Let’s take a circumstance in which such an additional force can be described by a potential energy, as would be true in the case of gravitation; we will let $\phi$ stand for the potential energy per unit mass. (For gravity, for instance, $\phi$ is just $gz$.) The force per unit mass is given in terms of the potential by $-\FLPgrad{\phi}$, and if $\rho$ is the density of the fluid, the force per unit volume is $-\rho\,\FLPgrad{\phi}$. For equilibrium this force per unit volume added to the pressure force per unit volume must give zero: \begin{equation} \label{Eq:II:40:1} -\FLPgrad{p}-\rho\,\FLPgrad{\phi}=\FLPzero. \end{equation} Equation ( 40.1 ) is the equation of hydrostatics. In general, it has no solution. If the density varies in space in an arbitrary way, there is no way for the forces to be in balance, and the fluid cannot be in static equilibrium. Convection currents will start up. We can see this from the equation since the pressure term is a pure gradient, whereas for variable $\rho$ the other term is not. Only when $\rho$ is a constant is the potential term a pure gradient. Then the equation has a solution \begin{equation*} p+\rho\phi=\text{const}. \end{equation*} Another possibility which allows hydrostatic equilibrium is for $\rho$ to be a function only of $p$. However, we will leave the subject of hydrostatics because it is not nearly so interesting as the situation when fluids are in motion.

If we take a small cube of water, what is the net force on it from the pressure? Since the pressure at any place is the same in all directions, there can be a net force per unit volume only because the pressure varies from one point to another. Suppose that the pressure is varying in the $x$-direction—and we take the coordinate directions parallel to the cube edges. The pressure on the face at $x$ gives the force $p\,\Delta y\,\Delta z$ (Fig. 40-3 ), and the pressure on the face at $x+\Delta x$ gives the force $-[p+(\ddpl{p}{x})\,\Delta x]\,\Delta y\,\Delta z$, so that the resultant force is $-(\ddpl{p}{x})\,\Delta x\,\Delta y\,\Delta z$. If we take the remaining pairs of faces of the cube, we easily see that the pressure force per unit volume is $-\FLPgrad{p}$. If there are other forces in addition—such as gravity—then the pressure must balance them to give equilibrium.

The pressure in a fluid may vary from place to place. For example, in a static fluid at the earth’s surface the pressure will vary with height because of the weight of the fluid. If the density $\rho$ of the fluid is considered constant, and if the pressure at some arbitrary zero level is called $p_0$ (Fig. 40-2 ), then the pressure at a height $h$ above this point is $p=p_0-\rho gh$, where $g$ is the gravitational force per unit mass. The combination \begin{equation*} p+\rho gh \end{equation*} is, therefore, a constant in the static fluid. This relation is familiar to you, but we will now derive a more general result of which it is a special case.

We begin by considering hydrostatics, the theory of liquids at rest. When liquids are at rest, there are no shear forces (even for viscous liquids). The law of hydrostatics, therefore, is that the stresses are always normal to any surface inside the fluid. The normal force per unit area is called the pressure. From the fact that there is no shear in a static fluid it follows that the pressure stress is the same in all directions (Fig. 40-1 ). We will let you entertain yourself by proving that if there is no shear on any plane in a fluid, the pressure must be the same in any direction.

We suppose that the elementary properties of water are already known to you. The main property that distinguishes a fluid from a solid is that a fluid cannot maintain a shear stress for any length of time. If a shear is applied to a fluid, it will move under the shear. Thicker liquids like honey move less easily than fluids like air or water. The measure of the ease with which a fluid yields is its viscosity. In this chapter we will consider only situations in which the viscous effects can be ignored. The effects of viscosity will be taken up in the next chapter.

The subject of the flow of fluids, and particularly of water, fascinates everybody. We can all remember, as children, playing in the bathtub or in mud puddles with the strange stuff. As we get older, we watch streams, waterfalls, and whirlpools, and we are fascinated by this substance which seems almost alive relative to solids. The behavior of fluids is in many ways very unexpected and interesting—it is the subject of this chapter and the next. The efforts of a child trying to dam a small stream flowing in the street and his surprise at the strange way the water works its way out has its analog in our attempts over the years to understand the flow of fluids. We have tried to dam the water up—in our understanding—by getting the laws and the equations that describe the flow. We will describe these attempts in this chapter. In the next chapter, we will describe the unique way in which water has broken through the dam and escaped our attempts to understand it.

40–2 The equations of motion

First, we will discuss fluid motions in a purely abstract, theoretical way and then consider special examples. To describe the motion of a fluid, we must give its properties at every point. For example, at different places, the water (let us call the fluid “water”) is moving with different velocities. To specify the character of the flow, therefore, we must give the three components of velocity at every point and for any time. If we can find the equations that determine the velocity, then we would know how the liquid moves at all times. The velocity, however, is not the only property that the fluid has which varies from point to point. We have just discussed the variation of the pressure from point to point. And there are still other variables. There may also be a variation of density from point to point. In addition, the fluid may be a conductor and carry an electric current whose density $\FLPj$ varies from point to point in magnitude and direction. There may be a temperature which varies from point to point, or a magnetic field, and so on. So the number of fields needed to describe the complete situation will depend on how complicated the problem is. There are interesting phenomena when currents and magnetism play a dominant part in determining the behavior of the fluid; the subject is called magnetohydrodynamics, and great attention is being paid to it at the present time. However, we are not going to consider these more complicated situations because there are already interesting phenomena at a lower level of complexity, and even the more elementary level will be complicated enough.

We will take the situation where there is no magnetic field and no conductivity, and we will not worry about the temperature because we will suppose that the density and pressure determine in a unique manner the temperature at any point. As a matter of fact, we will reduce the complexity of our work by making the assumption that the density is a constant—we imagine that the fluid is essentially incompressible. Putting it another way, we are supposing that the variations of pressure are so small that the changes in density produced thereby are negligible. If that is not the case, we would encounter phenomena additional to the ones we will be discussing here—for example, the propagation of sound or of shock waves. We have already discussed the propagation of sound and shocks to some extent, so we will now isolate our consideration of hydrodynamics from these other phenomena by making the approximation that the density $\rho$ is a constant. It is easy to determine when the approximation of constant $\rho$ is a good one. We can say that if the velocities of flow are much less than the speed of a sound wave in the fluid, we do not have to worry about variations in density. The escape that water makes in our attempts to understand it is not related to the approximation of constant density. The complications that do permit the escape will be discussed in the next chapter.

In the general theory of fluids one must begin with an equation of state for the fluid which connects the pressure to the density. In our approximation this equation of state is simply \begin{equation*} \rho=\text{const}. \end{equation*} This then is the first relation for our variables. The next relation expresses the conservation of matter—if matter flows away from a point, there must be a decrease in the amount left behind. If the fluid velocity is $\FLPv$, then the mass which flows in a unit time across a unit area of surface is the component of $\rho\FLPv$ normal to the surface. We have had a similar relation in electricity. We also know from electricity that the divergence of such a quantity gives the rate of decrease of the density per unit time. In the same way, the equation \begin{equation} \label{Eq:II:40:2} \FLPdiv{(\rho\FLPv)}=-\ddp{\rho}{t} \end{equation} expresses the conservation of mass for a fluid; it is the hydrodynamic equation of continuity. In our approximation, which is the incompressible fluid approximation, $\rho$ is a constant, and the equation of continuity is simply \begin{equation} \label{Eq:II:40:3} \FLPdiv{\FLPv}=0. \end{equation} The fluid velocity $\FLPv$—like the magnetic field $\FLPB$—has zero divergence. (The hydrodynamic equations are often closely analogous to the electrodynamic equations; that’s why we studied electrodynamics first. Some people argue the other way; they think that one should study hydrodynamics first so that it will be easier to understand electricity afterwards. But electrodynamics is really much easier than hydrodynamics.)

We will get our next equation from Newton’s law which tells us how the velocity changes because of the forces. The mass of an element of volume of the fluid times its acceleration must be equal to the force on the element. Taking an element of unit volume, and writing the force per unit volume as $\FLPf$, we have \begin{equation*} \rho\times(\text{acceleration})=\FLPf. \end{equation*} We will write the force density as the sum of three terms. We have already considered the pressure force per unit volume, $-\FLPgrad{p}$. Then there are the “external” forces which act at a distance—like gravity or electricity. When they are conservative forces with a potential per unit mass, $\phi$, they give a force density $-\rho\,\FLPgrad{\phi}$. (If the external forces are not conservative, we would have to write $\FLPf_{\text{ext}}$ for the external force per unit volume.) Then there is another “internal” force per unit volume, which is due to the fact that in a flowing fluid there can also be a shearing stress. This is called the viscous force, which we will write $\FLPf_{\text{visc}}$. Our equation of motion is \begin{equation} \label{Eq:II:40:4} \rho\times(\text{acceleration})= -\FLPgrad{p}-\rho\,\FLPgrad{\phi}+\FLPf_{\text{visc}}. \end{equation}

For this chapter we are going to suppose that the liquid is “thin” in the sense that the viscosity is unimportant, so we will omit $\FLPf_{\text{visc}}$. When we drop the viscosity term, we will be making an approximation which describes some ideal stuff rather than real water. John von Neumann was well aware of the tremendous difference between what happens when you don’t have the viscous terms and when you do, and he was also aware that, during most of the development of hydrodynamics until about 1900, almost the main interest was in solving beautiful mathematical problems with this approximation which had almost nothing to do with real fluids. He characterized the theorist who made such analyses as a man who studied “dry water.” Such analyses leave out an essential property of the fluid. It is because we are leaving this property out of our calculations in this chapter that we have given it the title “The Flow of Dry Water.” We are postponing a discussion of real water to the next chapter.

If we leave out $\FLPf_{\text{visc}}$, we have in Eq. (40.4) everything we need except an expression for the acceleration. You might think that the formula for the acceleration of a fluid particle would be very simple, for it seems obvious that if $\FLPv$ is the velocity of a fluid particle at some place in the fluid, the acceleration would just be $\ddpl{\FLPv}{t}$. It is not—and for a rather subtle reason. The derivative $\ddpl{\FLPv}{t}$, is the rate at which the velocity $\FLPv(x,y,z,t)$ changes at a fixed point in space. What we need is how fast the velocity changes for a particular piece of fluid. Imagine that we mark one of the drops of water with a colored speck so we can watch it. In a small interval of time $\Delta t$, this drop will move to a different location. If the drop is moving along some path as sketched in Fig. 40-4, it might in $\Delta t$ move from $P_1$ to $P_2$. In fact, it will move in the $x$-direction by an amount $v_x\,\Delta t$, in the $y$-direction by the amount $v_y\,\Delta t$, and in the $z$-direction by the amount $v_z\,\Delta t$. We see that, if $\FLPv(x,y,z,t)$ is the velocity of the fluid particle which is at $(x,y,z)$ at the time $t$, then the velocity of the same particle, at the time $t+\Delta t$ is given by $\FLPv(x+\Delta x,y+\Delta y,z+\Delta z,t+\Delta t)$—with \begin{equation*} \Delta x=v_x\,\Delta t,\quad \Delta y=v_y\,\Delta t,\quad \text{and}\quad \Delta z=v_z\,\Delta t. \end{equation*} From the definition of the partial derivatives—recall Eq. (2.7)—we have, to first order, that \begin{align*} \FLPv(x+\;&v_x\,\Delta t,y+v_y\,\Delta t,z+v_z\,\Delta t,t+\Delta t)\\[2pt] &=\FLPv(x,y,z,t)+ \ddp{\FLPv}{x}\,v_x\,\Delta t+ \ddp{\FLPv}{y}\,v_y\,\Delta t+ \ddp{\FLPv}{z}\,v_z\,\Delta t+ \ddp{\FLPv}{t}\,\Delta t. \end{align*} \begin{gather*} \FLPv(x+v_x\Delta t,\,y+v_y\Delta t,\,z+v_z\Delta t,\,t+\Delta t)=\\[1.5ex] \FLPv(x,y,z,t)\!+\! \ddp{\FLPv}{x}\,v_x\Delta t+\! \ddp{\FLPv}{y}\,v_y\Delta t+\! \ddp{\FLPv}{z}\,v_z\Delta t+\! \ddp{\FLPv}{t}\,\Delta t. \end{gather*} The acceleration $\Delta\FLPv/\Delta t$ is \begin{equation*} v_x\,\ddp{\FLPv}{x}+v_y\,\ddp{\FLPv}{y}+v_z\,\ddp{\FLPv}{z}+\ddp{\FLPv}{t}. \end{equation*} We can write this symbolically—treating $\FLPnabla$ as a vector—as \begin{equation} \label{Eq:II:40:5} (\FLPv\cdot\FLPnabla)\FLPv+\ddp{\FLPv}{t}. \end{equation} Note that there can be an acceleration even though $\ddpl{\FLPv}{t}=\FLPzero$ so that velocity at a given point is not changing. As an example, water flowing in a circle at a constant speed is accelerating even though the velocity at a given point is not changing. The reason is, of course, that the velocity of a particular piece of water which is initially at one point on the circle has a different direction a moment later; there is a centripetal acceleration.

The rest of our theory is just mathematical—finding solutions of the equation of motion we get by putting the acceleration (40.5) into Eq. (40.4). We get \begin{equation} \label{Eq:II:40:6} \ddp{\FLPv}{t}+(\FLPv\cdot\FLPnabla)\FLPv= -\frac{\FLPgrad{p}}{\rho}-\FLPgrad{\phi}, \end{equation} where viscosity has been omitted. We can rearrange this equation by using the following identity from vector analysis: \begin{equation*} (\FLPv\cdot\FLPnabla)\FLPv=(\FLPcurl{\FLPv})\times\FLPv+ \tfrac{1}{2}\FLPgrad{(\FLPv\cdot\FLPv)}. \end{equation*} If we now define a new vector field $\FLPOmega$, as the curl of $\FLPv$, \begin{equation} \label{Eq:II:40:7} \FLPOmega=\FLPcurl{\FLPv}, \end{equation} the vector identity can be written as \begin{equation*} (\FLPv\cdot\FLPnabla)\FLPv=\FLPOmega\times\FLPv+ \tfrac{1}{2}\FLPgrad{v^2}, \end{equation*} and our equation of motion (40.6) becomes \begin{equation} \label{Eq:II:40:8} \ddp{\FLPv}{t}+\FLPOmega\times\FLPv+ \frac{1}{2}\,\FLPgrad{v^2}= -\frac{\FLPgrad{p}}{\rho}-\FLPgrad{\phi}. \end{equation} You can verify that Eqs. (40.6) and (40.8) are equivalent by checking that the components of the two sides of the equation are equal—and making use of (40.7).

The vector field $\FLPOmega$ is called the vorticity. If the vorticity is zero everywhere, we say that the flow is irrotational. We have already defined in Section 3-5 a thing called the circulation of a vector field. The circulation around any closed loop in a fluid is the line integral of the fluid velocity, at a given instant of time, around that loop: \begin{equation*} (\text{Circulation})=\oint\FLPv\cdot d\FLPs. \end{equation*} The circulation per unit area for an infinitesimal loop is then—using Stokes’ theorem—equal to $\FLPcurl{\FLPv}$. So the vorticity $\FLPOmega$ is the circulation around a unit area (perpendicular to the direction of $\FLPOmega$). It also follows that if you put a little piece of dirt—not an infinitesimal point—at any place in the liquid it will rotate with the angular velocity $\FLPOmega/2$. Try to see if you can prove that. You can also check it out that for a bucket of water on a turntable, $\FLPOmega$ is equal to twice the local angular velocity of the water.

If we are interested only in the velocity field, we can eliminate the pressure from our equations. Taking the curl of both sides of Eq. (40.8), remembering that $\rho$ is a constant and that the curl of any gradient is zero, and using Eq. (40.3), we get \begin{equation} \label{Eq:II:40:9} \ddp{\FLPOmega}{t}+\FLPcurl{(\FLPOmega\times\FLPv)}=\FLPzero. \end{equation} This equation, together with the equations \begin{equation} \label{Eq:II:40:10} \FLPOmega=\FLPcurl{\FLPv} \end{equation} and \begin{equation} \label{Eq:II:40:11} \FLPdiv{\FLPv}=0, \end{equation} describes completely the velocity field $\FLPv$. Mathematically speaking, if we know $\FLPOmega$ at some time, then we know the curl of the velocity vector, and we also know that its divergence is zero, so given the physical situation we have all we need to determine $\FLPv$ everywhere. (It is just like the situation in magnetism where we had $\FLPdiv{\FLPB}=0$ and $\FLPcurl{\FLPB}=\FLPj/\epsO c^2$.) Thus, a given $\FLPOmega$ determines $\FLPv$ just as a given $\FLPj$ determines $\FLPB$. Then, knowing $\FLPv$, Eq. (40.9) tells us the rate of change of $\FLPOmega$ from which we can get the new $\FLPOmega$ for the next instant. Using Eq. (40.10), again we find the new $\FLPv$, and so on. You see how these equations contain all the machinery for calculating the flow. Note, however, that this procedure gives the velocity field only; we have lost all information about the pressure.