User Account

All Pages

differential equation, Mathematics

Assignment Help:
Suppose a fluid (say, water) occupies a domain D? R^(3 ) and has velocity field V=V(x, t). A substance (say, a day) is suspended into the fluid and will be transported by the fluid as well as diffused within it; let u= u(x,t) be the concentration of the substance( mass per unit volume). Let Ø= Ø(x,t) be the concentration flux (mass per unit area per unit time, analogous to heat flux). Let B(x) be a ball of radius r > 0 contained in D. Derive the conservation law.

?_(B (x))¦u_(t ) dV= -?_(? B(x))¦?Ø.n_0 dS.?

Fick''s law for diffusion states that the concentration flux due to diffusion is proportional to the gradient of the concentration flux due to diffusion is proportional to the gradient of the concentration. Deduce that Ø= -k?+uV.
Apply the divergence theorem to the conservation law and substitute the flux formula to arrive at the diffusion- transport equation:
u_t= ?.(k?u)+ ?.(uV).
in the absence of diffusion (k=0), this is the higher- dimensional transport equation. If the fluid is motionless, it is called the diffusion equation.
Suppose a fluid (say, water) occupies a domain D? R^(3 ) and has velocity field V=V(x, t). A substance (say, a day) is suspended into the fluid and will be transported by the fluid as well as diffused within it; let u= u(x,t) be the concentration of the substance( mass per unit volume). Let Ø= Ø(x,t) be the concentration flux (mass per unit area per unit time, analogous to heat flux). Let B(x) be a ball of radius r > 0 contained in D. Derive the conservation law.

?_(B (x))¦u_(t ) dV= -?_(? B(x))¦?Ø.n_0 dS.?

Fick''s law for diffusion states that the concentration flux due to diffusion is proportional to the gradient of the concentration flux due to diffusion is proportional to the gradient of the concentration. Deduce that Ø= -k?+uV.
Apply the divergence theorem to the conservation law and substitute the flux formula to arrive at the diffusion- transport equation:
u_t= ?.(k?u)+ ?.(uV).
in the absence of diffusion (k=0), this is the higher- dimensional transport equation. If the fluid is motionless, it is called the diffusion equation.

Related Discussions:- differential equation

Fractions Word Problem, 1/8 of the passengers of a train were children.If t...

1/8 of the passengers of a train were children.If there were 40 children travelling in the train on a certain day,how many adults were there in that train that day?

Probability, TWO PERSONS A AND B AGREE TO MEET AT A PLACE BTWEEN 11 TO 12 N...

TWO PERSONS A AND B AGREE TO MEET AT A PLACE BTWEEN 11 TO 12 NOON.  THE FIRST ONE TOARRIVE WAITS FOR 20 MIN AND THEN LEAVE. IF THE TIME OF THIR ARRIVAL BE INDEPENDET AND AT RNDOM,T

What is a scatter diagram, What is a scatter diagram? A scatter diagram...

What is a scatter diagram? A scatter diagram is a graphical representation of data points for a particular sample. Choosing a dissimilar sample or enlarging the original one ca

Can u please tell me how to solve, a triangle with side lengths in the rati...

a triangle with side lengths in the ratio 3:4:5 is inscribed in a circle

Find a quadratic polynomial having a and ß, If α,β are the zeros of a Quadr...

If α,β are the zeros of a Quadratic polynomial such that α + β = 24, α - β = 8. Find a Quadratic polynomial having α and β as its zeros.

Patrice has worked a certain how many hours has she worked, Patrice has wor...

Patrice has worked a certain amount of hours so far this week. Tomorrow she will work four more hours to finish out the week along with a total of 10 hours. How many hours has she

Actual implicit solution, y 2 = t 2 - 3 is the actual implicit solution t...

y 2 = t 2 - 3 is the actual implicit solution to y'= t/y, y(2) = -1. At such point I will ask that you trust me that it is actually a solution to the differential equation. You w

Set builder notation, A={2,3,5,7,11} B={1,3,5,7,9} C={10,20,30,40,......100...

A={2,3,5,7,11} B={1,3,5,7,9} C={10,20,30,40,......100} D={8,16,24,32,40} E={W,O,R,K} F={Red,Blue,Green} G={March,May} H={Jose,John,Joshua,Javier} I={3,6,9,12,15}

Working definition of limit - sequences and series, Working Definition of L...

Working Definition of Limit 1. We state that if we can create an as close to L like we want for all adequately large n.  Alternatively, the value of the a n 's approach

Calculate the radius of the circle, In the figure, ABCD is a square inside ...

In the figure, ABCD is a square inside a circle with centre O. The Centre of the square coincides with O & the diagonal AC is horizontal of AP, DQ are vertical & AP = 45 cm, DQ = 2

Write Your Message!

Captcha
Order Assignment

Featured Services

Order Now

Popular Subjects

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

Submit Assignment

Academics

Major Subjects

Majors

Get In Touch

whatsapp: +91-977-207-8620

Phone: +91-977-207-8620

Email: [email protected]

TERMS & POLICIES

HELP & SUPPORT

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd