D.P. Mason

Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics

Abstract The conservation laws for second order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches. The direct method, Noether’s theorem, the characteristic method, the variational approach (multiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a non-linear field equation describing the relaxation to a Maxwellian distribution. The conservation laws for the non-linear diffusion equation for the spreading of an axisymmetric thin liquid drop, the system of two partial differential equations governing flow in a laminar two-dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches.

Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids

An example from a perfect fluid FRW space‐time is presented to show that a conformal Killing vector (CKV) need not map fluid flow lines into fluid flow lines. Kinematic properties of the Lie derivative along a CKV of timelike and spacelike unit vectors are derived and applied to the fluid unit four‐velocity vector. Dynamic properties of special conformal Killing vectors (SCKV) in a fluid with anisotropic pressure and vanishing energy flux are obtained using Einstein’s field equations. It is shown that a SCKV maps both fluid flow lines and integral curves of na into themselves, where na is the unit spacelike vector of anisotropy. The relation between the anisotropic pressure components and the energy density is considered. By means of an example from a radiationlike viscous fluid FRW space‐time it is shown that the dynamic results depend crucially on the vanishing of the energy flux vector. The extension of the dynamic results to a fluid with arbitrary stress tensor and zero energy flux vector is examined.

Spacelike conformal Killing vectors and spacelike congruences

Necessary and sufficient conditions are derived for space‐time to admit a spacelike conformal motion with symmetry vector parallel to a unit spacelike vector field na. These conditions are expressed in terms of the shear and expansion of the spacelike congruence generated by na and in terms of the four‐velocity of the observer employed at any given point of the congruence. It is shown that either the expansion or the rotation of this spacelike congruence must vanish if Dna/dp =0, where  p denotes arc length measured along the integral curves of na, and also that there exist no proper spacelike homothetic motions with constant expansion. Propagation equations for the projection tensor and the rotation tensor are derived and it is proved that every isometric spacelike congruence is rigid. Fluid space‐times are studied in detail. A relation is established between spacelike conformal motions and material curves in the fluid: if a fluid space‐time admits a spacelike conformal Killing vector parallel to na and ...

Non-linear diffusion of an axisymmetric thin liquid drop: group-invariant solution and conservation law

The non-linear diffusion equation describing the axisymmetric spreading of a thin incompressible liquid drop under gravity on a horizontal plane is considered. A group-invariant solution is derived by finding a linear combination of the three Lie point symmetries admitted by the non-linear diffusion equation which conserves the total volume of the liquid drop and which satisfies the boundary condition of vanishing thickness at the rim. It is shown that conservation of the total volume of the liquid drop and the existence of a certain conservation law for the differential equation impose the same condition on the constants in the linear combination of the three Lie point symmetries.

Group Invariant Solution and Conservation Law for a Free Laminar Two-Dimensional Jet

Abstract A group invariant solution for a steady two-dimensional jet is derived by considering a linear combination of the Lie point symmetries of Prandtl’s boundary layer equations for the jet. Only two Lie point symmetries contribute to the solution and the ratio of the constants in the linear combination is determined from conservation of total momentum flux in the downstream direction. A conservation law for the differential equation for the stream function is derived and it is shown that the Lie point symmetry associated with the conservation law is the same as that which generates the group invariant solution. This establishes a connection between the conservation law and conservation of total momentum flux.

On spacelike congruences in general relativity

The theory of spacelike congruences in general relativity is briefly reviewed and the physical interpretation of the rotation tensor Rab, the expansion E, and the shear tensor Sab, of the curves is discussed. It is proved that if the unit tangent vector to any curve of the congruence is everywhere orthogonal to the 4‐velocity field ua of a self‐gravitating fluid, then observers comoving with the fluid can be employed along a curve of the congruence if and only if the curves are material curves in the fluid. A congruence of vortex lines is studied in detail. Starting from the Ricci identity for ua and using Einstein’s equations, general expressions in terms of the kinematic quantities and fluid variables are derived for Rab, C, and Sab for a vortex congruence. It is found that E and Sab depend explicitly on the gravitational field only through the magnetic part of the Weyl tensor, and Rab only through a term proportional to the total energy flux qa derived from Einstein’s equations. With the aid of Maxwell...

Ricci collineation vectors in fluid space‐times

The properties of fluid space‐times that admit a Ricci collineation vector (RCV) parallel to the fluid unit four‐velocity vector ua are briefly reviewed. These properties are expressed in terms of the kinematic quantities of the timelike congruence generated by ua. The cubic equation derived by Oliver and Davis [Ann. Inst. Henri Poincare 30, 339 (1979)] for the equation of state p=p(μ) of a perfect fluid space‐time that admits an RCV, which does not degenerate to a Killing vector, is solved for physically realistic fluids. Necessary and sufficient conditions for a fluid space‐time to admit a spacelike RCV parallel to a unit vector na orthogonal to ua are derived in terms of the expansion, shear, and rotation of the spacelike congruence generated by na. Perfect fluid space‐times are studied in detail and analogues of the results for timelike RCVs parallel to ua are obtained. Properties of imperfect fluid space‐times for which the energy flux vector qa vanishes and na is a spacelike eigenvector of the aniso...

Kinematics and dynamics of conformal collineations in relativity

Anisotropic fluids in general relativity that admit a conformal collineation, a generalization of a conformal motion, are considered. By investigating the kinematic properties of such fluids, and then using the field equations, some recent results on the restrictions imposed by a conformal collineation symmetry are generalized.

Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

Abstract The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a thirdorder ordinary differential equation which depends on a parameter a. For special values of a the third-order ordinary differential equation admits a three-dimensional symmetry Lie algebra L 3. For solvable L 3 the equation is integrated by quadrature. For non-solvable L 3 the equation reduces to the Chazy equation. The Chazy equation is reduced to a first-order differential equation in terms of differential invariants which is transformed to a Riccati equation. In general the third-order ordinary differential equation admits a two-dimensional symmetry Lie algebra L 2. For L 2 the differential equation can only be reduced to a first-order equation. The invariant solutions of the third-order ordinary differential equation are also derived.

Group Invariant Solution for a Two-Dimensional Turbulent Free Jet described by Eddy Viscosity

Abstract The group invariant solution for the stream function and the effective viscosity of a two-dimensional turbulent free jet are derived. Prandtl’s hypothesis is not imposed. When the eddy viscosity is constant across the jet it is found that the mean velocity profile is the same as that of a laminar jet in agreement with Görtler (1942). When the eddy viscosity decreases across the jet it is found that the jet is narrower due to the decrease in the effective viscosity of the mean flow.