Understand the foremost Economic Theory of Engineering …. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Study of Navier-Stokes Equations

Differential Momentum Conservation Equations for Fluid Flows It is a must to explore the most important microscopic relation.

Philosophy of Science The goal which physical science has set itself is the simplest and most economical abstract expression of facts. The human mind, with its limited powers, attempts to mirror in itself the rich life of the world, of which it itself is only a small part……. In reality, the law always contains less than the fact itself. A Law does not reproduce the fact as a whole but only in that aspect of it which is important for us, the rest being intentionally or from necessity omitted.

Newtonian (linear) Viscous Fluid "Newton's law of viscosity" is a theory in physics named after English physicist Sir Isaac Newton. This was published in Philosophiae Naturalis Pricipia Mathematica in The original law was modified as, under conditions of steady streamline flow, the shearing stress needed to maintain the flow of the fluid is proportional to the velocity gradient in a direction to the direction of flow Newton’s Law of viscosity is the most Economic solution to highly complex truth.

Relation between Stress & Strain on A General Plane Axes These stress and strain components on a general plane must obey stokes laws, and hence

Newtonian (linear) Viscous Fluid: Compare stokes equations with Newton’s Law of viscosity. The linear coefficient K is equal to twice the ordinary coefficient of viscosity, K = 2 .

A virtual Viscosity The coefficient C 2, is new and independent of  and may be called the second coefficient of viscosity. In linear elasticity, C 2, is called Lame's constant and is given the symbol, which is adopted here also. Since is associated only with volume expansion, it is customary to call it as the coefficient of bulk viscosity.

General deformation law for a Newtonian (linear) viscous fluid: This deformation law was first given by Stokes (1845).

Thermodynamic Pressure Vs Mechanical Pressure Stokes (1845) pointed out an interesting consequence of this general Equation. By analogy with the strain relation, the sum of the three normal stresses  xx,  yy and  zz is a tensor invariant. Define the mechanical pressure as the negative one-third of this sum. Mechanical pressure is the average compression stress on the element.

Stokes Hypothesis The mean pressure in a deforming viscous fluid is not equal to the thermodynamic property called pressure. This distinction is rarely important, since  v is usually very small in typical flow problems. But the exact meaning of mechanical pressure has been a controversial subject for more than a century. Stokes himself simplified and resolved the issue by an assumption: This relation, frequently called the Stokes’ relation,. This is truly valid for monoatomic gases Above equation leads to

The Controversy Stokes hypothesis simply assumes away the problem. This is essentially what we do in this course. The available experimental evidence from the measurement of sound wave attenuation, indicates that for most liquids is actually positive. is not equal to -2  /3, and often is much larger than . The experiments themselves are a matter of some controversy.

Thus spake : Ernst Mach In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations. It is highly an economical reason to think that the fastness of a flying machine is described in terms of velocity (km/hr) !!!!

Incompressible Flows Again this merely assumes away the problem. The bulk viscosity cannot affect a truly incompressible fluid. In fact it does affect certain phenomena occurring in nearly incompressible fluids, e.g., sound absorption in liquids. Meanwhile, if .v  0, that is, compressible flow, we may still be able to avoid the problem if viscous normal stresses are negligible. This is the case in boundary-layer flows of compressible fluids, for which only the first coefficient of viscosity  is important. However, the normal shock wave is a case where the coefficient cannot be neglected. The second case is the above-mentioned problem of sound- wave absorption and attenuation.

Bulk Viscosity Coefficient The second viscosity coefficient is still a controversial quantity. Truly saying, may not even be a thermodynamic property, since it is found to be frequency-dependent. The disputed term, divv, is almost always so very small that it is entirely proper simply to ignore the effect of altogether. Collect more discussions on This topic and submit as an assignment: Date of submission: 22 nd September 2015.

The Navier-Stokes Equations The desired momentum equation for a general linear (newtonian) viscous fluid is now obtained by substituting the stress relations, into Newton's law. The result is the famous equation of motion which bears the names of Navier (1823) and Stokes (1845). In scalar form, we obtain

These are the Navier-Stokes equations, fundamental to the subject of viscousfluid flow. Considerable economy is achieved by rewriting them as a single vector equation, using the indicia1 notation:

Incompressible Flow If the fluid is assumed to be of constant density, divv vanishes due to the continuity equation. The vexing coefficient disappears from Newton's law. NS Equations are not greatly simplified, if the first viscosity  is allowed to vary with temperature and pressure. This leads to assumption of  is constant, many terms vanish. A much simpler Navier-Stokes equation for constant viscosity is

Incompressible NS Equations in Cylindrical Coordinate system Navier- Stokes equation in r-direction: Navier- Stokes equation in  -direction:

Navier- Stokes equation in z-direction:

Fluid Mechanics Made Easy Incompressibility is an excellent point of departure in the theory of incompressible viscous flow. It is essential to remember that it assumes constant viscosity. For non-isothermal flows, it may be a rather poor approximation. This approximation is highly objectionable, particularly for liquids, whose viscosity is often highly temperature-dependent. For gases, whose viscosity is only moderately temperature- dependent, this is a good approximation This fails only when compressibility becomes important, i.e., when  v  0.