## Dz Sentence Examples

- If we suppose that the force impressed upon the element of mass D dx dy
**dz**is**DZ**dx dy**dz**, being everywhere parallel to the axis of Z, the only change required in our equations (I), (2) is the addition of the term Z to the second member of the third equation (2). - (b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 =
**dz**' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i =**dZ**/dy (b 2 v 2 +n 2)', , 2 = -**dZ**/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy**dz** - These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation p = pk =RA (6) where 0 denotes the absolute temperature; and then d9 d p R
**dz**-**dz**(p) n+ 1' so that the temperature-gradient deldz is constant, as in convective equilibrium in (I I). - From the gas-equation in general, in the atmosphere n d dp _ I dp 1 de _ d0 de i de (8) z p
**dz-edz-p-edz-k-edz**' which is positive, and the density p diminishes with the ascent, provided the temperature-gradient de/**dz**does not exceed elk. - Then dp/
**dz**=kdp/**dz**= P, = Poe ik, p - po= kpo(ez Ik -1); (16) and if the liquid was incompressible, the depth at pressure p would be (p - po) 1po, so that the lowering of the surface due to compression is ke h I k -k -z= 1z 2 /k, when k is large. - On another physical assumption of constant cubical elasticity A, dp = Ad p /P, (p - po)IA= lo g (P/Po), (18) dp _ A dp (I 1 zd p
**dz**- P ' A Po-p -z, I - p -k, A kPo ' (19) (3) P dx Pdy Pdz -., (I) When the density p is f un dp/ iform, this becomes, as before in (2) § 9 P pp ==Pzz++pao constant. - The integral equation of continuity (I) may now be written l f fdxdydz+ff (lpu+mpv+npdso, (4) which becomes by Green's transformation (dt +d
**dz**dy dx (p u) + d (p v) + d (p w) l I dxdydz - o, dp leading to the differential equation of continuity when the integration is removed. - Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is - f'fpuq cos OdS = - f f (lpu 2 -+mpuv+npuw)dS, (1) which by Green's transformation is (d(uiu 2) dy
**dz**dxdydz. - The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy
**dz**dx / leading to the differential equation of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy +**dz**with two similar equations. - These equations may be simplified slightly, using the equation of continuity (5) § for dpu dpu 2 dpuv dpuw dt dx + dy +
**dz**=p Cat +uax+vay+waz? - Dp dpu dpv dpw -z)' reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form du du du du d dt +udx+vdy +wdz =X -n dx' with the two others dv dv dv dv i dp dt +u dx +v dy +w
**dz**- Y -P d y' dw dw dw Z w dw i d p dt +u dx +v dy +wd - -P**dz**. - To determine the component acceleration of a particle, suppose F to denote any function of x, y, z, t, and investigate the time rate of F for a moving particle; denoting the change by DF/dt, DF = 1t F(x+uSt, y+vIt, z+wSt, t+St) - F(x, y, z, t) dt at = d + u dx +v dy+ w
**dz**and D/dt is called particle differentiation, because it follows the rate of change of a particle as it leaves the point x, y, z; but dF/dt, dF/dx, dF/dy, dF/**dz**(2) represent the rate of change of F at the time t, at the point, x, y, z, fixed in space. - = dx dy
**dz**the equations of motion may be Written du - 2v? - 2wr { a 0, dt2WE+2UC+
**dz**= o, dw dt - 2un+2v+ dH = 0, where H = fdp/p +V +1q 2, (7) 2 2 +v 2 2 (8) and the three terms in H may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and Zq 2 is replaced by 1q 2 1 g. - Eliminating H between (5) and (6) DS du dv dw (du dv d1zv dt u dx n dx udx' 5 -,
**dzi**=°' and combining this with the equation of continuity Dp du dv dw p iit dx+dy+**dz**= °' (10) D i du n dv dw_ dt (p p dx p dx p dx - o, with two similar equations. - D o, dx dy
**dz**dx dy**dz**so that, at any instant, the surfaces over which tk and m are constant intersect in the vortex lines. **Dz**=o.- Equation (5) becomes, by a rearrangement, dK dmdm dm din dx dt +u dx + dy +Zee
**dz**+ dx (dt +u dx +v dy +w d) = o,. - D - K dK dK _ dK dK dK ?dx n dyd °, udx
**dz**- ° and K=fdp/o+V+2q 2 =H (3) is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid. - The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are du du, du d i g d g 2 _ dH dx +v dy +
**dz**- 2v? - = 0,
**dz**= O. - So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, Du + dQ =o, Dv+dQ =o, Dw + dQ dt dx dt dy dt
**dz**and taking dx, dy,**dz**in the direction of u, v, w, and dx: dy:**dz**=u: v: w, (udx + vdy + wdz) = Du dx +u 1+.. - Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are 1, m, n, k=lu+mv+nw; (2) and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u', v', w')dt; so that Dk dl, do dt dt dt dt + dtw +1 (?t +u, dx +v, dy +w,
**dz**) +m (d +u dx + v dy +w'**dz**) dw, dw +n (dt ?dx+v?dy +w**dz**) But as 1, m, n are the direction cosines of a line fixed in space, dl= m R-n Q, d m = nP-lR an =1Q-mP dt dt ' dt ' so that Dk __ du, du, du, du dt l (dt -vR+ wQ+u + v dy + w**dz**) +m(.. - U '= - dx -md x, ' - dy -m dy, w = -
**dz-mdz**' as in § 25 (I), a first integral of the equations in (5) may be written dp V + 2q 2 - d - n dt +14-14) (dx + m**dz**) +(v-v') (+m) +(w - w) (+m) =F(t), (7) in which d4, do, d? - Dt-(u)dy- (w-w)
**dz**= d - (U-yR+zQ) dy - (V-zP+xR)d -(W-xQ+yP) d z (8) is the time-rate of change of 49 at a point fixed in space, which is left behind with velocity components u-u', v-v', w-w'. - D t dy
**dz**where _ oo abcdA A, B ' C ' - (a 2 +A, b 2 ±x, A, c 2 +A) P P 2 = 4(a 2 -F-A) (b 2 ±A) (c2+A). - Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy,
**dz**respectively, its centre having co-ordinates (x, y, z). - Hence the total flux is - (+ d2V d 2 V d2V dye + dz2) dy
**dz**, dx2 and by the previous theorem this must be equal to 4'rrp dxdydz. - We have by partial integration ff1 fV dd - ' 2 dy JJ dx y JJ y dxd
**dz**= V - d**dzdxd****dz**, and Itwo (similar equations in y and z. - I by the whole area B"
**DZ'VO**under the isothermal 9"D and the adiabatic**DZ**', bounded by the axes of pressure and volume. **DZ**, representing the loads taken in their order.- ZpzL(z)
**dz**, where, in general, we must suppose p a function of z. - Hence if we write K =27rf o,/i(z)
**dz**, H =27rf o zi/i(z)**dz**, the pressure of a column of the fluid itself terminating at the surface will be p2{K+1H(I/Rid-I/R2)}, and the work done by the attractive forces when a particle m is brought to the surface of the fluid from an infinite distance will be mp{K+zH(I/Ri+I/Ro)} If we write (.0 J then 27rmpo(z) will express the work done by the attractive forces, while a particle m is brought from an infinite distance to a distance z from the plane surface of a mass of the substance of density p and infinitely thick. - Hence the surface-tension =e - =47rp 2 (f 0(z)
**dz**- ce(c)). - Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)
**dz**(27) When c is greater than e this is equivalent to 2H in the equation of Laplace. - If the density be a, the attraction between the whole of one side and a layer upon the other distant z from the plane and of thickness
**dz**is 27r6 2 P(z)**dz**, reckoned per unit of area. - The expression for the intrinsic pressure is thus simply K= 2 iro 2 f 1,G(z)
**dz**(28) In Laplace's investigation o- is supposed to be unity. - If a i, a 2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area 21ra l a fZ '(z)
**dz**, (30) or 27ra l 02 0(z), if we write f 4,(z)**dz**=0(z) (31) The work required to produce the separation in question is thus 2 7ru l a o 0 (z)**dz**; (32) and for the tension of a liquid of density a we have T = a f o 0 (z)**dz**. - For f0(z)
**dz**=0(z).z - fz d**dz**)**dz**=0(z).z+fz4,(z)**dz**. - Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)
**dz**f: (z)**dz**, (34) and T= ref: z1,1,(z)**dz**(35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H. - Integrating by parts, we get J l'(z)d z = zI, G (z) + 3 z 3 I I (z) 3 f z3Cb(z)
**dz**, fzqi(z)**dz**= J z21 '(z) + k z41 I (z) + a fz4(1)(z)**dz**. - In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write f o (z)
**dz**= 3f2' z 3 4(z)**dz**, f o z(z)**dz**= 'a' z4 cb(z)d z; (36) so that Ko = 3 f o z3 ?(z)**dz**, To = $?