Name: parabolic flow in circular tubes
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 23 s
Channel: JoVE

Hagen-Poiseuille akışı olarak bilinen, sabit yarıçaplı R olan düz silindirik bir tüpten geçen sabit, sıkıştırılamaz akışı düşünün.
Silindirik geometri, radyal, açısal ve eksenel ölçümler için r, θ ve z koordinatlarının kullanılmasına izin verir.
Tüpün eksenine paralel olarak akan sıvı, tamamen eksenel bir hız bileşenine sahiptir ve yalnızca radyal mesafe ile değişir.
Bu koşullar Navier-Stokes denklemlerini basitleştirerek, hızın merkezde en yüksek olduğunu ve duvarlara doğru azaldığını ve sınırda sıfıra ulaştığını gösteren parabolik bir hız profilini ortaya çıkarır.
Bu, sıvı parçacıklarının organize katmanlar halinde düzgün bir şekilde hareket ettiği laminer akışı gösterir.
Bu profili tüpün enine kesitine entegre etmek, akış hızını Q tüpün yarıçapının, sıvı viskozitesinin ve basınç gradyanının dördüncü kuvvetine bağlı olarak ifade eden Poiseuille yasasını verir.
Q'dan hesaplanan ortalama hız V, tüpün merkezinde gözlemlenen maksimum hızın yarısıdır.
Bu hız dağılımı, akışın merkezde en yüksek olduğunu ve duvarda parabolik olarak sıfıra düştüğünü doğrular.

dairesel borularda sabit, laminer akış

Steady, Laminar Flow in Circular Tubes

Hagen-Poiseuille akışı, sabit yarıçapı R olan silindirik bir borudan geçen viskoz bir akışkanın sabit, sıkıştırılamaz akışını tanımlar. Bu akış profili genellikle kılcal damarlar gibi dar kanallardaki akışkan taşınımını anlamak için uygulanır. Laminer akışın temel bir örneği olarak hizmet eder. Bu modelde, boru içindeki radyal (r), açısal (θ) ve eksenel (z) boyutları tanımlamak için silindirik koordinatlar (r,θ,z) kullanılır. Hagen-Poiseuille akışı için hız profili tamamen ekseneldir, yani hız vektörü yalnızca z ekseni boyunca işaret eder ve yalnızca borunun merkezinden radyal mesafe r ile değişir.

 Akışkan hareketinin yönetici denklemleri olan Navier-Stokes denklemleri bu koşullar altında basitleşir. Açısal veya radyal hız bileşenleri olmadan, eksenel hız v_z(r) yalnızca r'nin bir fonksiyonu olarak ortaya çıkar. İndirgenmiş Navier-Stokes denklemlerini çözmek parabolik bir hız profili verir:

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">


 Bu profil, viskoz sürüklenmenin akışkan hızını merkezdeki maksimumdan tüp duvarındaki sıfıra düşürdüğünü gösterir. Kaymama koşulu olarak bilinen sınırdaki bu sıfır hız, akışkan ile tüp duvarı arasındaki sürtünmeden kaynaklanır. Sonuç olarak akış laminerdir ve akışkan parçacıkları yanal karışma olmadan paralel katmanlarda hareket eder.

 Hız profilini tüpün enine kesit alanı üzerinden entegre etmek, Poiseuille yasası olarak bilinen debiyi Q'yu verir:

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">


 Burada ΔP, tüpün uzunluğu L boyunca basınç farkıdır ve μ, akışkanın dinamik viskozitesidir. Bu ilişki, Q'nun borunun yarıçapına karşı oldukça hassas olduğunu ve R'nin dördüncü kuvvetiyle arttığını gösterir. Bundan, yarıçaptaki ufak bir artışın bile akış hızını önemli ölçüde artırdığı sonucu çıkar; bu, akışkan taşıma uygulamalarında kritik bir ilkedir.

 Poiseuille yasasından, akışkanın ortalama hızı V şu şekilde ifade edilebilir:

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

 Bu ortalama hız, borunun merkezindeki maksimum hızın tam olarak yarısıdır; bu da parabolik hız dağılımını doğrular ve silindirik bir borudaki laminer akışın öngörülebilir, katmanlı yapısını gösterir.

Dairesel Borularda Sabit, Laminer Akış

Hagen-Poiseuille akışı olarak bilinen, sabit yarıçaplı R olan düz silindirik bir tüpten geçen sabit, sıkıştırılamaz akışı düşünün.
Silindirik ...

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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

parabolic flow in circular tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries.

Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves only along the z-axis and changes with distance r from the center of the tube.

Under these conditions, the Navier-Stokes equations become simpler. With no angular or radial velocity components, the axial velocity vz(r) depends only on r. Solving the reduced equations gives a parabolic velocity profile. The fluid moves fastest at the center and slows toward the wall because viscous drag reduces the speed.

At the tube wall, the velocity is zero. This is the no-slip condition, which comes from friction between the fluid and the wall. The result is laminar flow, where fluid particles travel in parallel layers with no lateral mixing.

Integrating the velocity profile over the tube cross section gives the volumetric flow rate Q. This is Poiseuille’s law. In this relation, ΔP is the pressure difference across tube length L, and μ is the fluid’s dynamic viscosity. The flow rate depends strongly on the tube radius and increases with the fourth power of R, so even a small increase in radius can greatly raise Q.

Poiseuille’s law also gives the mean velocity V of the fluid. The average velocity is exactly half of the maximum velocity at the tube center. This matches the parabolic profile and shows the predictable, layered nature of flow in a circular tube.

Parabolic Flow in Circular Tubes

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Explore how Hagen-Poiseuille flow creates a parabolic velocity profile in circular tubes, with no-slip at the wall and Poiseuille’s law for flow rate.

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JoVE Core Civil Engineering provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Parabolic Flow in Circular Tubes

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