THERMODYNAMICS AND KINETIC THEORY in Java Print Denso QR Bar Code in Java THERMODYNAMICS AND KINETIC THEORY THERMODYNAMICS AND KINETIC THEORYQR Code maker with javagenerate, create qr-codes none for java projectsFig. 5.4 Forces acting on an element of fluid. Bar Code maker on javausing java toassign bar code with asp.net web,windows applicationrespectively denoted by F1 and G. Thus we can write read bar code with javaUsing Barcode scanner for Java Control to read, scan read, scan image in Java applications.p dX 1 dX 2 dX 3 Control qr code 2d barcode image in c#generate, create qr code iso/iec18004 none with visual c#.net projects(F1 + G) dX 1 dX 2 dx 3 Control qr code 2d barcode size with .netqr code size for .netTherefore Newton's second law for a fluid element takes the form Qr-codes barcode library in .netusing .net framework torender qr code jis x 0510 with asp.net web,windows application:t + QR-Code barcode library in visual basic.netgenerate, create qrcode none in visual basic.net projects)U = F1+ G Control upc symbol size for java upc code size with java(5.104)Control code 128 size with javato attach code-128c and code 128b data, size, image with java barcode sdkThus the derivation of the Navier-Stokes equation reduces to the derivation of a definite expression for G. Let uS choose a coordinate system such that the fluid element under consideration is a cube with edges along the three coordinate axes, as shown in Fig. 5.4. The six faces of this cube are subjected to forces exerted by neighboring fluid elements. The force on each face is such that its direction is determined by the direction of the normal vector to the face. That is, its direction depends on which side of the face is considered the "outside." This is physically obvious if we remind ourselves that this force arises from hydrostatic pressure and viscous drag. Let Ti be the force per unit area acting on the face whose normal lies along the x i axis. Then the forces per unit area acting on the two faces normal to the Xi axis are, respectively (see Fig. 5.4),Java gs1-128 generating with javausing java todraw gs1128 with asp.net web,windows applicationTi ,Control ean 128 image with javausing java touse ucc ean 128 with asp.net web,windows applicationaT (T + ax;Control data matrix 2d barcode size for javato generate barcode data matrix and 2d data matrix barcode data, size, image with java barcode sdkdX i Render cbc on javausing barcode creation for java control to generate, create rm4scc image in java applications.(i=1,2,3)Control ean-13 supplement 5 size with .net ean13 size with .net(5.105)Control pdf-417 2d barcode image in visual basic.netgenerate, create pdf-417 2d barcode none on visual basic projectsThe total force acting on the cube by neighboring fluid elements is then given by GS1 - 12 barcode library in visual basic.netgenerate, create upc barcodes none in visual basic.net projects(5.106)Control qr code data with .net qr code 2d barcode data with .netTRANSPORT PHENOMENA Matrix Barcode maker for vb.netusing .net framework torender 2d barcode in asp.net web,windows applicationWe denote the components of the vectors T1 , T z, T3 as follows:Control pdf 417 data in .net barcode pdf417 data with .netT1 Tz Control 3 of 9 barcode size in .net code 39 full ascii size on .net(P11' P1Z , P13 ) (PZ1 ' Pzz , P23 ) (P31 , P32 , P33 )(5.107)Then (5.108)G=-V' p With this, (5.104) becomes (5.109)p(~ at + u V')U = V' . P (5.110)which is of the same form as (5.22) if we set F1 = pF1m, where F is the external force per molecule and m is the mass of a molecule. To derive the Navier-Stokes equation, we only have to deduce a more explicit form for Pij We postulate that (5.110) is valid, whatever the coordinate system we choose. It follows that Pi} is a tensor. We assume the fluid under consideration to be isotropic, so that there can be no intrinsic distinction among the axes Xl' Xz, x 3 Accordingly we must havePH = Pzz = P33 == P (5.111) where P is by definition the hydrostatic pressure. Thus Pi} can be written in the form (5.112) Pi} = ~i}P + Pi}where Pi} is a traceless tensor, namely," pI ..., II (5.113)This follows from the fact that (5.113) is true in one coordinate system and that the trace of a tensor is independent of the coordinate system. Next we make the physically reasonable assumption that the fluid element under consideration, which is really a point in the fluid, has no intrinsic angular momentum. This assumption implies that Pij , and hence Pi}' is a symmetric tensor: (5.114) To see this we need only remind ourselves of the meaning of, for example, P1z . A glance at Fig. 5.5a makes (5.114) obvious. Finally we incorporate into Pi} the empirical connection between the shear force applied to a fluid element and the rate of deformation of the same fluid element. A shear force F per unit area acting parallel to a face of a cube of fluidTHERMODYNAMICS AND KINETIC THEORY Fig. 5.5. Nonrotation of fluid element implies P{2Pli . Fig. 5.5b Deformation of fluid element due to shear force. tends to stretch the cube into a parallelopiped at a rate given by R' = p.(dcf>/dt), where p. is the coefficient of viscosity and cf> is the angle shown in Fig. 5.5b. Consider now the effect of P12 on one fluid element. It can be seen from Fig. 5.5c, where P{2 is indicated in its positive sense in accordance with (5.105), thatP2l = _p.(dcf>l + dcf>2) dt dt p. ( x ~ _p.(au 1 + au l ) aX l aX 1 (5.115)In general we have au.) + a x~ (5.116)To make Pi} traceless we must take -Jl [(au au) ax~ + ax~ 3~ij\7 u (5.117)Fig. 5.5e P{2 as shear force. TRANSPORT PHENOMENA Therefore p . = l).p IJ IJ au [( ax. 2 -l)'\7(5.118)which is identical in form to (5.75). This completes the phenomenological derivation, which makes it plausible that the Navier-Stokes equation is valid for dilute gas and dense liquid alike.5.9 EXAMPLES IN HYDRODYNAMICS To illustrate the mathematical techniques of dealing with the equations of hydrodynamics (5.110)-(5.102), we consider two examples of the application of the Navier-Stokes equation to a liquid.