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Many documents write about "flow of incompressible fluids". The flow rate is proportional to the pressure gradient so that the flow is from the higher pressure zone to a lower pressure zone. But an incompressible fluid cannot have a pressure gradient. First we must agree on the meaning of pressure. Pressure is the repelling force between the particles of the fluid. In order to have a repelling force, the spacing between the particles must be not be the equilibrium spacing. For example, in the Lennard-Jones (LJ) model of a liquid, the potential energy, `V`, as a function of spacing, `r`, is
`V(r)=4V_(min)[(sigma/r)^12-(sigma/r)^6]`
where the value of r at the minimum of V is `r_0=2^(1/6)sigma`. In order to be repelling, `r` must be slightly less than `r_0`. The Lennard-Jones model is more realistic than a simple symmetrical force and potential model because, at smaller spacings, the electron clouds of all atoms and molecules are affected by the Pauli Exclusion Principle where electrons cannot partially occupy the same space as other electrons. The potential at `r_0` is then`V(r_0)=4V_(min)[(2^(-1/6))^12-(2^(-1/6))^6]=4[1/4-1/2]=- V_(min) `
Thus, to have positive pressure, the average spacing must be less than `r_0`. T he numerical definition, in 3D, of the pressure is force per unit area and has units of `"Newtons per" \ meter^2`. Then, for our LJ model, if particle spacing is just a bit less than `r_(min")` we can write the differential of the force between 2 particles as:`delta F=-(d^2V)/(dr^2)=-4(V_(min))/(r_0^2)[144(sigma/r_0)^12-36(sigma/r_0)^6](r-r_0)`
where `r` is a small negative displacement from `r_0`. We need the pressure which is the force per unit area and the area from the particle spacing is just `r_0^2` so the pressure, `P`, is just`P=(delta F)/(r_0^2)=-4(V_(min))/(r_0^4)[144(sigma/(r_0))^12-36(sigma/r_0)^6](r-r_0)\ \ \ (1)`
We can simplify this expression by substituting the value of `(sigma/r_0)^6=1/2` so the term in the square brackets becomes`[144(sigma/(r_0))^12-36(sigma/r_0)^6]=[144/4-36/2]=36-18=18`
One of the great features of this animation is that, right after you press Start, you can see the motion of a velocity change wave and a pressure wave both of which should be at the speed of sound. First we'll repeat the very short derivation of the speed of sound given in Wikipedia which results from Newton's equations of motion. It starts with the expression
`rho (dv)/dt=-(dP)/dx`
where `rho` is the mass density `(kg)/m^3` and 'P` is pressure. From that expression, the chain rule is applied to convert to the form`dP=-(rho dv)dx/dt=-(rho dv)v`
where we recognize that `dx/dt=v` . Next is an assumption that only minimal mass motion occurs during one cycle of the sound wave which means that '(rho v)=constant' and therefore`v drho =-rho dv`
In this animation some particle motion in the direction of propagation does occur but the initial speed of particle motion compared to the speed of the sound is very small. Using this latter expression in the former one we find`dP=-(rho dv)dx/dt=(drho v)v`
and this results in the expression for the speed of a sound wave:`v^2=(dP)/(drho)`
Now there are two types of sound wave:1. Transverse where the particle motion is perpendicular to the direction of propagation
2. Longitudinal or compressional where the particle motion is parallel to the direction of propagation
This animation will be of type 2 since the initial pressure gradient is parallel to the long axis of the container.In equation 1, we already have an expression for the pressure in terms of the Lennard-Jones potential .
`P=-4*18(V_(min))/(r_0^4)(r-r_0)\ \ \ (1)`
For our model the density, `rho` increases as `1/r` as the pressure increases. To get the derivative `(dP)/(drho)` we need to take the one more differential of the potential. Using the chain rule we have:`(dP)/(drho)=(dP)/(dr)(dr)/(drho)\ \ \ (2)`
Again using the chain rule:`(drho)/(dr)=-m(dn/dr)`
where m is the mass of a single particle and n is the number of particles per unit volume. A good approximation for `n` is just:`n=1/r_0^3`
Then:`(drho)/(dr)=-3(m/r_0^4)`
Substituting this expression into equation 2 we have for `v^2`:`v^2=(dP)/(drho)=((dP)/(dr))/((drho)/(dr))=-((dP)/(dr))/(3m/(r_0)^4)=24 V_(min)/m\ \ \(3)`
where `m` is the mass of one particle.Units check: Recall that a potential has units of ` m l^2/t^2` where `l` is a length unit and `t` is a time unit so that `(dP)/(drho)` has units `l^2/t^2` as would be expected for a velocity squared.
Since the number of particles required is large, they are shown at 1/5 of their actual size on Canvas 1. For better viewing of the relative motion of the particles the leading edge section of the particle array is shown full size on Canvas 2. Canvas 1 has three moving averaged histogram plots :
1. The density (green) Vs x position of the particle array
2. The x component of the velocity Vs x position of the array
3. The x component of the force or acceleration Vs x position of the array
The pressure gradient is generated by reducing the x component of the initial spacing of the columns of the array thereby causing the particles to repel each other along the x direction. For the "sloped" gradient, the initial spacing is minimum at the left and increases linearly to the non-repelling spacing at the right side of the array. For the "stepped" gradient the initial spacing on the left HALF of the array is that dictated by the full gradient`r=r_0(1-g)`
where g is what I call the Gradient on the slider. In order to more easily see tne spacing (or 1/density variation) gradient the plot, the plots are a moving average lagged by 10 bins. The initial gradient that propels the particles is reduced as the particles move away from the left hand side of the container and therefore the particle force is maximum at the start and diminishes gradually. As a result of this force reduction, the fluid breaks up into segments with spaces between the segments. This is the same as cavitation (bubbling) of a pumped fluid.Eventually the fluid becomes unstable, particles are lost from the boundaries and the animation stops. A new animation can be started by changing a slider setting and pressing Start button again.
The the initial velocity and force pulse speeds are dependent, as computed above, on the square root of `V_(min)` . It turns out that, unexpectedly, the speeds are dependent linearly on the gradient. It was to be expected that the column speed would be linearly dependent on the gradient but not the pressure pulse since the latter should be dependent on the intrinsic properties of the material which is a Lennard-Jones medium. We will leave that mystery for a later study.