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The definition of incompressibility is that the density of a fluid parcel (a volume element) does not change (i.e., is constant); this is your first equation: $$ \frac{D}{Dt}\rho(\mathbf x_0,t)=0 $$ which leads to the solenoidal constraint, $\nabla\cdot\mathbf u=0$. As for your "counter example," there is no issues because the density field can actually ...


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Here's a way to get a decent distance estimate for solids, liquids or gases.: for solids or liquids, you can get the number density, $n$, of atoms or molecules (as needed), from the density, Avogadro's number ($6.02E23$) and the molecular weight ($\rho,N_a,W$: $$n = \frac{\rho N_a}{W}$$ for a gas with pressure, temperature and the boltzmann constant ...


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Your experience with more resistance in water of depth of ~ 6ft is probably due to increased resistance of the boat against the bow wave which itself is beginning to "feel bottom". There is an excellent post on the Earth Science Stack that explains what it means to "feel bottom" here. If you can estimate the size of the bow wave (wavelength) then possibly ...


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The Magnus effect was discovered when an explanation for the low precision of guns was needed. It affects the cylindrical, pointed grenades just as much as any ball. It does not matter how long the rotating body is: Once it rotates, it will create a low pressure area on one side orthogonal to the crosswind direction and a corresponding high pressure area ...


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I believe that when an oblong ball tumbles with backspin its range is increased due to the magnus effect. Most studies of oblong balls analyze spinning rotation about the long axis, but there is some evidence that tumbling about the short axis may enhance the Magnus effect: "The Magnus effect is also exploited in a number of Nature’s designs. Many seed ...


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I agree with user3823992 that it was incorrect to neglect the pressure differential. With the steady sinusoidal body force that's given, it's basically a hydrostatics problem with the pressure differential balancing the body force. Consider the Navier-Stokes momentum equation: $$ \frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v} \cdot ...


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It is an approximation. If the object were bobbing up and down in the water at high enough speed, then there would at least be an appreciable drag force which would, to first approximation, be \begin{align} \mathbf F_d = -b\mathbf v \end{align} where $b$ is a constant and $\mathbf v$ is the velocity of the object relative to the water. If the speed ...


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First of all, make use of your knowledge in circular motion. Also, first consider things in polar coordinates. Imagine a point in the fluid a distance $r$ from the centre of rotation and an angle $\theta$ rotate anticlockwise from the +ve x direction. You can use your knowledge of circular motion to find the tangential velocity, and use trig to get it into ...


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The pressure of falling water can vary by a LOT - it depends in detail on the shape of the interface between the water and the surface it hits. When a perfectly spherical drop of water hits a hard surface, there will actually be a short moment in time when the contact point between the water and the surface travels faster than the speed of sound in water - ...


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Each kilogram of water that falls from 10 m will have the kinetic energy of about 100J. I can't tell you the pressure since you didn't specify the area. Shorter falls will have less energy.


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Use a blunt trailing edge in XFoil, and for the real thing round it any way you want. If you increase local curvature close to the trailing edge, your local flow will simply separate. Then it is irrelevant for the flow whether the trailing edge is rounded or blocky - the separated air will fill up the void and for the outer flow the exact shape of the ...


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You need to consider an impedance discontinuities at the ends of the tube. In simplified model the duct has non-zero impedance, ending of the stopped pipe has infinite impedance and open end zero impedance. Therefore the reflection occurs and a standing wave can be created. Number, frequencies and amplitudes of the modes depend on the parameters of the ...


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Let's review the linearisation and go to the further details. Just the pressure might be not enough. Take the momentum equation: $$ -\frac{1}{\rho}\nabla p = \frac{\partial \vec{v}}{\partial t}+\vec{v}\cdot\nabla\vec{v} $$ Here we have to eliminate the convective part $\vec{v}\cdot\nabla\vec{v}$. Usually the argumentation is that changes of the velocity ...


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Water forms close to perfect spheres in zero gravity due to it's surface tension. There's a variety of videos of water in the space station. Ice, assuming you start with one of those balls of water, you have to ask first, would it freeze outside in (say, the temperature of the station is dropped below 0 C), or would it freeze inside-out, say you stick a ...


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Sound will behave just like on earth provided it has a medium to travel through (Astronauts on the ISS can communicate normally; watch some videos)


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The key to flow of macroscopic particles is to make sure they don't "clump" or jam. I don't know what is propelling them down your pipe (gravity? Air flow?) but that will affect the answer. In general, adding some vibration keeps particles flowing freely; a larger pipe diameter with minimal obstructions / bends is the other thing. Making the pipe diameter ...


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I don't think creating a vortex effect will increase the velocity of flow, because it would direct movement in a direction other than forward. Smooth bore gunbarrels, for example, have greater muzzle velocity than rifled gunbarrels (assuming all other variables are equal). You might want to maximize laminar flow and to minimize turbulence inside the tube. ...


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The shape of the spout of your milk jug makes the milk from the edges flow towards the center - but as this means that the profile is trying to get narrower, the milk "has to go somewhere" and makes the jet wider in the other direction. However, surface tension is pulling back on the liquid (it would prefer the jet to be a perfect circle) so the liquid ...


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This may not be quite what you're after, but "Relativistic Hydrodynamics" by Rezzolla and Zanotti covers (relativistic) hydrodynamics in the language of differential geometry. This is a graduate level textbook on hydrodynamics in the context of general relativity (hence the differential geometry). It covers kinetic theory (including quantum effects), ...


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Astute observation, and very interesting phenomena. I do know for a fact that the reason the column of water narrows as it gets further from the faucet is due to acceleration by gravity and an increasing fluid velocity. The higher the velocity, the lower the pressure. Since the pressure outside everywhere along the column is essentially the same, the column ...


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Obviously darkness of lake water depends on depth of lake, impurity in water and many other things. But my answer to the question What causes the surface of lake to appear darker in some places? is Its depends on two things, [1] Position of observer [2] Position of sun in the sky. If sun is nearer to horizon then the amount of light, reflected from ...


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It is an interplay between the wind and the shoreline, and basic laws of reflection. As you can see in your photo, where the water surface is still, you see a reflected image of the skyline - lighter for the sky, darker for the buildings. Where the water surface ripples, you get reflections "from everywhere" - some from the sky, some from buildings, etc. ...


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It depends. It could be wind, simply the lighter section is rougher and the waves scatter light back to you while the flatter section appears darker because the light is scattered in a different direction. It can also happen where waters mix. A fresh water stream merges into an ocean, a flowing river meets a shallow stagnant area or (as below) river water ...


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From Wikipedia, the free encyclopedia A sonic black hole (sometimes called a dumb hole) is a phenomenon in which phonons (sound perturbations) are unable to escape from a fluid that is flowing more quickly than the local speed of sound. They are called sonic, or acoustic, black holes because these trapped phonons are analogous to light in astrophysical ...


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There are two primary factors that allow the cochlea to isolate frequencies. These are generally referred to as passive and active properties: tl;dr version: The passive properties are due to the mechnical properties of one of the membranes in the cochlea, the basilar membrane, primarily the width and stiffness at a given point. The active properties are ...


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Altitude can indeed have such an effect. As your linked article explains, one can get a rough sense of the aerodynamic force on a spherical ball by neglecting viscosity (i.e., model air as a bunch of ballistic particles that do not drag on one another), in which case the formula is1 $$ F = \frac{16\pi^2}{3} C_l \rho \omega v r^3. $$ The important point is ...


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I believe that yes, it could. However you must also take into consideration that air density may not be the only apprehension that a player is dealing with throughout a game. As to answer your other question multiple world stadiums are covered at a sea level however the highest was located at Estadio Da Baixada, Curitiba which was 920m (3,018 ft). Depending ...


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My understanding is this. I invite a discussion on this answer. While pouring the milk from the glass, Lets say "N" milk-molecules is reaching the air in the open space say "S". As the milk is more viscous fluid the milk-molecules are interested in coming as close to each other and hence it converges and hence the space in which the milk travels is reduced ...


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Wings provide lift because they direct air downwards. They direct air downwards in two ways. In part, the bottom of the wing slopes downward a bit and just pushes the air down as it moves forward through the air. But this is a small effect. The top of the wing is more important. The top of the wing pulls the air down partially by providing a ramp. The rear ...


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When you have a pipette, the liquid is usually kept in place by the surface tension. So there are several things: the weight of the column of liquid results in a pressure difference of $\Delta P = \rho g h$ between the top and bottom of the column this pressure difference will be supported in part by surface tension of the liquid, and in part by an ...


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You are confusing yourself. The statement $P - P_0$ would remain the same is false. Why would it remain the same? There is a certain amount of compressed gas inside the bubble, and there is a force that maintains it compressed. In the first case, this force is just the surface tension. In the second case, it is the surface tension reduced by the ...


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Excitation is due to interaction (exchange of momentum, changes of volume and force) of the droplets with the liquid surface. The air volume in the bottle has its own characteristic response which emphasizes particular frequencies of the excitation. During filling, the air volume decreases and small things tend to vibrate more at higher frequencies. A small ...


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Benard convection involves fluid flow on a horizontal plane heated from below. - Scholarpedia (peer-reviewed open access online encyclopedia) Ever since antiquity, geometers have known that only three regular polygons can tile a plane, without overlap or extra space. These are the equilateral triangle, the square, and the hexagon. Interference patterns on ...


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With a basic understanding of physical geometry we can understand this difference. A soliton is a KIND of pseudosphere created by the rotation of a tactrix, however, this particular kind of pseudosphere is harmonically generated by the negation of itself. We can say that if the natural growing pseudosphere has a one dimensional rotational property then the ...


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You say you know the temperature and the volume of the water. Assuming that you also know the mass of the water, you know both temperature and specific volume (the inverse of the the density). You just need a thermodynamic table for the properties of liquid water like this one from NIST to find the pressure. If you actually don't have enough information to ...


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The equations above thus represent conservation of mass, momentum, and energy. Mass density, Flow velocity and pressure are the so-called physical variables, while mass density, momentum density and total energy density are the so-called conserved variables. So three unknowns, right? Actually, there are four variables: density, velocity, pressure ...


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Both are of course connected, but denote very different concepts. Your definition of variance is fine with me. It tells us, on average, how much the flow deviates from its average. Intermittency is much harder to explain. I'll do my best here: In a scale invariant flow, the $n$'th order correlation function will take a scaling form, $$\Delta\Psi(n,r) = ...


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Your equations are correct. It's just that you have a degree of freedom. You can obtain a relationship between the force you exert and the velocity you obtain by modeling the piston itself: $M_p\frac{\partial^2 V_a}{\partial t^2}=S_ap_a$, where $M_p$ is the velocity of the pistol and assuming the friction forces are negligible.


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If the fluid inside the shell is frictionless and without bubbles or sloshing, it may be treated as a solid which does not rotate, but just slides down the ramp. This provides one component of instantaneous acceleration, which you would add to the instantaneous acceleration of the hollow ball. It seems to me that with this method, you needn't combine the ...


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Essentially a fixed-wing aircraft flies because it moves through the air and has a fixed wing which is angled to the direction of airflow. A component of the drag force acting on the wing acts in the direction (up) opposite to the direction (down) of the aircraft's weight force. An aeroplane wing acts like a weather vane responding to the relative flow of ...


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Actually here $\rho$ and ${\bf v}$ are function of $(t,\vec{x}) \in \mathbb R \times \mathbb R^3$, as it is usual in the so-called Eulerian description of a continuous body. There is no reference to the curves describing the evolutions of the particles of the fluid $\vec{x}_{\vec{x_0}}= \vec{x}_{\vec{x_0}}(t)$ as instead, it is the standard in the so-called ...


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From the chain rule we have, $\frac{d \rho}{dt} = \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x}\frac{dx}{dt} + \frac{\partial \rho}{\partial y}\frac{dy}{dt} + \frac{\partial \rho}{\partial z}\frac{dz}{dt}$ $\therefore \frac{d \rho}{dt} = \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x} v_x + \frac{\partial ...


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$$E_{\mathrm{mass}}=\frac{m R_n + \rho_a c_p \left(\delta e \right) g_a }{\lambda_v \left(m + \gamma \right) }$$ Where $E_{mass}$ = mass of the evaporated water $m$ = Slope of the saturation vapor pressure curve $(Pa K−1)$ $R_n$ = Net irradiance $(W m−2)$ $ρ_a$ = density of air $(kg m−3)$ $c_p$ = heat capacity of air $(J kg−1 K−1)$ $g_a $= momentum ...


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Yes, the exact solution is known. The general spherically symmetric metric is $$g=-B(r)\mathrm{d}t^2+A(r)\mathrm{d}r^2+r^2\mathrm{d}\Omega^2.$$ The solution for $A(r)$ is $$A(r)=\left[1-\frac{2G\mathcal{M}(r)}{r}\right]^{-1},\quad\mathcal{M}(r)=\int^r \rho \,\mathrm{d}V=\int_0^r 4\pi r'^2\rho(r')\,\mathrm{d}r.$$ The solution for $B(r)$ is ...


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Follow two stream lines, one above the wing and one below. You can ignore the height difference and assume that under the wing you have atmospheric pressure and normal stream speed. The two stream lines follow the Bernoulli principle and thus $$ \frac{1}{2} \rho v_{under}^2 + P_{under} = \frac{1}{2} \rho v_{top}^2 + P_{top} $$ Since you know the weight ...


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If the plane is just flying at constant altitude then a vertical force balance requires that lift from the wings be equal to the plane's weight. The lift force, $L$ comes from a pressure difference above and below the wing so that $$ L = (p_1-p_2)A = mg $$ You can use the Bernoulli equation assuming a negligible difference in height to express the pressure ...


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To get the ice cube to twirl by pure twirling of the glass, there need to be viscous stresses applied to the ice cube from the spinning water. To simplify matters, lets say that the water in your glass is perfectly still before you start twirling, that the ice cube is away from the edge of the glass, and that the twirling is initiated smoothly so that ...


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Considering your comment "Actually it is just an air cavity made inside the water with no boundary material": $P_{in} = P_{out} + 2\gamma /R$ where $R$ is the radius of the air cavity and $\gamma$ is the surface tension of the air-water interface, which is 0.072 N/m at 298K. See Internation Tables of the Surface Tension of Water for values at other ...


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Let us make an mental experiment. Suppose that there is no friction between the glass and the water inside. In this case, if you rotate the glass, the water inside and the ice cube will just stay still. This is because the angular momentum of the water and of the ice cube is conserved, since in this case no torque is applied to the water, i.e., there is no ...


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If the liquid were really rotating in the glass, the ice cubes would rotate with it. What you are (probably) seeing is the superposition of two perpendicular resonant waves sloshing back and forth, but not rotating. Here is how it works: Imagine that, instead of moving the glass in a circle, you just move it back and forth in an east/west direction. This ...



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