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Navier Stokes Equations | Fluid Mechanics for Mechanical Engineering PDF Download
A general way of deriving the Navier-Stokes equations from the basic laws of physics
- Consider a general flow field as represented in Fig. 25.1.
- Imagine a closed control volume,
within the flow field. The control volume is fixed in space and the fluid is moving through it. The control volume occupies reasonably large finite region of the flow field. - A control surface , is defined as the surface which bounds the volume .
- According to Reynolds transport theorem, "The rate of change of momentum for a system equals the sum of the rate of change of momentum inside the control volume and the rate of efflux of momentum across the control surface".
- The rate of change of momentum for a system (in our case, the control volume boundary and the system boundary are same) is equal to the net external force acting on it.
Now, we shall transform these statements into equation by accounting for each term,
Rate of change of momentum inside the control volume
(25.1)
(25.2)
(25.3)
- Body force acting on the control volume
is the body force per unit mass.
- Finally, we get,
or
( 25.5)
We know that 
is the general form of mass conservation equation (popularly known as the continuity equation), valid for both compressible and incompressible flows.
- Invoking this relationship in Eq. (25.5), we obtain
- Equation (25.6) is referred to as Cauchy's equation of motion in equation is the stress tensor
- After having substituted we get
(25.8)
From Stokes's hypothesis we get,
(25.9)
Invoking above two relationships into Eq.( 25.6) we get
(25.10)
This is the most general form of Navier-Stokes equation.
Exact Solutions Of Navier-Stokes Equations
Consider a class of flow termed as parallel flow in which only one velocity term is nontrivial and all the fluid particles move in one direction only.
- We choose to be the direction along which all fluid particles travel , i.e. . Invoking this in continuity equation, we get
to be the direction along which all fluid particles travel , i.e. 
. Invoking this in continuity equation, we get
- Now. Navier-Stokes equations for incompressible flow become
So, we obtain
(25.11)
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FAQs on Navier Stokes Equations - Fluid Mechanics for Mechanical Engineering
| 1. What are the Navier-Stokes equations? |  |
Ans. The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances. They are widely used in mechanical engineering to model fluid flow, including the flow of gases and liquids. The equations take into account factors such as viscosity, pressure, and density to determine the velocity and pressure distributions within a fluid.
| 2. How are the Navier-Stokes equations derived? | |
Ans. The Navier-Stokes equations are derived from the fundamental laws of physics, namely the conservation of mass, momentum, and energy. By applying these principles to a control volume within a fluid, and accounting for the effects of viscosity and external forces, the equations can be derived. The derivation involves solving for the rate of change of momentum and the conservation of mass and energy.
| 3. What are the applications of the Navier-Stokes equations in mechanical engineering? | |
Ans. The Navier-Stokes equations have numerous applications in mechanical engineering. Some common applications include the design and analysis of fluid machinery such as pumps, turbines, and compressors. They are also used in the design of pipes and channels for fluid transport, as well as in the study of heat transfer in fluids. Additionally, the Navier-Stokes equations are used in computational fluid dynamics (CFD) simulations to model and analyze fluid flow in various engineering systems.
| 4. Are the Navier-Stokes equations solvable for all fluid flow problems? | |
Ans. No, the Navier-Stokes equations are notoriously difficult to solve analytically for most practical fluid flow problems. They are nonlinear partial differential equations, and their solutions often require numerical methods or simplifying assumptions. Solving the Navier-Stokes equations accurately and efficiently is a challenging task, especially for complex flow problems with turbulence or multiphase flow. However, various numerical techniques and computational tools have been developed to tackle these challenges.
| 5. What are the limitations of the Navier-Stokes equations? | |
Ans. The Navier-Stokes equations have certain limitations that should be considered. They assume that the fluid is continuous, incompressible, and Newtonian (with constant viscosity). These assumptions may not hold true in certain cases, such as when dealing with highly compressible fluids, non-Newtonian fluids, or rarefied gases. Additionally, the equations do not fully capture the effects of turbulence, which is a complex phenomenon in fluid flow. However, turbulence models can be used in conjunction with the Navier-Stokes equations to approximate turbulent flow behavior.
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