Te Mātāpono me te Whārite a Bernoulli

Ngā Rauemi o te Mātāpono me te Whārite a Bernoulli

KIna tere rawa tā tātou eke motopaika, ka pupuhi ngā kākahu e mau ana tātou i muri. Ki te kore koe e mōhio ki te eke motopaika, kia aro ki ō mātua, ki ō hoa rānei e eke motopaika ana. Ko te tuara o ō rātou kākahu ka pupuhi i muri ina tere te haere o te motopaika. I ētahi wā, ina kaha te pupuhi o te hau, ka taea e te tatau o te whare te kati i a ia anō. Ahakoa kei waho te hau i te whare, i te mea kei roto te tatau i te whare.

Ka taea te whakamārama i tēnei mā te whakamahi i te mātāpono a Bernoulli. I kitea e Daniel Bernoulli (1700–1782) tētahi mātāpono ka taea te whakamahi hei whakamārama ētahi o ngā mea o runga ake nei.

Te Mātāpono a Bernoulli

E ai ki te mātāpono a Bernoulli, ki te tere te rere o te wai, ka iti te pēhanga wai. I tetahi atu taha, ki te iti te tere o te rere o te wai, ka teitei te pēhanga. Ina tere te neke o te motopaika, ka teitei te tere o te hau i mua me te taha o tō tinana. Nō reira, ka iti te pēhanga hau. Ka āraia te tuara o tō tinana e te mua o tō tinana, nō reira kāore te tere o te hau i muri i tō tinana e rerekē ki te teitei (i muri tonu i tō tinana). Nō reira, ka nui ake te pēhanga hau i muri i tō tinana. Nā te mea he rerekē te pēhanga hau, ki te nui ake te pēhanga hau i muri tonu i tō tinana, ka panaia tō hāte e te hau ki muri, ka āhua rite ki te pupuhi i muri.

Ā, me pēhea te tatau e kati ana i a ia anō ina pupuhi kaha te hau i waho? Ka tere ake te neke o te hau i waho i te hau i roto. Nō reira, he iti iho te pēhanga hau i waho i te pēhanga hau i roto. Nā tēnei rerekētanga pēhanga, ki te teitei ake te pēhanga hau i roto, ka panaia te tatau ki waho. Arā, ka neke te tatau mai i tētahi wāhi he teitei ake te pēhanga hau ki tētahi wāhi he iti iho te pēhanga hau.

Te whārite a Bernoulli

I mua, i ako tātou mō te mātāpono a Bernoulli. I whakawhanakehia hoki e Bernoulli tēnei mātāpono mā te tau. Hei whakaputa i te whārite Bernoulli, ka whakaarohia e tātou he rere wai pumau, he wai laminar, he kore e taea te pēhi, he iti te matotoru, ka taea te wareware.

I roto i te kōrero mō te Whārite Tonutanga, i ako mātou ka rerekē anō te tere o te rere o te wai i runga i te horahanga whakawhiti o te ngongo rere. I runga i te mātāpono Bernoulli i whakamāramahia i runga ake nei, ka rerekē anō te pēhanga wai i runga i te tere o te rere o te wai. Ka rerekē anō te pēhanga wai i runga i te teitei o te wai. Ka taea te tiki i te whanaungatanga i waenga i te pēhanga, te tere rere, me te teitei o te rere mai i te whārite Bernoulli.

He mea tino nui te whārite Bernoulli nā te mea ka taea te whakamahi hei tātari i te rere o ngā waka rererangi, ngā tipu hiko wai, ngā pūnaha paipa, me ētahi atu. Hei whakamahi whānui i te whārite Bernoulli ka puta mai i a tātou, ka whakaarohia e tātou ka rere te wai i roto i tētahi ngongo rere me ngā horahanga whakawhiti-kore ōrite me ngā teitei rerekē hoki (tirohia te pikitia i raro nei). Hei whakaputa i te whārite Bernoulli, ka whakamahia e tātou te ariā mahi me te pūngao ki te wai i roto i te rohe ngongo rere. Muri iho, ka tatauhia e tātou te nui o te wai me te mahi i mahia hei neke i te wai.

Ko te tae mātotoru o te ngongo rere i te ahua e tohu ana i te rere o te wai, ko te tae mā ia e tohu ana i te korenga o te wai.

Ka rere te wai i te horahanga whakawhiti-wāhanga 1 (taha maui) i te tawhiti L₁ ā, ka akiaki i te wai i te wāhanga whakawhiti 2 (taha matau) kia neke i te tawhiti L₂Nā te mea he iti ake te horahanga whakawhiti-wāhanga 2 i te taha matau, he nui ake te tere rere o te wai i te taha matau o te ngongo rere (Kia maumahara ki te whārite haere tonu). Ka puta he rerekētanga pēhanga i waenga i te whakawhiti-wāhanga 2 (te taha matau o te ngongo rere) me te whakawhiti-wāhanga 1 (te taha maui o te ngongo rere) – Kia maumahara ki te mātāpono a Bernoulli. Ko te wai i te taha maui o te whakawhiti-wāhanga 1 ka whakaputa i te pēhanga P₁ i runga i te wai i te taha matau, ā, e mahi ana i ngā mahi e whai ake nei:

Kātahi ka puta te whārite W₁ ka taea te tuhi penei:

W₁ = wh₁ A₁ L₁

I te wāhanga whakawhiti 2 (te taha matau o te ngongo rere), ko te mahi i mahia ki te wai ko:

W₂ = − p₂ A₂ L₂

Ko te tohu kino e tohu ana kei te ritenga kē te kaha e whakamahia ana ki te ahunga o te nekehanga. Nō reira, ka mahi te wai ki te taha matau o te wāhanga 2. Hei tāpiri, ka mahi anō hoki te kaha ā-papa ki te wai. I roto i te take i runga ake nei, ka nekehia tētahi papatipu wai mai i te wāhanga 1 mā te tawhiti L.₁ ki te whakawhiti i te wāhanga 2 tae noa ki te L₂, ko te rōrahi o te wai i te wāhanga whakawhiti 1 (A₁ L₁) = te rōrahi o te wai i te wāhanga whakawhiti 2 (A₂ L₂Ko te mahi e mahia ana e te kaha ā-papa ko:

W₃ = − mg (h₂ - h₁)

W₃ = − mgh₂ + mgh₁)

W₃ = mgh₁ - mgh₂

Ko te tohu kino nā te rere o te wai ki runga, e anga whakarara ana ki te ahunga o te kaha ā-papa. Nō reira, ko te mahi katoa i mahia ki te wai, e whakaaturia ana i te pikitia i runga ake nei, koia tēnei:

W = W₁ + W₂ + W₃

W = P₁ A₁ L₁ - P₂ A₂ L₂ + mgh₁ - mgh₂

E ai ki te ariā mahi-pūngao, ko te katoa o te mahi i mahia i runga i tētahi pūnaha he rite ki te huringa o tōna pūngao nekeneke. Nō reira, ka taea e tātou te whakakapi i te Mahi (W) ki te huringa o te pūngao nekeneke (EK).₂ ‐ EK₁).

Ka taea e tātou te tuhi anō i te whārite i runga ake nei penei:

W = P₁ A₁ L₁ - P₂ A₂ L₂ + mgh₁ - mgh₂

EK₂ ‐ EK₁ =P₁ A₁ L₁ - P₂ A₂ L₂ + mgh₁ - mgh₂

1⁄2 mv₂² – 1⁄2 mv₁² =P₁ A₁ L₁ - P₂ A₂ L₂ + mgh₁ - mgh₂

Te papatipu o te wai e rere ana i te tawhiti L₁ i te wāhanga whakawhiti A₁ = te papatipu o te wai e rere ana i te tawhiti L₂ (wāhanga whakawhiti A₂). Ko tētahi papatipu wai, me kī ko m, he A te rōrahi.₁L₁ me A₂ L₂ kei hea a A₁ L₁ =A₂ L₂ (L₂ roa atu i te L₁).

Nā, ka whakakapia, ka whakakapia rānei a m i te whārite i runga ake nei ki a m = ρ AL:

Koinei te whārite Bernoulli. I ahu mai i a mātou te whārite Bernoulli i runga i te mātāpono mahi-pūngao, nō reira he momo o te Ture Tiaki Pūngao.

Ngā Mōhiohio:

P = Pēhanga

ρ = Te mātotoru o te wai

v = Te tere rere o te wai

g = Te whakaterenga nā te kaha ā-papa

h = Teitei o te ngongo/paipa rere mai i te mata o te whenua

Ka taea e ngā taha maui me ngā taha matau o te whārite Bernoulli i runga ake nei te kōrero mō ngā pūwāhi e rua i te taha o te ngongo rere, nō reira ka taea e tātou te tuhi anō i te whārite i runga ake nei:

E kī ana tēnei whārite he ōrite te uara o te tapeke o ngā rahinga i roto i te whārite puta noa i te ngongo rere.

Nā, me arotake tātou i te whārite a Bernoulli mō ētahi take.

Te Whārite a Bernoulli mō ngā Wai e Okioki ana

Ko tētahi tauira motuhake o te whārite a Bernoulli ko ngā wai e okioki ana (ngā wai pūmau). Ina okioki te wai, kāore he tere. Nō reira, ko v₁ = v₂ = 0. Mō te wai kore e neke, ka taea e tātou te whakatakoto i te whārite Bernouli penei:

Mena h₂ - h₁ = h, kātahi ka taea te tuhi i tēnei whārite penei:

p₁ -p₂ = ρ g (h₂ - h₁)

p₁ -p₂ = ρ gh

Te whārite a Bernoulli mō ngā ngongo rere, ngā paipa rānei o te teitei ōrite

Mena he ōrite te teitei o te ngongo rere, o te paipa rānei, ka hurihia te whārite Bernoulli ki: