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

1, 5 hockey pucks and three hockey sticks cost $23. 5 hockey pucks and 1 ho...

Advanced abstract algebra, prove that J[i] is an euclidean ring

Turning fractions into percents, i don''t understand how

Logistic marketing, supply chain management

Algebra 2 Appendix F, I have an algebra assignment I need help with, you ha...

Simulation and Modeling, I have some sample simulation and modeling practic...

Calculas, Q1: Find three positive numbers whose sum is 54 and whose product...

Using pythagorean theorem solve z 2 = ( x + y )2 + 3502, Two people on bik...

Tests for an ideal index number, Tests for an Ideal Index Number 1. F...

Applying percents, $1,575 computer,15% discount

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