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Chapter 6: Problem 24

In Section \(6.3,\) we derived the differential equation(s) of linear momentum by considering the motion of a fluid element. Derive the linear momentum equation(s) by considering a small control volume, like we did for the continuity equation in Section 6.2.

Short Answer

Expert verified
The differential form of the linear momentum equations for a fluid can be derived using the concept of control volume and principles of conservation of linear momentum and Reynold's transport theorem. The final form of the equations is: \(\int_{Volume} BdV + \int_{Surface}(Pn + \tau \cdot n)dA = \frac{d}{dt} \int_{Volume} \rho V dV + \int_{Surface} \rho V (V \cdot n) dA\).

Step by step solution

01

Identify Fluid Properties

Recognize the fluid properties which will remain constant within the small control volume. These properties are typically mass, momentum, and energy. In this case, we are interested in linear momentum, which can be expressed as the product of density (\( \rho \)) and velocity (V).
02

Apply Reynold's Transport Theorem

Apply the Reynold's transport theorem to the control volume. The theorem expresses the rate of change of a property (in this case linear momentum) in a control volume as the sum of the rate of change within the volume and the net flux of the property across the volume's boundaries. Mathematically, this is written as: \(\frac{d}{dt} \int_{Volume} \rho V dV = - \int_{Surface} \rho V (V \cdot n) dA\).
03

Apply Conservation of Momentum

The principle of Conservation of Momentum states that the net force acting on the fluid within the control volume equals the rate of change of linear momentum of the fluid within the volume plus the net outward flux of linear momentum across the volume boundaries. It can be expressed as: \( net\,force = \frac{d}{dt} \int_{Volume} \rho V dV + \int_{Surface} \rho V (V \cdot n) dA\).
04

Express the Net Force

Net force acting on the fluid in the control volume can be represented as the sum of body forces and surface forces. Body forces act on the volume of the fluid (gravity for instance), while surface forces act on the fluid along the boundaries of the control volume. Surface forces could be pressure forces or shear stresses, which can be represented in terms of pressure (P) and stress tensor (\( \tau \)): \(\int_{Volume} BdV + \int_{Surface}(Pn + \tau \cdot n)dA\). Here B is body force per unit volume.
05

Linear Momentum Equations

Combining the equations from steps 3 and 4 gives the differential form of the linear momentum equations: \(\int_{Volume} BdV + \int_{Surface}(Pn + \tau \cdot n)dA = \frac{d}{dt} \int_{Volume} \rho V dV + \int_{Surface} \rho V (V \cdot n) dA\).

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