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

Surface area with parametric equations, Surface Area with Parametric Equati...

Surface Area with Parametric Equations In this final section of looking at calculus applications with parametric equations we will take a look at determining the surface area o

Functions, the function g is defined as g:x 7-4x find the number k such tha...

the function g is defined as g:x 7-4x find the number k such that kf(-8)=f- 3/2

Velocity of a particle, A particle moves along a straight line so that afte...

A particle moves along a straight line so that after t secs its distance from fixed point O on the line is given by s=(t-1)^2(t-2).find the distance from O when the velocity is zer

Estimation of difference among two means, Estimation of difference among tw...

Estimation of difference among two means We know that the standard error of a sample is given by the value of the standard deviation (σ) divided by the square root of the numbe

Relative maximum point, Relative maximum point The above graph of the ...

Relative maximum point The above graph of the function slopes upwards to the right between points C and A and thus has a positive slope among these two points. The function ha

Evaluate limit in indeterminate form, Evaluate following limits. S...

Evaluate following limits. Solution In this case we also contain a 0/0 indeterminate form and if we were actually good at factoring we could factor the numerator & den

Analysis, Ask question #Minimum 1Let X be a topological space, let p ? X, a...

Ask question #Minimum 1Let X be a topological space, let p ? X, and let F and ? be C-valued functions on X that are continuous at p. Then the functions F + ?, F?, |F|, ReF and ImF

Integration, ((1/x^1/2-(x-1)^1/2)+(1/(5-3(x-1)^2)^1/2)

((1/x^1/2-(x-1)^1/2)+(1/(5-3(x-1)^2)^1/2)

Kfso, poijn jjjnjb jbjdbjbj

poijn jjjnjb jbjdbjbj

Probability that a leap year will have 53 sunday?explain, A leap year has 3...

A leap year has 366 days, therefore 52 weeks i.e. 52 Sunday and 2 days. The remaining 2 days may be any of the following : (i) Sunday and Monday (ii) Monday and Tuesday (iii)

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