}.t 'V u=O in Java

Paint qr barcode in Java }.t 'V u=O
}.t 'V u=O
Qr Barcode barcode library in java
using barcode generating for java control to generate, create qr barcode image in java applications.
'V 2 u
Barcode barcode library on java
use java bar code implementation tointegrate bar code with java
= - 'VP
recognizing barcode in java
Using Barcode scanner for Java Control to read, scan read, scan image in Java applications.
Control qr bidimensional barcode data with visual c#
to add qr code iso/iec18004 and qrcode data, size, image with c#.net barcode sdk
with the boundary condition that the fluid sticks to the sphere. Let us translate the coordinate system so that the sphere is at rest at the origin while the fluid at infinity flows with uniform constant velocity uo. The equations (5.133) remain invariant under the translation, whereas the boundary conditions become
.net Vs 2010 qr-code encoding for .net
using barcode implement for .net control to generate, create qr bidimensional barcode image in .net applications.
.net Framework qrcode integrating with vb.net
use vs .net quick response code writer tomake qr code for visual basic
u(r) ~
Control qr codes data with java
qr code data with java
Bar Code barcode library for java
use java barcode printer toaccess bar code in java
Taking the divergence of both sides of the first equation of (5.133), we obtain "V 2 p = 0 (5.135) Thus the pressure, whatever it is, must be a linear superpoSItIOn of solid harmonics. A systematic way to proceed would be to write P as the most general superposition of solid harmonics and to determine the coefficient by requiring that (5.133) be satisfied. We take a short cut, however, and guess that P is, apart from an additive constant, a pure solid harmonic of order 1:
Java 2d matrix barcode writer for java
use java 2d barcode generator toencode 2d barcode for java
cos () Po + J-tP l - 2r
Pdf417 2d Barcode barcode library for java
using java touse pdf 417 on asp.net web,windows application
Java barcode creation with java
using barcode integration for java control to generate, create barcode image in java applications.
where Po and Pi are constants to be determined later. With this, the problem reduces to solving the inhomogeneous Laplace equation
Java identcode encoding with java
use java identcode generator tocompose identcode with java
"V 2U = P1"V - r2
Add ean13+2 with .net c#
using barcode creator for web form crystal control to generate, create gtin - 13 image in web form crystal applications.
cos ()
ReportingService Class 2d matrix barcode generation in .net
use ms reporting service matrix barcode integrating toproduce 2d barcode for .net
Deploy bar code on .net
generate, create bar code none in .net projects
subject to the conditions
Control image for word documents
using word toassign ucc ean 128 in asp.net web,windows application
"V u=o [u(r)]
VS .NET code 3/9 recognizer on .net
Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications.
r=a =
Code 39 Full ASCII maker on .net
using .net toproduce barcode 3 of 9 for asp.net web,windows application
Control code128b image with microsoft excel
using barcode printing for microsoft excel control to generate, create code-128 image in microsoft excel applications.
u(r) ~
A particular solution of (5.137) is
u = - Pi r 2 "V cos () = _ Pi 1 6 r2 6
3r~) 3
where denotes the unit vector along the z axis, which lies along uo. It is easily verified that (5.139) solves (5.137), if we note that l/r and z/r 3 are both solid harmonics. Thus,
2 = U1
:1 [- 3"V 2( :: )]
= P 1"V
= P1"V c:s
() 2
The complete solution is obtained by adding an appropriate homogeneous solution to (5.139) to satisfy (5.138). By inspection we see that the complete solution is
where we have set
to have 'V u = o. We now calculate the force acting on the sphere by the fluid. By definition the force per unit area acting on a surface whose normal point along the x) axis is - T) of (5.107). It follows that the force per unit area acting on a surface element of the sphere is
f = (
-T1 + -T2 + -T3 r r r
-f P
where f is the unit vector in the radial direction and total force experienced by the sphere is
P is given by (5.118). The
where dS is a surface element of the sphere and the integral extends over the entire surface of the sphere. Thus it is sufficient to calculate f for r = a. The vector f P has the components
A.... 1 1 au) ( r P).= -x.p.. = -xJ [ i)p-II. ( -ax. I r J JI r JI r
au i + - )] ax.
Xi }.t a a -P - - [ -(xu) - u + x - u ] r r ax I J J I J ax. I J
where P is given by (5.136) and (5.142), and u is given by (5.141). Since u = 0 when r = a, we only need to consider the first and the last terms in the bracket. The first term is zero at r = a by a straightforward calculation. At r = a the second term is found to be
1 -[(r 'V)u] r~a r
- - - -fu - r-a
3 Uo
cos ()
When this is substituted into (5.145), the second term exactly cancels the dipole part of fP, and we obtain
The constant Po is unknown, but it does not contribute to the force on the sphere. From (5.144) we obtain
which is Stokes' law.
The validity of (5.141) depends on the smallness of the material derivative of u as compared to /L'V 2U. Both these quantities can be computed from (5.141). It is then clear that we must require
Thus Stokes' law holds only for small velocities and small radii of the sphere. A more elaborate treatment shows that a more accurate formula for F' is
= 67T/LaU o
+ 8" -/L- + ...
3 puoa
The pure number puoa//L is called the Reynolds number. When the Reynolds number becomes large, turbulence sets in and streamline motion completely breaks down.
5.1 Make order-of-magnitude estimates for the mean free path and the collision time for
(a) Hz molecules in a hydrogen gas in standard condition (diameter of Hz (b) protons in a plasma (gas of totally ionized Hz) at T
= 2.9 A); 10 5 K, n = 1015
protons/cm3 , 0 = 'lTrz, where r = e 2 /kT; (c) protons in a plasma at the same density as (b) but at T = 10 7 K, where thermonuclear reactions occur; (d) protons in the sun's corona, which is a plasma at T = 10 6 K, n = 10 6 protons/cc; (e) slow neutrons of energy 0.5 MeV in 238U (0"" 'lTr z , r"" 10- 13 cm).