Login

Create Account
Customer Service Chat
+14156709189
info@expertsmind.com
Submit Homework/Assignment
Get quote & make Payment
Get Solution
differential equation, Mathematics
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.
Posted Date: 3/23/2013 7:10:39 PM  Location : United States
Ask an Expert
Related Discussions:
differential equation, Assignment Help, Ask Question on differential equation, Get Answer, Expert's Help, differential equation Discussions
Write discussion on differential equation
Your posts are moderated
Write your message here..
Related Questions
Prove that 2b33abc+a2d=0, If the ratios of the polynomial ax 3 +3bx...
If the ratios of the polynomial ax 3 +3bx 2 +3cx+d are in AP, Prove that 2b 3 3abc+a 2 d=0 Ans: Let p(x) = ax 3 + 3bx 2 + 3cx + d and α , β , r are their three Z
Chain rule, Chain Rule : If f(x) and g(x) are both differentiable func...
Chain Rule : If f(x) and g(x) are both differentiable functions and we describe F(x) = (f. g)(x) so the derivative of F(x) is F′(x) = f ′(g(x)) g′(x). Proof We will s
Regrouping addition, how do you regroup?
how do you regroup?
Arithmetic progression (a.p.), A series is said to be in Arithmetic...
A series is said to be in Arithmetic Progression (A.P.) if the consecutive numbers in the series differs by a constant value. This constant value is referre
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
Hyperbolic paraboloid three dimensional space, Hyperbolic Paraboloid Thre...
Hyperbolic Paraboloid Three Dimensional Space The equation which is given here is the equation of a hyperbolic paraboloid. x 2 / a 2  y 2 / b 2 = z/c Here is a dia
Fraction.., 7
7
Matrices, what is the business application of matrices
what is the business application of matrices
Equations and Inequalities, Write an algebraic expression for “Julie runs t...
Write an algebraic expression for “Julie runs three miles less than twice the number of miles,
Derivatives of trig functions, Derivatives of Trig Functions In this s...
Derivatives of Trig Functions In this section we will see derivatives of functions other than polynomials or roots of polynomials. We'll begin this process off through taking
Assignment Help
Accounting Assignment Help
Economics Assignment Help
Finance Assignment Help
Statistics Assignment Help
Physics Assignment Help
Chemistry Assignment Help
Math Assignment Help
Biology Assignment Help
English Assignment Help
Management Assignment Help
Engineering Assignment Help
Programming Assignment Help
Computer Science Assignment Help
ExpertsMind Services
Online Tutoring
Projects Assistance
Exam Preparation
Coursework Help
Programming Help
IT Services
Why Us ?
~Experienced Tutors
~24x7 hrs Support
~Plagiarism Free
~Quality of Work
~Time on Delivery
~Privacy of Work