Name: laminar flow profile between parallel plates
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 17 s
Channel: JoVE

İki paralel plaka arasındaki sabit, laminer akış, genellikle basit akışkan sistemlerini modellemek için kullanılan dar, sınırlı bir kanaldaki sıvı hareketini tanımlar.
İki sabit plaka arasındaki akış için, Navier-Stokes denklemleri, hızın akış yönü boyunca ivme olmaksızın plakalara sadece dik olarak değişmesi nedeniyle basitleştirilir.
Bu, sıvının merkezde en hızlı hareket ettiği ve kaymama durumu olarak bilinen sürtünme nedeniyle plakalarda sıfıra yavaşladığı parabolik bir hız profiline yol açar.
Plakalar arasındaki akış hızı, basınç gradyanına, akışkan viskozitesine ve plaka ayrımına bağlıdır; boşluğu genişletmek veya basınç gradyanını artırmak akışı artırır.
Ortalama akış hızı aynı zamanda basınç düşüşü ile de ilgilidir ve kanalın merkezindeki tepe hızı ortalama hızın 1.5 katıdır.
Bu analiz yalnızca laminer koşulları gösteren düşük Reynolds sayıları için geçerli kalır ve türbülanslı, yüksek Reynolds sayısı akışları için başarısız olur.
Bu analizin uygulamaları, laminer koşulların mevcut olduğu su arıtma tesislerinde sulama kanalları, yeraltı suyu akışı ve çökeltme havuzlarını içerir.

paralel levhalar arasında sabit, laminer akış

Steady, Laminar Flow Between Parallel Plates

Paralel levhalar arasında sabit, laminer akışı anlamak, çeşitli su iletim ve drenaj sistemlerinde yaygın olarak bulunan dar dikdörtgen kanallardaki akışı analiz etmek ve tasarlamak için önemlidir. Navier-Stokes denklemleri akışkan hareketini yönetir ve doğrusal olmamaları nedeniyle genellikle çözülmesi zordur. Ancak, paralel levhalar arasındaki sabit laminer akış gibi belirli durumlarda basitleştirmeler mümkündür. Bu senaryo için sabit, sıkıştırılamaz, laminer akış varsayıyoruz.

 X ekseni boyunca hizalanmış ve y ekseni levhalara dik olacak şekilde '2h' mesafe aralıklı iki yatay levha düşünüldüğünde, akışkan hareketinin paralel olduğu varsayılır. Burada, süreklilik denklemi x ekseni boyunca hızda bir değişiklik olmadığını gösterir ve Navier-Stokes denklemleri aşağıdaki gibi azalır:

<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;">


 Y ve z yönlerindeki denklemler bu eksenler boyunca bir basınç gradyanı göstermez, bu da basıncın y yönünde hidrostatik olarak değiştiğini gösterir ve entegrasyonun şunu vermesine olanak tanır:

<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;">


 X ekseni boyunca bir basınç gradyanı olan paralel plakalar için denklem basitleşir:

<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;">


 İki kez entegre ederek ve sınır koşullarını uygulayarak (y = +h ve y = -h'de u = 0), şunu elde ederiz:

<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;">

 Bu ifade, plakalar arasındaki boşluktaki hız profilinin parabolik olduğunu ve plakaların ortasında y = 0'da maksimum hızın olduğunu ortaya koyar. Plakaların birim genişliğindeki akışın debisi, boşluk boyunca hız profilini entegre ederek elde edilir.

 Paralel plakalar arasındaki laminer akışı anlamak, yeraltı suyu akışında, atık su arıtımında ve yağlamada pratik uygulamalara sahiptir. Örneğin, bu model, suyun plakalar arasındaki akış gibi toprak parçacıkları arasında hareket ettiği toprak katmanlarındaki akışı yaklaşık olarak hesaplayabilir. Debisini ve hız dağılımını bilmek, mühendislerin verimli drenaj sistemleri tasarlamalarına ve yeraltı suyu hareketini tahmin etmelerine yardımcı olur.

Paralel Levhalar Arasında Sabit, Laminer Akış

İki paralel plaka arasındaki sabit, laminer akış, genellikle basit akışkan sistemlerini modellemek için kullanılan dar, sınırlı bir kanaldaki sıvı hareketini tanımlar.
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Education

JoVE Core

Civil Engineering

İnşaat Mühendisliği

Unit 1

Ünite 1

Differential Analysis of Fluid Flow

Akışkan Akışının Diferansiyel Analizi

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

laminar flow profile between parallel plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings.

The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem becomes simpler because the flow is steady, incompressible, and laminar. Two horizontal plates are placed along the x-axis, and the gap between them is 2h. The y-axis is perpendicular to the plates, and the fluid moves parallel to the plates.

With these assumptions, the continuity equation shows that the velocity does not change along the x-axis. The Navier-Stokes equations also simplify in the y and z directions. They show no pressure gradient along those axes, so pressure changes hydrostatically in the y-direction. After integration, the x-direction equation includes the pressure gradient along the plates.

Solving that equation requires integrating twice and applying the no-slip boundary conditions, where u = 0 at y = +h and y = -h. The result is a parabolic velocity profile across the gap. The fluid moves fastest at y = 0, the midpoint between the plates. The volumetric flow rate for a unit width of the plates is then found by integrating the velocity profile across the space.

This flow model has practical uses in groundwater flow, wastewater treatment, and lubrication. It can also represent water moving through soil layers, where the flow resembles movement between plates. Engineers use the velocity distribution and flow rate to help design drainage systems and predict groundwater movement.

Laminar Flow Profile Between Parallel Plates

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Learn how steady laminar flow between parallel plates leads to a parabolic velocity profile, pressure-gradient behavior, and flow-rate calculations.

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civil engineering

Laminar Flow Profile Between Parallel Plates

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