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(3.29) by the density n:mndu¼ qnðE þ uBÞdtð3:30ÞThis is, however, not a convenient form to use. In Eq. (3.29), the time derivative isto be taken at the position of the particles. On the other hand, we wish to have anequation for fluid elements fixed in space, because it would be impractical to dootherwise. Consider a drop of cream in a cup of coffee as a fluid element. As thecoffee is stirred, the drop distorts into a filament and finally disperses all over thecup, losing its identity. A fluid element at a fixed spot in the cup, however, retains itsidentity although particles continually go in and out of it.To make the transformation to variables in a fixed frame, consider G(x, t) to beany property of a fluid in one-dimensional x space. The change of G with time in aframe moving with the fluid is the sum of two terms:d Gðx; tÞ ∂G ∂G dx ∂G∂G¼þ¼þ uxdt∂t∂x ∂t∂t∂xð3:31ÞThe first term on the right represents the change of G at a fixed point in space, andthe second term represents the change of G as the observer moves with the fluid intoa region in which G is different.
In three dimensions, Eq. (3.31) generalizes todG ∂G¼þ ðu ∇ ÞGdt∂tð3:32ÞThis is called the convective derivative and is sometimes written DG/Dt. Note that(u · ∇) is a scalar differential operator. Since the sign of this term is sometimes asource of confusion, we give two simple examples.Figure 3.1 shows an electric water heater in which the hot water has risen to thetop and the cold water has sunk to the bottom. Let G(x, t) be the temperature T; ∇Gis then upward.
Consider a fluid element near the edge of the tank. If the heaterelement is turned on, the fluid element is heated as it moves, and we have dT/dt > 0.If, in addition, a paddle wheel sets up a flow pattern as shown, the temperature in afixed fluid element is lowered by the convection of cold water from the bottom. Inthis case, we have ∂T/∂x > 0 and ux > 0, so that u ∇T > 0. The temperaturechange in the fixed element, ∂T/∂t, is given by a balance of these effects,∂T dT¼∂tdtu ∇TIt is quite clear that ∂T/∂t can be made zero, at least for a short time.ð3:33Þ583 Plasmas as FluidsFig.
3.1 Motion of fluidelements in a hot waterheaterFig. 3.2 Direction of thesalinity gradient at themouth of a riverAs a second example we may take G to be the salinity S of the water near themouth of a river (Fig. 3.2). If x is the upstream direction, there is normally agradient of S such that ∂S/∂x < 0.
When the tide comes in, the entire interfacebetween salt and fresh water moves upstream, and ux > 0. Thus∂S¼∂tux∂S>0∂xð3:34Þmeaning that the salinity increases at any given point. Of course, if it rains, thesalinity decreases everywhere, and a negative term dS/dt is to be added to themiddle part of Eq. (3.34).As a final example, take G to be the density of cars near a freeway entrance atrush hour. A driver will see the density around him increasing as he approaches the3.3 The Fluid Equation of Motion59crowded freeway.
This is the convective term ðu ∇ ÞG. At the same time, the localstreets may be filling with cars that enter from driveways, so that the density willincrease even if the observer does not move. This is the ∂G/∂t term. The totalincrease seen by the observer is the sum of these effects.In the case of a plasma, we take G to be the fluid velocity u and write Eq. (3.30)as∂umnþ ðu ∇ Þu ¼ qnðE þ uBÞ∂tð3:35Þwhere ∂u/∂t is the time derivative in a fixed frame.3.3.2The Stress TensorWhen thermal motions are taken into account, a pressure force has to be added tothe right-hand side of Eq.
(3.35). This force arises from the random motion ofparticles in and out of a fluid element and does not appear in the equation for asingle particle. Let a fluid element Δx Δy Δz be centered at (x0, 12Δy, 12Δz) (Fig. 3.3).For simplicity, we shall consider only the x component of motion through the facesA and B.
The number of particles per second passing through the face A withvelocity vx isΔnv vx Δy Δzwhere Δnv is the number of particles per m3 with velocity vx:ððΔnv ¼ Δvx f vx ; v y ; vz dv y dvzFig. 3.3 Origin of the elements of the stress tensor603 Plasmas as FluidsEach particle carries a momentum mvx. The density n and temperature KT in eachcube is assumed to have the value associated with the cube’s center. The momentum PA+ carried into the element at x0 through A is thenhi1PAþ ¼ Σ Δnv mv2x Δy Δz ¼ Δy Δz mv2x n2x0 Δxð3:36ÞThe sum over Δnv results in the average v2x over the distribution.
The factor 1/2comes from the fact that only half the particles in the cube at x0 Δx are goingtoward face A. Similarly, the momentum carried out through face B ishi1PBþ ¼ Δy Δz mv2x n2x0Thus the net gain in x momentum from right-moving particles isPAþPBþ ¼ Δy Δz 12 mhnv2xix0 Δx¼ Δy Δz 12 mð ΔxÞ∂ 2nvx∂xh i nv2xx0ð3:37ÞThis result will be just doubled by the contribution of left-moving particles, sincethey carry negative x momentum and also move in the opposite direction relative tothe gradient of nv2x . The total change of momentum of the fluid element at x0 istherefore∂ðnmux ÞΔx Δy Δz ¼∂tm∂ 2nvx Δx Δy Δz∂xð3:38ÞLet the velocity vx of a particle be decomposed into two parts,vx ¼ ux þ vxrux ¼ v xwhere ux is the fluid velocity and vxr is the random thermal velocity.
For aone-dimensional Maxwellian distribution, we have from Eq. (1.7)1mv2xr212ð3:39Þ¼ KTEquation (3.38) now becomes∂ðnmux Þ ¼∂tmi∂h 2n ux þ 2uv xr þ v2xr ¼∂xm ∂KTn u2x þ∂xm3.3 The Fluid Equation of Motion61We can cancel two terms by partial differentiation:mn∂ux∂n¼þ mux∂t∂tmux∂ðnux Þ∂xmnux∂ux∂x∂ðnKT Þ∂xð3:40ÞThe equation of mass conservation1∂n ∂þ ðnux Þ ¼ 0∂t ∂xð3:41Þallows us to cancel the terms nearest the equal sign in Eq. (3.40). Defining thepressurep nKTð3:42Þwe have finally∂ux∂uxmnþ ux¼∂t∂x∂p∂xð3:43ÞThis is the usual pressure-gradient force.
Adding the electromagnetic forces andgeneralizing to three dimensions, we have the fluid equation∂umnþ ðu ∇ Þu ¼ qnðE þ uBÞ∂t∇pð3:44ÞWhat we have derived is only a special case: the transfer of x momentum bymotion in the x direction; and we have assumed that the fluid is isotropic, so that thesame result holds in the y and z directions. But it is also possible to transfery momentum by motion in the x direction, for instance.
Suppose, in Fig. 3.3, thatuy is zero in the cube at x ¼ x0 but is positive on both sides. Then as particles migrateacross the faces A and B, they bring in more positive y momentum than they takeout, and the fluid element gains momentum in the y direction. This shear stresscannot be represented by a scalar p but must be given by a tensor P, the stresstensor, whose components Pi j ¼ mn vi v j specify both the direction of motion andthe component of momentum involved. In the general case the term ∇ p isreplaced by ∇ P.We shall not give the stress tensor here except for the two simplest cases.
Whenthe distribution function is an isotropic Maxwellian, P is written1If the reader has not encountered this before, it is derived in Sect. 3.3.5.623 Plasmas as Fluids0pP¼@000p0100Apð3:45Þ∇ P is just ∇p. In Sect. 1.3, we noted that a plasma could have two temperatures T⊥and T║ in the presence of a magnetic field.
In that case, there would be twopressures p⊥ ¼ nKT ⊥ and pk ¼ nKT k . The stress tensor is then0p⊥P¼@ 000p⊥0100 Apkð3:46Þwhere the coordinate of the third row or column is the direction of B. This is stilldiagonal and shows isotropy in a plane perpendicular to B.In an ordinary fluid, the off-diagonal elements of P are usually associated withviscosity.
When particles make collisions, they come off with an average velocityin the direction of the fluid velocity u at the point where they made their lastcollision. This momentum is transferred to another fluid element upon the nextcollision. This tends to equalize u at different points, and the resulting resistance toshear flow is what we intuitively think of as viscosity. The longer the mean freepath, the farther momentum is carried, and the larger is the viscosity.
In a plasmathere is a similar effect which occurs even in the absence of collisions. The Larmorgyration of particles (particularly ions) brings them into different parts of theplasma and tends to equalize the fluid velocities there. The Larmor radius ratherthan the mean free path sets the scale of this kind of collisionless viscosity. It is afinite-Larmor-radius effect which occurs in addition to collisional viscosity and isclosely related to the vE drift in a nonuniform E field (Eq.
(2.58)).3.3.3CollisionsIf there is a neutral gas, the charged fluid will exchange momentum with it throughcollisions. The momentum lost per collision will be proportional to the relativevelocity u u0, where u0 is the velocity of the neutral fluid. If τ, the mean free timebetween collisions, is approximately constant, the resulting force term can beroughly written as mn(u u0)/τ. The equation of motion (3.44) can be generalized to include anisotropic pressure and neutral collisions as follows:∂umnþ ðu ∇ Þu ¼ qnðE þ u BÞ∂t∇Pmnðu u0 Þτð3:47ÞCollisions between charged particles have not been included; these will be treatedin Chap.
5.3.3 The Fluid Equation of Motion3.3.463Comparison with Ordinary HydrodynamicsOrdinary fluids obey the Navier–Stokes equationρ∂uþ ðu ∇ Þu ¼∂t∇ p þ ρν∇2 uð3:48ÞThis is the same as the plasma equation (3.47) except for the absence of electromagnetic forces and collisions between species (there being only one species). Theviscosity term ρν∇2 u, where ν is the kinematic viscosity coefficient, is just thecollisional part of ∇ P ∇ p in the absence of magnetic fields. Equation (3.48)describes a fluid in which there are frequent collisions between particles.
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