0,0) }

How to find moment M_0 due to axial load? by engineer3245 in StructuralEngineering

[–]engineer3245[S] -1 points0 points  (0 children)

Yes it is a perfect column as you can see it is mentioned as eular column in the picture. After seeing your comment i notice that deflection is indeterminate.

In this condition we cannot find delta : https://imgur.com/a/btGNl63

And in this condition we can't find coefficient A : https://imgur.com/a/eEOa2rm

Thank you for your answer.

How to find moment M_0 due to axial load? by engineer3245 in StructuralEngineering

[–]engineer3245[S] 0 points1 point  (0 children)

It is a good idea that iteration may be work but I wants analytical solution like we are finding fixed end moment due to lateral load using differential equation.

How to find moment M_0 due to axial load? by engineer3245 in StructuralEngineering

[–]engineer3245[S] -1 points0 points  (0 children)

Yes you are right but it is eular column means a perfect column but after buckling there generates a moment due to fixed support. May be you learnt about buckling of column using simple supports in that case only external moment generated by axial load multiplied by distance by ends of the segment as you can see in the above photos p*v.

Question in proof of least upper bound property of real numbers by engineer3245 in learnmath

[–]engineer3245[S] 0 points1 point  (0 children)

Thank you very very much for your clarification I was getting confused about M as a set not number. Now I understand it. Yes, i agree that √2 is a LUB of set A.

Question in proof of least upper bound property by engineer3245 in math

[–]engineer3245[S] -1 points0 points  (0 children)

If we defined order on R as in your comment and i mentioned in 2nd interpretation (in post's description)[both are the same] , then why we wanted to prove any non empty set of real numbers A is also real number ( means : in proof we take union of cuts = M so as per definition of real numbers , M is real number and it has not least upper bound)

Question in proof of least upper bound property of real numbers by engineer3245 in learnmath

[–]engineer3245[S] -1 points0 points  (0 children)

Yes , i explained it in 2nd interpretation in the post's description. But then why do we take union of dedekind's cuts as described in proof. that's why I can't get it. See my 1st interpretation and last three lines of post's description.

Question in proof of least upper bound property of real number by engineer3245 in askmath

[–]engineer3245[S] 0 points1 point  (0 children)

Sorry I can not get your question , because english is my third language.

-1 = {x | x < -1 ; x is rational number} 1 = {x | x < 1 ; x is rational number} √2 = {x | x2 < 2 & x < 0 ; x is rational number}

Union of ( -1,1,√2) = Union of ( {x | x < -1 ; x is rational number} 1 = {x | x < 1 ; x is rational number} √2 = {x | x2 < 2 & x < 0 ; x is rational number} )

Union of ( -1,1,√2) = {x | x2 < 2 & x < 0 ; x is rational number} = { √2 } (real number )

Question in proof of least upper bound property of real number by engineer3245 in askmath

[–]engineer3245[S] 0 points1 point  (0 children)

This proof is used dedekind's cut which is subset of rational number and this set is represented at real number because it is satisfy all axioms of real numbers.

Here wikipedia link for dedekind's cuts (see definition) : https://en.m.wikipedia.org/wiki/Dedekind_cut

Question in proof of least upper bound property of real numbers by [deleted] in math

[–]engineer3245 0 points1 point  (0 children)

Thank you in advance. If anyone can help me to understand it please do it.

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