Lecture 3

Question 1

How is the mass flow rate in and out of a system written WRT x-direction? How does this change for other directions?

Answer 1

dxdz for in, and dx)dydz changes for other directions

Question 2

How do you calculate net mass flow rate for the x-direction? What is the equation for this in the continuity equation?

Answer 2

Net = out - in

Mass in element = rho * volume , where dxdydz is volume

Question 3

What’s the equation for net mass flow rate out for all directions?

Answer 3

Sum of net flow for all directions

Question 4

Rate of mass increase through system equation?

Answer 4

Question 5

How do you get to the continuity equation?

Answer 5

Net flow out = Net mass decrease

RHS negative due to decreasing mass Divide both sides by dxdydz Set = 0

Question 6

Tensor notation for continuity equation

Answer 6

First term = change in density (mass) Second term = net mass flow across fluid element

Question 7

What is the divergence operator?

Answer 7

Question 8

What does it mean if the divergence form of the continuity equation is greater than 0?

Answer 8

The system is expanding; if < 0, it’s contracting.

Question 9

What is the divergence form of the continuity equation also known as?

Answer 9

Vector form

Question 10

What is the LAGRANGIAN form of the continuity equation?

Answer 10

Question 11

What does the total derivative expand into?

Answer 11

Question 12

What are the definitions of substantial (total), local, and convective derivatives?

Answer 12

Total: rate of change of time following a moving fluid element

Local: rate of change of time at fixed point

Convective: rate of change of time due to movement of the fluid

Question 13

What is the symbol for local and convective derivatives?

Answer 13

Local: ∂/∂t

Convective: u(∂/∂x)+v(∂/∂y)+w(∂/∂z)