1. Consider the diffusion of an impurity (like ink) into a fluid (like water). a. Derive the...

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1. Consider the diffusion of an impurity (like ink) into a fluid (like water). a. Derive the continuity equation ap A -+ Vrj • j = V at using the conservation of the mass of ink, where p is the density of the impurity and / is the current density. (This was done quickly in class. You may expand on the class notes or consult a book or the web. List your reference.) b. When the fluid is motionless, Fick's law gives the current / = —DV.p where D is the diffusion coefficient. Explain why this formula makes sense. Why is there a minus sign? c. Use these equations to obtain the diffusion equation. d. Now suppose the fluid is (lowing with velocity V(F,t). How does this modify the current? How does this change the diffusion equation?2. Consider the flow of heat into a solid block of iron that is heated at one point. The heat equation is a classic partial differential equation that describes the temperature T(1.,t) as a function of position and time throughout the block. Derive the heat equation by considering the continuity equation for energy conservation. Assume that the energy per unit volume satisfies £ = c„T and that the amount of energy that flows through a unit area per unit time satisfies Fourier's law j =
	
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David · Accepted Answer

1. a) Continuity equation.
 
 
Lets consider a rectangular volume of the fluid in rectangular Cartesian coordinate system, where 
fluid enters from the side ABCD with a velocity u and density ρ, the velocity of the fluid leaving from 
the side EFGH is   
  
  
    and the density
  
    . [Neglecting higher order terms] 
Therefore, rate of mass entering the volume from the ABCD side is ρu dy dz, 
And hence, rate of mass leaving out through the side EFGH is  
= (  
  
  
   ) (
  
   )  = (
  
 (  )   )         [Neglecting higher order terms in dx] 
This calculation is done considering mass flow through x direction only. 
Similarly, efflux and influx will happen through the y and z direction also.

1. Consider the diffusion of an impurity (like ink) into a fluid (like water). a. Derive the continuity equation ap A -+ Vrj • j = V at using the conservation of the mass of ink, where p is the...

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