By Bernhard Weigand
This booklet describes valuable analytical tools by means of utilizing them to real-world difficulties instead of fixing the standard over-simplified school room difficulties. The ebook demonstrates the applicability of analytical equipment even for complicated difficulties and courses the reader to a extra intuitive realizing of techniques and solutions.
Although the answer of Partial Differential Equations through numerical tools is the normal perform in industries, analytical tools are nonetheless very important for the serious evaluation of effects derived from complicated desktop simulations and the advance of the underlying numerical innovations. Literature dedicated to analytical equipment, even though, frequently makes a speciality of theoretical and mathematical elements and is as a result lifeless to so much engineers. Analytical equipment for warmth move and Fluid movement difficulties addresses engineers and engineering students.
The moment version has been up to date, the chapters on non-linear difficulties and on axial warmth conduction difficulties have been prolonged. And labored out examples have been included.
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Additional info for Analytical Methods for Heat Transfer and Fluid Flow Problems
110), one notices that a physically plausible solution occurs for both þk2 and Àk2 . In order to investigate this problem further, we analyse in more detail the solutions for the function Gð~yÞ. From Eq. 110) we get G001 ð~yÞ þ k2 G1 ð~yÞ ¼ 0 for þk2 ð2:111Þ G002 ð~yÞ À k2 G2 ð~yÞ ¼ 0 for Àk2 ð2:112Þ which gives rise to the following two possible solutions G1 ð~yÞ ¼ C3 cosðk~yÞ þ C4 sinðk~yÞ ð2:113Þ G2 ð~yÞ ¼ C3 coshðk~yÞ þ C4 sinhðk~yÞ ð2:114Þ If we now reconsider the problem to be solved (Eqs.
3 Separation of Variables 39 where the pressure gradient in the axial direction is constant for a hydrodynamically fully developed flow. The boundary conditions are the no slip conditions at all boundaries of the channel. Thus wðb=2; yÞ ¼ 0; wðx; c=2Þ ¼ 0; wðÀb=2; yÞ ¼ 0 wðx; Àc=2Þ ¼ 0 ð2:130Þ ¼ À1=l @p=@z, it can be seen If we introduce into Eq. 129) the abbreviation K that the problem for determining the fully developed velocity proﬁle is identical to the heat conduction problem in a plate containing a heat sink with constant sink intensity (the derivation of the fully developed velocity ﬁeld in a rectangular channel is given for example in Spurk (1987)).
110). 36 2 Linear Partial Differential Equations For the function Fð~xÞ, one obtains from Eq. 110) F 00 ð~xÞ À k2 Fð~xÞ ¼ 0 ð2:115Þ Fð~xÞ ¼ C5 coshðk~xÞ þ C6 sinhðk~xÞ ð2:116Þ which has the solution Combining the solutions for F and G leads to the following expression for Hh Hh ¼ ðC3 cosðk~yÞ þ C4 sinðk~yÞÞðC5 coshðk~xÞ þ C6 sinhðk~xÞÞ ð2:117Þ This expression has to satisfy the boundary conditions given by Eq. 107) À Á À Á 1 À ~y2 ; Hh ðÀA; ~yÞ ¼ ÀK=2 1 À ~y2 Hh ðA; ~yÞ ¼ ÀK=2 Hh ð~x; À1Þ ¼ 0 Hh ð~x; 1Þ ¼ 0; ð2:107Þ Applying the two boundary conditions for ﬁxed values of ~y, the following two equations are obtained C3 cosðkÞ þ C4 sinðkÞ ¼ 0 ð2:118Þ C3 cosðkðÀ1ÞÞ þ C4 sinðkðÀ1ÞÞ ¼ 0 Because cosðkÞ ¼ cosðÀkÞ and sinðkÞ ¼ À sinðÀkÞ, one obtains from the above equations that C4 = 0 and that C3 cosðkÞ ¼ 0 ð2:119Þ From this equation it follows that k¼ 2n À 1 p; 2 n ¼ 1; 2; 3; .
Analytical Methods for Heat Transfer and Fluid Flow Problems by Bernhard Weigand