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Physics of Continuous Media Problems and Solutions in Electromagnetism, Fluid Mechanics and MHD

113 SOLVED PROBLEMS

Question: 4.3.7

Derive the singular ideal MHD equilibrium, which is formed under the dipole-type deformation of the magnetic field with the neutral X-point (see Problem 4.3.6). ...

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The respective solution containing the current she...
Question: 5.1.3

A plug of length L and radius a, made from elastic material with Young’s modulus E and Poisson’s ratio σ, is squeezed into an absolutely rigid cylindrical channel so that the pressure p acts on the lateral surface of the plug. What minimum pushing force, Fmin, is required to move the plug, if the ...

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The problem becomes non-trivial because of the non...
Question: 5.1.2

An isotropic elastic material occupies the half-space z > 0, with its boundary surface z = 0 being “glued” to an absolutely rigid medium lying beneath this surface. Find the reflection coefficient for a longitudinal wave with the angle of incidence θl at the boundary plane (Figure 5.2). ...

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In a general case, both longitudinal (wave 2) and ...
Question: 5.2.8

A nematic liquid crystal with the equal elastic moduli (K1 = K3 = K) occupies the domain inside a cylindrical tube of radius R. At the tube surface the directors are oriented parallel to the boundary and lie in the plane perpendicular to the axis of the tube (see Figure 5.12). Determine the ground ...

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It follows from the definition of the problem, tha...
Question: 5.2.7

Determine the force acting on a straight disclination with the Frank index m, if it is located at a distance l from the boundary plane of the nematic with the fixed orientation of directors (see Figure 5.11). ...

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This problem can be solved by exploring the above ...
Question: 5.2.6

Two parallel straight disclinations with the Frank indices m1 = m2 = 1 are present in a nematic with equal elastic moduli (K1 = K2 = K). Plot the resulting field of directors and derive the interaction force per unit length of the disclinations. ...

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If the separation of the two parallel disclination...
Question: 5.2.5

Determine the field of directors n(r) for a straight disclination with the Frank index m in a nematic, when the moduli K1 and K3 are equal to each other: K1 = K3 = K. ...

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If the disclination line is directed along the z-a...
Question: 5.2.4

If the external magnetic field applied to a cholesteric is directed perpendicular to the twist axis of the latter, it has the unwinding effect on the spatial orientation of its molecules. Determine the critical magnetic field, under which the ground state of the cholesteric corresponds to a uniform ...

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If the twist axis of the cholesteric is directed a...
Question: 5.2.3

Determine the equilibrium field of directors for the setup of Problem 5.2.2, when the external magnetic field is only slightly above the critical one, i.e., (B − Bcr) ≪ Bcr. ...

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If (B −B_{cr}) ≪ B_{cr}, the deform...
Question: 5.2.2

The domain between two parallel glass plates is filled with a nematic. Due to its relatively strong interaction with the glass, all directors are constrained to lie parallel to the glass surfaces at the boundaries (see Figure 5.7). Derive the magnitude of an external magnetic field which makes the ...

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The presence of an external magnetic field affects...
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