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    Lecture 4 - Fluid Dynamics - Linear Momentum Conservation

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    Apdo Mustafa
    The document discusses the conservation of momentum, also known as the momentum equation. It states that the time rate of change of the linear momentum of a system is equal to the sum of external forces acting on the system. Using the Reynolds transport theorem, it derives momentum equations in the x, y, and z directions for steady and unsteady flow. It then applies these equations to the specific case of one-dimensional, steady flow in a channel to determine the total wall forces exerted on the fluid flow, neglecting body forces.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Lecture 4 - Fluid Dynamics - Linear Momentum Conservation

    Uploaded by

    Apdo Mustafa
    30 views5 pages
    The document discusses the conservation of momentum, also known as the momentum equation. It states that the time rate of change of the linear momentum of a system is equal to the sum of external forces acting on the system. Using the Reynolds transport theorem, it derives momentum equations in the x, y, and z directions for steady and unsteady flow. It then applies these equations to the specific case of one-dimensional, steady flow in a channel to determine the total wall forces exerted on the fluid flow, neglecting body forces.

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    Lecture 4_Fluid dynamics_linear momentum conservation

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    The document discusses the conservation of momentum, also known as the momentum equation. It states that the time rate of change of the linear momentum of a system is equal to the sum of external forces acting on the system. Using the Reynolds transport theorem, it derives momentum equations in the x, y, and z directions for steady and unsteady flow. It then applies these equations to the specific case of one-dimensional, steady flow in a channel to determine the total wall forces exerted on the fluid flow, neglecting body forces.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    30 views5 pages

    Lecture 4 - Fluid Dynamics - Linear Momentum Conservation

    Uploaded by

    Apdo Mustafa
    The document discusses the conservation of momentum, also known as the momentum equation. It states that the time rate of change of the linear momentum of a system is equal to the sum of external forces acting on the system. Using the Reynolds transport theorem, it derives momentum equations in the x, y, and z directions for steady and unsteady flow. It then applies these equations to the specific case of one-dimensional, steady flow in a channel to determine the total wall forces exerted on the fluid flow, neglecting body forces.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 5

    13/07/1443

    2 -Conservation of momentum (vector) (Momentum equation)

    a- Conservation of linear momentum

    the time rate of change of the linear momentum of the


    system is equal to the sum of external forces acting on
    the system
    𝑑
    (M𝑉)= 𝐹
    𝑑𝑡

    Using reynolds transport theorem

    𝑑𝑋 𝜕
    = 𝑥𝜌𝑑𝑉 + 𝑥𝜌𝑉. 𝑑𝐴 & 𝑋 = 𝑀𝑉 𝑎𝑛𝑑 𝑥 = 𝑉
    𝑑𝑡 𝜕𝑡 𝐶𝑉 𝐶𝑆

    𝜕 𝑑𝑀𝑉
    𝜕𝑡 𝐶𝑉
    𝑉𝜌𝑑𝑉 + 𝐶𝑆
    𝑉𝜌𝑉 . 𝑑𝐴 = 𝑑𝑡

    𝜕
    𝜕𝑡 𝐶𝑉
    𝑉𝜌𝑑𝑉 + 𝐶𝑆
    𝑉𝜌𝑉 . 𝑑𝐴 = 𝐹
    𝜕
    When the flow is steady
    𝜕𝑡 𝐶𝑉
    𝑉𝜌𝑑𝑉 =0
    let 𝑉= 𝑉𝑥 𝑖+ 𝑉𝑦 𝑗+ 𝑉𝑧 𝑘
    then 𝑉𝜌𝑉 . 𝑑𝐴 = 𝐹
    𝐶𝑆 𝐹 = 𝐹𝑥 𝑖+ 𝐹𝑦 𝑗+ 𝐹𝑧 𝑘
    Momentum equation in x-direction

    𝑉 𝜌 𝑉. 𝑑𝐴 = 𝐹𝑥 𝑚° 𝑉𝑥 − 𝑚° 𝑉𝑥 = 𝐹𝑥
    𝐶𝑆 𝑥
    𝑜𝑢𝑡 𝑖𝑛
    Momentum equation in y-direction
    𝑉 𝜌𝑉. 𝑑𝐴 = 𝐹𝑦 𝑚° 𝑉𝑦 − 𝑚° 𝑉𝑦 = 𝐹𝑦
    𝐶𝑆 𝑦
    𝑜𝑢𝑡 𝑖𝑛
    Momentum equation in Z-direction
    𝑉 𝜌𝑉. 𝑑𝐴 = 𝐹𝑧 𝑚° 𝑉𝑧 − 𝑚° 𝑉𝑧 = 𝐹𝑧
    𝐶𝑆 𝑧
    𝑜𝑢𝑡 𝑖𝑛

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    13/07/1443

    Body forces Surface forces


    (Gravity, Electrical, ….) (Normal, p, Tangential, 𝜏)

    Assume steady, one dimensional flow in a


    channel, determine the total wall forces
    exerted on the fluid flow neglecting body
    forces.
    𝑚° 𝑉𝑥 − 𝑚° 𝑉𝑥 = 𝐹𝑥
    𝑜𝑢𝑡 𝑖𝑛
    𝑚° 𝑉𝑜𝑢𝑡 − 𝑚° 𝑉𝑖𝑛 = 𝐹𝑏𝑥 + 𝑃𝑖𝑛 𝐴𝑖𝑛 − 𝑃𝑜𝑢𝑡 𝐴𝑜𝑢𝑡
    𝐹𝑏𝑥 = [𝑚° 𝑉𝑜𝑢𝑡 + 𝑃𝑔𝑜𝑢𝑡 𝐴𝑜𝑢𝑡 ] − [𝑚° 𝑉𝑖𝑛 +𝑃𝑔𝑖𝑛 𝐴𝑖𝑛 ]

    The inlet and outlet pressures are gauge pressure to take the
    effect of control volume

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