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Study of Navier-Stokes Equations

Study of Navier-Stokes Equations. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi. Understand the Mathematical Behaviour of Real Flows……. General deformation law for a Newtonian (linear) viscous fluid:. This deformation law was first given by Stokes (1845).

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Study of Navier-Stokes Equations

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  1. Study of Navier-Stokes Equations P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Understand the Mathematical Behaviour of Real Flows……

  2. General deformation law for a Newtonian (linear)viscous fluid: This deformation law was first given by Stokes (1845). Initial form of N-S equations:

  3. 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 zzis 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.

  4. 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: Above equation leads to This relation, frequently called the Stokes’ relation,. This is truly valid for monoatomic gases

  5. 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.

  6. 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.

  7. 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 Births of N-S Equations & Bulk Viscosity prepare a report : Date of submission: 29tthSeptember 2016.

  8. 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

  9. Tensor Notation for Fluid Flow Analysis • The tensor Notation is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics. • Enables to recall all conservation laws of fluid mechanics without memorizing any single equation. • All the quantities encountered in fluid dynamics are redefined as tensors.

  10. Einstein Notation : 1916 http://www.continuummechanics.org/cm/tensornotationbasic.html Range convention: Whenever a subscript appears only once in a term, the subscript takes all possible values. E.g. in 3D space: Summation convention: Whenever a subscript appears twice in the same term the repeated index is summed over the index parameter space. E.g. in 3D space:

  11. Scalar Product : Work & Energy • Scalar or dot product of two vectors results in a scalar quantity . • Apply the Einstein's summation convention to work or energy scalars. Rearrange the unit vectors and the components separately:

  12. Kronecker delta • In Cartesian coordinate system, the scalar product of two unit vectors is called Kronecker delta, which is: Using the Kronecker delta,

  13. Vector or Cross Product : Creation of Torque • The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors. Apply the index notation With ijk as the permutation symbol with the following definition

  14. Using the above definition, the vector product is given by:

  15. Scalar Triple product For every three vectors A, B and C, combination of dot and cross products gives

  16. Non repeated subscripts Non repeated subscripts remain fixed during the summation. E.g. in 3D space one for each i = 1, 2, 3 and j is the dummy index.

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