Differential Equations in Engineering by Goyal Nupur;Kulczycki Piotr;Ram Mangey;

Author:Goyal, Nupur;Kulczycki, Piotr;Ram, Mangey;
Language: eng
Format: epub
Publisher: Taylor & Francis Group
Published: 2021-07-02T00:00:00+00:00

5.3 Solution for Model 1: Initially, Volume Fraction Varies in the Vertical Direction

Let us assume the boundary conditions for nanoparticle volume fraction and temperature to be

(5.6)

At the basic state, let the variables for the system vary as

(5.7)

Using Eqs. (5.1)−(5.6), we obtain

(5.8)

The expression of pressure at the basic state is not required explicitly but can be obtained from Eq. (5.4). Let us assume that the system is disturbed slightly, and write

(5.9)

Using Eqs. (5.8) and (5.9) in Eqs. (5.1)–(5.5), we obtain

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

Partially differentiate Eq. (5.12) with respect to y and Eq. (5.13) with respect to x, and subtract to eliminate the pressure term, to obtain

(5.15)

Let us write disturbances as being exponential in time and periodic in the x-direction as

(5.16)

where n and α are growth rate and wave number, respectively.

The partial differential equations (5.11), (5.14), and (5.15) reduce to the ordinary and we obtain

(5.17)

(5.18)

(5.19)

Substitute growth rate n = 0 for non-oscillatory motions, simplify Eqs. (5.17)−(5.19), and reduce to a non-dimensional form by substituting α = αd and y = yd, to obtain

(5.20)

where the Rayleigh number

For the free−free boundaries [7]

(5.21)

Consider a trial solution to be V = sinπ y to satisfy Eq. (5.21), and hence Eq. (5.20) gives

(5.22)

For a regular fluid , Eq. (5.22) coincides with the expression given by Chandrasekhar [7]. It is worth mentioning that the nanoparticle parameters contribute to the reduction in the value of Ra (Eq. 5.22), with the destabilizing influence of nanoparticles being greater for a top-heavy (ϕ 0 > ϕ 1) case than for a bottom-heavy (ϕ 0 < ϕ 1) arrangement.

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