How Does Fluid Dynamics Equation Manipulation Work?

[Halitus]

I am not entirely sure how some of these steps work.

What I have so far (pardon any latex mistakes, I am new to it)

K=\frac{\Delta P}{\Lambda Vol /Vol}

change in pressure over volumetric strain = K the bulk modulus

\rho Vol'= d \rho / dt

(\frac{d P}{d t}/\frac{d \rho}{d t}*)\rho=K

I am unsure about this step del P to dP/dt i understand but not the strain term.

\frac{d P}{d t}= \frac{K}{p} * \frac{\rho}{dt} = K Vol'

Vol'=div(V)

V is velocity

\frac{d P}{d t} = K div (V)

because

\frac{d P}{d t} =div(M*grad(P))

div(M*grad(P))= K div (V)

These last two steps I don't understand apparently \frac{d P}{d t} =div(M*grad(P)) is some kind of identity. And because I don't understand this step I don't understand what value M should have.

Any light on these would be awesome thanks. Sorry its kinda messy I am very new to latex

 

[jvicens]

.
Thank you for your post. I can understand how some of these steps may be confusing, especially if you are new to using LaTeX. Let me try to break down the steps for you and provide some explanation.

First of all, the equation you have written for the bulk modulus is correct, where K represents the bulk modulus and \Delta P is the change in pressure over a change in volumetric strain (\Lambda Vol /Vol). This is a measure of how much a material will compress or expand under different pressures.

Next, you have correctly identified that \rho Vol' represents the change in density over time (d \rho / dt). This is because density can change over time due to changes in pressure or temperature.

The next step is where things may start to get a bit confusing. The equation \frac{d P}{d t}= \frac{K}{p} * \frac{\rho}{dt} is actually the equation for the continuity equation, which is used to describe the conservation of mass in a fluid. In this equation, \frac{d P}{d t} represents the change in pressure over time, \frac{K}{p} represents the bulk modulus divided by the density, and \frac{\rho}{dt} represents the change in density over time. This equation is used to describe the relationship between pressure, density, and velocity in a fluid.

Moving on to the next step, where you have written Vol'=div(V), this represents the divergence of velocity (V). The divergence of a vector field is a measure of how much the field is spreading out or converging at a particular point. In this case, it is used to describe the change in velocity over time.

Finally, the equation \frac{d P}{d t} =div(M*grad(P)) is known as the Navier-Stokes equation, which is a fundamental equation in fluid dynamics. In this equation, M represents the dynamic viscosity of the fluid, which is a measure of how easily the fluid can flow. This value is dependent on the properties of the fluid, such as its density and viscosity.

I hope this explanation has helped to clarify some of the steps for you. If you have any further questions or need more clarification, please don't hesitate to ask. it is important to fully understand the equations and steps we use in our research, so it is great that you are seeking clarification. Keep up the good work!