The 7 reference contexts in paper P. Vlasov A., П. Власов А. (2016) “Влияние равномерного движения границы на температурное поле полупространства, подверженного нагреву внешним тепловым потоком // The Influence of the Boundary Motion on Temperature Field in the Half-Space Under External Heat Flux” / spz:neicon:technomag:y:2014:i:8:p:101-109

  1. Start
    1521
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    ›ÚË Ô‰ÒÚ‡‚ÎÂÌˡ ÏÓ„ÛÚ · ̊Ú ̧ ËÒÔÓÎ ̧ÁÓ‚‡Ì ̊ Ì ÚÓÎ ̧ÍÓ ‰Îˇ Ôӂ‰ÂÌˡ Ô‡‡ÏÂÚË ̃ÂÒÍÓ„Ó ‡Ì‡ÎËÁ‡ ÚÂÏÔÂ‡ÚÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ò ˆÂÎ ̧ ̨ ÓÔÚËÏËÁ‡ˆËË Â ÚÂÔÎÓ‚ÓÈ Á‡ ̆ËÚ ̊, ÌÓ Ë ‰Îˇ ÚÂÒÚËÓ‚‡Ìˡ ‚ÌÓ‚ ̧ ÒÓÁ‰‡‚‡ÂÏ ̊ı ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ı ÍÓÏÔÎÂÍÒÓ‚. —ÎÓÊÌÓÒÚË, ‚ÓÁÌË͇ ̨ ̆Ë ÔË ÔÓËÒÍ ڇÍËı  ̄ÂÌËÈ, ıÓÓ ̄Ó ÓÔËÒ‡Ì ̊ ‚ ÎËÚÂ‡ÚÛÂ
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    [1]
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    . —ΉÛÂÚ Á‡ÏÂÚËÚ ̧, ̃ÚÓ ‚Ó ÏÌÓ„Ëı Ô‡ÍÚË ̃ÂÒÍË ‚‡ÊÌ ̊ı ËÌÊÂÌÂÌ ̊ı Á‡‰‡ ̃‡ı ÔÓˆÂÒÒ ÙÓÏËÓ‚‡Ìˡ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ‚ ËÁÛ ̃‡ÂÏÓÈ ÍÓÌÒÚÛ͈ËË ÒÓÔÓ‚Óʉ‡ÂÚÒˇ ÛÌÓÒÓÏ ̃‡ÒÚË ‚ ̆ÂÒÚ‚‡ Ò ÔÓ‚ÂıÌÓÒÚË, ̃ÚÓ ÔË‚Ó‰ËÚ Í ËÁÏÂÌÂÌË ̨  „‡Ìˈ ‚Ó ‚ÂÏÂÌË.
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  2. Start
    1980
    Prefix
    ÕÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ÔÓ‰‚ËÊÌÓÒÚË „‡Ìˈ ËÁÛ ̃‡ÂÏÓÈ Ó·Î‡ÒÚË ÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÒÎÓÊÌˇÂÚ ÔÓÎÛ ̃ÂÌË ‡Ì‡ÎËÚË ̃ÂÒÍËı  ̄ÂÌËÈ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃ ‰‡Ê ‚ ÔÓÒÚÂÈ ̄Ëı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ
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    [1,2,3,4,5]
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    . “‡Í, ‚ ‡·ÓÚ [6] Á‡‰‡ ̃‡ ̇ıÓʉÂÌˡ ÚÂÏÔÂ‡ÚÛ ̊ ÔÓ‰‚ËÊÌÓÈ „‡Ìˈ ̊ ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡ ‚ ÛÒÎӂˡı ÌÂÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÔÎÓÓ·ÏÂ̇ Ò ‚Ì ̄ÌÂÈ Ò‰ÓÈ Ò‚Â‰Â̇ Í Á‡‰‡ ̃  ̄ÂÌˡ Û‡‚ÌÂÌˡ ¬ÓÎ ̧ÚÂ‡. ÷ÂÎ ̧ ̨ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ̊ ˇ‚ΡÂÚÒˇ ËÁÛ ̃ÂÌË ‚ÎˡÌˡ ‡‚ÌÓÏÂÌÓ„Ó ‰‚ËÊÂÌˡ „‡Ìˈ ̊ Ú‚Â‰Ó„Ó Ú·, ÏÓ‰ÂÎËÛÂÏÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚ÓÏ, ̇ ÔÓˆÂÒÒ ÙÓÏËÓ‚‡Ìˡ Â„Ó ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ‚ ÛÒÎӂˡı ‚ÓÁ‰ÂÈÒڂˡ ÒÚ
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  3. Start
    2007
    Prefix
    ÕÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ÔÓ‰‚ËÊÌÓÒÚË „‡Ìˈ ËÁÛ ̃‡ÂÏÓÈ Ó·Î‡ÒÚË ÒÛ ̆ÂÒÚ‚ÂÌÌÓ ÛÒÎÓÊÌˇÂÚ ÔÓÎÛ ̃ÂÌË ‡Ì‡ÎËÚË ̃ÂÒÍËı  ̄ÂÌËÈ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃ ‰‡Ê ‚ ÔÓÒÚÂÈ ̄Ëı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ [1,2,3,4,5]. “‡Í, ‚ ‡·ÓÚÂ
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    [6]
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    Á‡‰‡ ̃‡ ̇ıÓʉÂÌˡ ÚÂÏÔÂ‡ÚÛ ̊ ÔÓ‰‚ËÊÌÓÈ „‡Ìˈ ̊ ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡ ‚ ÛÒÎӂˡı ÌÂÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÔÎÓÓ·ÏÂ̇ Ò ‚Ì ̄ÌÂÈ Ò‰ÓÈ Ò‚Â‰Â̇ Í Á‡‰‡ ̃  ̄ÂÌˡ Û‡‚ÌÂÌˡ ¬ÓÎ ̧ÚÂ‡. ÷ÂÎ ̧ ̨ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ̊ ˇ‚ΡÂÚÒˇ ËÁÛ ̃ÂÌË ‚ÎˡÌˡ ‡‚ÌÓÏÂÌÓ„Ó ‰‚ËÊÂÌˡ „‡Ìˈ ̊ Ú‚Â‰Ó„Ó Ú·, ÏÓ‰ÂÎËÛÂÏÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚ÓÏ, ̇ ÔÓˆÂÒÒ ÙÓÏËÓ‚‡Ìˡ Â„Ó ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ‚ ÛÒÎӂˡı ‚ÓÁ‰ÂÈÒڂˡ ÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÔÎÓ‚Ó
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  4. Start
    3669
    Prefix
    ÔË Í‡Ê‰ÓÏ ÙËÍÒËÓ‚‡ÌÌÓÏ Á̇ ̃ÂÌËËFo>0ÙÛÌÍˆËˇθ(ξ,Fo)ËÌÚ„ËÛÂχ Ò Í‚‡‰‡ÚÓÏ ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈ ÔÂÂÏÂÌÌÓÈξ∈[V0Fo,+∞). ÓÏ ÚÓ„Ó, ·Û‰ÂÏ Ô‰ÔÓ·„‡Ú ̧, ̃ÚÓ ‰Îˇ Í‡Ê‰Ó„Ó Á̇ ̃ÂÌˡξÙÛÌÍˆËˇθ(ξ,·)ˇ‚ΡÂÚÒˇ ÓË„Ë̇ÎÓÏ ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇
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    [7]
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    . ƒÎˇ  ̄ÂÌˡ ‡ÒÒχÚË‚‡ÂÏÓÈ Á‡‰‡ ̃Ë Û‰Ó·ÌÓ Ò‰ÂÎ‡Ú ̧ Á‡ÏÂÌÛ ÔÂÂÏÂÌÌ ̊ı x=ξ−V0Fo, τ= Fo, ÔÂÂȉˇ ‚ ÒËÒÚÂÏÛ ÍÓÓ‰Ë̇Ú, Ò‚ˇÁ‡ÌÌÛ ̨ Ò ‰‚ËÊÛ ̆ÂÈÒˇ „‡ÌˈÂÈ ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡. ¬ ̋ÚÓÏ ÒÎÛ ̃‡Â ÏÓ‰ÂÎ ̧ (1){(3) ÔËÏÂÚ ÒÎÂ‰Û ̨ ̆ËÈ ‚ˉ: ∂θ(x, τ) ∂τ = ∂2θ(x, τ) ∂x2 +V0 ∂θ(x, τ) ∂x , x >0, τ >0,(5) θ(x,0) = 0, x >0,(6) ∂θ(x, τ) ∂x ∣ ∣ ∣ ∣ x=0 =−Q, τ>0,(7) ‡ ÛÒÎÓ‚Ë (4) Á‡ÔË ̄ÂÚÒˇ ‚ ‚ˉ θ(x, τ) ∣ ∣
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  5. Start
    5090
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    “‡ÍËÏ Ó·‡ÁÓÏ,  ̄ÂÌË Á‡‰‡ ̃Ë (9)|(11) ÏÓÊÌÓ Ô‰ÒÚ‡‚ËÚ ̧ ‚ ‚ˉ u(x, s) = Q s(V1+ √ V21+s) e−x(V1+ √ V21+s).(13) ÕÂÔÓÒ‰ÒÚ‚ÂÌÌ ̊È ÔÂÂıÓ‰ ÓÚ ËÁÓ·‡ÊÂÌˡu(x, s), ÓÔ‰ÂΡÂÏÓ„Ó ‡‚ÂÌÒÚ‚ÓÏ (13), Í ÓË„Ë̇ÎÛθ(x, τ)Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ Ú‡·Îˈ <ËÁÓ·‡ÊÂÌË | ÓË„Ë̇Î> ‚ÓÁÏÓÊÂÌ
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    [8]
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    , ÌÓ ÌÂ Ó ̃ÂÌ ̧ Û‰Ó·ÂÌ. œÓ ̋ÚÓÏÛ ‰Îˇ ̇ıÓʉÂÌˡ  ̄ÂÌˡ Á‡‰‡ ̃Ë (5)|(8) ‚ÓÒÔÓÎ ̧ÁÛÂÏÒˇ ÙÓÏÛÎÓÈ Ó·‡ ̆ÂÌˡ [7] θ(x, τ) = 1 2πi σ+i∞∫ σ−i∞ esτu(x, s)ds.(14) ‘ÛÌÍˆËˇu(x, s)ËÏÂÂÚ ‰‚ ÓÒÓ· ̊ ÚÓ ̃ÍËs= 0Ës=−V21=−V20/4, ÍÓÚÓ ̊ ˇ‚Ρ ̨ÚÒˇ ÔÓÒÚ ̊Ï ÔÓÎ ̨ÒÓÏ Ë ÚÓ ̃ÍÓÈ ‚ÂÚ‚ÎÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ, ÔÓ ̋ÚÓÏÛ ‡ÒÒÏÓÚËÏ Ô‰ÒÚ‡‚ÎÂÌÌ ̊È Ì‡ ËÒ.1ÍÓÌÚÛK, „‰Â x A1A2, x A5A6Ë x A3A4, | ‰Û„Ë ÓÍÛÊÌÓÒÚÂÈ ‡‰ËÛÒÓ‚RËρÒ
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  6. Start
    5199
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    “‡ÍËÏ Ó·‡ÁÓÏ,  ̄ÂÌË Á‡‰‡ ̃Ë (9)|(11) ÏÓÊÌÓ Ô‰ÒÚ‡‚ËÚ ̧ ‚ ‚ˉ u(x, s) = Q s(V1+ √ V21+s) e−x(V1+ √ V21+s).(13) ÕÂÔÓÒ‰ÒÚ‚ÂÌÌ ̊È ÔÂÂıÓ‰ ÓÚ ËÁÓ·‡ÊÂÌˡu(x, s), ÓÔ‰ÂΡÂÏÓ„Ó ‡‚ÂÌÒÚ‚ÓÏ (13), Í ÓË„Ë̇ÎÛθ(x, τ)Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ Ú‡·Îˈ <ËÁÓ·‡ÊÂÌË | ÓË„Ë̇Î> ‚ÓÁÏÓÊÂÌ [8], ÌÓ ÌÂ Ó ̃ÂÌ ̧ Û‰Ó·ÂÌ. œÓ ̋ÚÓÏÛ ‰Îˇ ̇ıÓʉÂÌˡ  ̄ÂÌˡ Á‡‰‡ ̃Ë (5)|(8) ‚ÓÒÔÓÎ ̧ÁÛÂÏÒˇ ÙÓÏÛÎÓÈ Ó·‡ ̆ÂÌˡ
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    [7]
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    θ(x, τ) = 1 2πi σ+i∞∫ σ−i∞ esτu(x, s)ds.(14) ‘ÛÌÍˆËˇu(x, s)ËÏÂÂÚ ‰‚ ÓÒÓ· ̊ ÚÓ ̃ÍËs= 0Ës=−V21=−V20/4, ÍÓÚÓ ̊ ˇ‚Ρ ̨ÚÒˇ ÔÓÒÚ ̊Ï ÔÓÎ ̨ÒÓÏ Ë ÚÓ ̃ÍÓÈ ‚ÂÚ‚ÎÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ, ÔÓ ̋ÚÓÏÛ ‡ÒÒÏÓÚËÏ Ô‰ÒÚ‡‚ÎÂÌÌ ̊È Ì‡ ËÒ.1ÍÓÌÚÛK, „‰Â x A1A2, x A5A6Ë x A3A4, | ‰Û„Ë ÓÍÛÊÌÓÒÚÂÈ ‡‰ËÛÒÓ‚RËρÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ Ò ˆÂÌÚ‡ÏË ‚ ÚÓ ̃ÍÂ−V21=−V20/4. œÓÒÍÓÎ ̧ÍÛ Ó·ıÓ‰ ÍÓÌÚÛ‡ ÒÓ‚Â ̄‡ÂÚÒˇ ÔÓÚË‚ ̃‡ÒÓ‚ÓÈ ÒÚÂÎÍË, ÚÓ
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  7. Start
    6142
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    ÃÓÊÌÓ ÔÓ͇Á‡Ú ̧, ̃ÚÓ ÒÛ ̆ÂÒÚ‚Û ̨ÚM0>0ËR0>0, Ú‡ÍËÂ, ̃ÚÓ ‰Îˇ ‚ÒÂıR>R0,x>0ÙÛÌÍˆËˇu(x, s) ÔËs, ÔË̇‰ÎÂʇ ̆ÂÏ ÓÍÛÊÌÓÒÚË{z∈C:|z+V21|=R}, Û‰Ó‚ÎÂÚ‚ÓˇÂÚ ÌÂ‡‚ÂÌÒÚ‚Û |u(x, s)|6 M0 R2 . ¬ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò ÎÂÏÏÓÈ ∆Ó‰‡Ì‡
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    [11]
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    ËÁ ̋ÚÓ„Ó ÒΉÛÂÚ, ̃ÚÓ I1= 1 2πi lim R→+∞ ∫ x A1A2∪ x A5A6 esτu(x, s)ds= 0.(16) ≈ÒÎËs∈ x A3A4, ÚÓs=ρeiφ−V21,ds=ρieiφdφ, Ë, Òӄ·ÒÌÓ (13), I2= 1 2πi lim ρ→0+ ∫ x A3A4 esτu(x, s)ds= = Qe−xV1 2πi ∫ −ππ ( lim ρ→0+ eρτexpiφe−x √ρexpiφ/2 (ρeiφ−V21) √ ρeiφ/2 ρieiφ ) dφ= 0.(17) ƒÎˇ ‚ ̊ ̃ËÒÎÂÌˡ ËÌÚ„‡Î‡ I3= 1 2πi lim R→+∞ ρ→0+ ∫ → A2A3∪ → A4A5 esτu(x, s)ds(18) ‚‰ÓÎ ̧ ‡ÁÂÁ‡ → A2A3∪ → A4A5҉·ÂÏ Ô‰‚
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