# Why can't water flow fast when the impact speeds up? (the incompressibility)

I'm a high school student studying physics in Korea! Please understand my poor English skills.

In the text below, the liquid is said to be destructive and incompressible. And he says, "I the flowing takes time, and if your speed of impact is too great, the water won’t be able to flow away fast enough, and so it pushes back at you." I don't know what this sentence means. Why can't water flow fast when the impact speeds up? Is this problem related to 'momentum & impulse'?

(The answer to this question I thought was, "Because of the incompressibility of the water, the impulse time is shortened and the force to receive increases.") • What is missing here, is the surface area that is hitting the water. Take a bucket, and fill it up with water. Punch through the water surface - it's not that difficult. However, try slapping the water surface with your palm. You'll feel a stronger force resisting you. So, the surface that is impacting the water level does matter, along with the speed of the impact. Nov 17, 2021 at 1:19
• The author is referring to fluid viscosity—resistance to flow. A key aspect of a viscous fluid is that its dynamic stiffness increases with increasing strain rate. Nov 17, 2021 at 3:49

It has a lot to do with inertia. Inertia a measure of how much an object will resist a change to its motion (stationary or moving) when acted upon by an external force. If the force is large, the object will resist it greatly and if it is smaller, the object will resist it less.

So if you increase your impact speed, the greater the water will impact you. This applies to all objects with mass. Whilst water is not easy to compress, the major reason why the larger a force that is applied to a body of water, the larger it's reaction force will be, is due mainly to the fact that is has mass and therefore inertia.

• Thank you for your help! Thanks to you, I learned a lot! Nov 21, 2021 at 13:01
• That’s ok and good luck with your studies. Cheers. Nov 21, 2021 at 20:25

The viscosity of the fluid actually determines if the fluid could flow "fast enough" or not.

Viscosity,$$\mu$$, is a property of fluid determined by the ratio of shear stress and velocity gradient: $$\begin{equation} \tau(\mathbf{u})=\mu\nabla\mathbf{u} \end{equation}$$ which is just the shear stress version of the familiar pressure, except the force is now parallel to the surface rather than perpendicular.

We can see that given a fixed shear stress, the acceleration,$$\nabla\mathbf{u}$$, decreases if the viscosity increases, meaning the fluid would not be able to flow "fast enough" if a large enough viscosity is given. Imagine something with super high viscosity, say, tar road. Punching tar road is more painful than punching a body of water because that tar can't deform "fast enough" and hence the impulse of your punch is very high, and hence the reaction of your force during the process of your "punching" is high as well.

To prove that the impulse of your punch is very high, imagine now that you are a perfectly spherical ball with a brain and nervous system and all that, and someone throw you into a pool of water for some reason. Stokes derived from the definition of the shear stress and told us that the reaction force(we usually call it the drag force) that you will experience is actually: $$\begin{equation} F_d=6\pi\mu r\mathbf{u} \end{equation}$$ where r is the radius of the ball.

From Newton's second law of motion,$$\mathbf{F}=m\mathbf{a}$$: $$\begin{equation} m\frac{d\mathbf{u}}{dt}=-6\pi\mu r\mathbf{u} \end{equation}$$ Now solve the equation, we can actually get the velocity as a function of time: $$\begin{equation} \mathbf{u}=\mathbf{u}_0 \mathrm{exp}(\frac{-6\pi\mu r}{m}t) \end{equation}$$ where $$\mathbf{u}_0$$ is the velocity when you hit the water.

And by the definition of impulse, we have: $$\begin{equation} \mathbf{J}=\int^{t_f}_0 6\pi\mu r\mathbf{u}dt=\int^{t_f}_0 \mathbf{u}_0 6\pi\mu r \mathrm{exp}(\frac{-6\pi\mu r}{m}t)dt=\mathbf{u}_0 m(1-\mathrm{exp}(\frac{-6\pi\mu r}{m}t_f)) \end{equation}$$ where $$t_f$$ is the time you stop moving, theoretically $$t_f\rightarrow\infty$$, but realistically, you cannot be in the water forever.

We see that if $$\mu$$ is high, the impulse is high, and vice versa. That's why punching tar road hurt much more than punching a body of water. So yes, it's related to impulse and momentum. And it's not that water can't flow fast when the impact speeds up(well, that is to assume that water is a perfect Newtonian liquid, but that's another story), but that impact is large when the fluid can't flow fast.

Inertia is part of the answer, yes. But inertia is caused only by gravity, whereas drag force actually takes the electromagnetic property of the material into account.

• Thank you for your help! Thanks to you, I learned a lot! Nov 21, 2021 at 13:01