differential equation, Suppose a fluid (say, water) occupies a domain D? R^(3 ) and has, Mathematics

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.

Write Your Message!

Name

Email id

Message

Verfication Code

User Account

All Pages

differential equation, Mathematics

Related Discussions:- differential equation

Trignometry, Sin3x ? Solution) THE FORMULA IS RIGHT ,SO sin3x=3sin...

Explain histogramsin details, Explain Histogramsin details? Another way...

Erp for a company, recomendation to a company to implement ERP to succeed

Wholenumberriddles, I am less than 100 the sum of my digits is 4 half of me...

Related to MCA, AskIf y=e^(a?sin?^(-1) x), prove that (1 – x2)yn+2 – (2n + ...

Parenteral calculations, 850ml is to be administered to a person over 8 hou...

All subjects, I need help. Is there anyone there to help me?

Add subtract fractions., how do you add and subtract mixed numbers with fra...

Mechanics, find the composition of the simple harmonic motion of the same p...

Frobenius number, Wht is Frobenius Number? Start discussion and problems so...

Write Your Message!

Featured Services

Online Tutoring

Project Development

Exam Preparation

Course Help

Assignment Help

Popular Subjects

Assured A++ Grade

Academics

Major Subjects

Majors

Get In Touch

TERMS & POLICIES

HELP & SUPPORT