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

Triangles, CM and RN are resp. the medians of triangle ABC and Triangle PQR...

CM and RN are resp. the medians of triangle ABC and Triangle PQR.if triangle ABC similar to Triangle PQR TRIANGLE AMC SIMILAR TO PNR

Integration by parts -integration techniques, Integration by Parts -Integra...

Integration by Parts -Integration Techniques Let's start off along with this section with a couple of integrals that we should previously be able to do to get us started. Fir

Calculus, I need help with my calculus

I need help with my calculus

Determine how many square centimeters, Determine how many square centimeter...

Determine how many square centimeters of paper are needed to make a label on a cylindrical can 45 cm tall with a circular base having diameter of 20 cm. Leave answer in terms of π.

Determine the measure of angle, Using the expample provided below, if m∠ABE...

Using the expample provided below, if m∠ABE = 4x + 5 and m∠CBD = 7x - 10, Determine the measure of ∠ABE. a. 155° b. 73° c. 107° d. 25° d. ∠CBD and ∠ABE are vert

Example of circles - common polar coordinate graphs, Example of Circles - C...

Example of Circles - Common Polar Coordinate Graphs Example: Graph r = 7, r = 4 cos θ, and r = -7 sin θ on similar axis system. Solution The very first one is a circle

Sum of a number of terms in a.p., We know that the terms in an ...

We know that the terms in an A.P. are given by a, a + d, a + 2d, a + 3d, ........ a + (n - 2)d, a + (n -  1)d The sum of all t

Fractions, how do you multiply fractions

how do you multiply fractions

#title.translating verbal expressions, what is the quotient of 3 and n?

what is the quotient of 3 and n?

Square of a number added to 25 equals 10 times the number, The square of a ...

The square of a number added to 25 equals 10 times the number. What is the number? Let x = the number.  The statement, "The square of a number added to 25 equals 10 times the n

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