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ME437/537 G. Ahmadi 1Brownian Motion When a small particle is suspended in a fluid, it subjected to the impact gas or liquid molecules. For ultra fine particles (colloids), the instantaneous momentum imparted to the particle varies random which causes the particle to move on an erotic path now known as Brownian motion. Figure 1 illustrates the Brownian motion process. Figure 1. Schematics of a Brownian motion process. The Brownian motion of a small particle in a stationary fluid in x-direction is governed by the following Langevin equation, )t(nudtdu=β+, (1) where u is the velocity of the particle, τ=πµ=β/1mC/d3c (2) and n(t) is a white noise excitation due to the impact of fluid molecules on the particle. The intensity of noise is specified by its spectral intensity given as mkT2Snnπβ= (3) where Kergk /1038.116−×= is the Boltzmann constant and T is the temperature. It should be emphasized that the Brown motion occurs in three dimensions and Equation (1) applies only to the x-component of the motion. For the stochastic equation given by (1), using the standard linear system analysis, it follows thatME437/537 G. Ahmadi 2 )(S|)(H|)(Snn2uuωω=ω , (4) where )(Suuωis the power spectrum of the velocity of the Brownian particle, and )(Hωis the system function given by β+ω=ωi1)(H . (5) Hence, 22uum/kT2)(Sβ+ωπβ=ω (6) The autocorrelation of the particle velocity field defined as )t(ut(u)(R )τ+=τ (with a bar standing for the expected value) is the inverse Fourier transform of the power spectrum function. i.e., ∫+∞∞−ωτωω=τ d)(Se21)(Ruuiuu (7) Hence ∫+∞∞−ωτ−ττπ=ω d)(Re1)(Suuiuu (8) From (6) and (7) it follows that ||uuemkT)(Rτβ−=τ (9) The mass diffusivity is defined as )t(xdtd21D2= for large t, (10) where x(t) is the position of particle given by ∫=t011dt)t(u)t(x (11) Using (11), one findsME437/537 G. Ahmadi 3 ∫∫τττ−τ=t0t02121uu2dd)(R)t(x (12) Changing variables, after some algebra it follows that ∫τττ−=t0uu2d)(R)t(2)t(x (13) Thus ∫∞ττ=0uud)(RD (14) Using (6) or (9) in (14), we find d3kTCmkTDcπµ=β= (15) Fokker-Planck Approach An alternative approach is to make use of the Fokker-Planck equation associated with the Langevin Equation given by (1). That is 22ufmkT)uf(utf∂∂β=β∂∂−∂∂ (16) The stationary solution to the Fokker-Planck equation given by (16) is given as kT2mu2em/kT21f−π=, (17) with kTum2= . Brownian Motion in a Force Field Consider the following Langevin equation: )t(nm)x(Fxx =−β+&&& (18) whereME437/537 G. Ahmadi 4 x)x(V)x(F∂∂−= (19) is a conservative force field. The corresponding Fokker-Planet equation for the transition probability density function is given as: 22xfmkT]f))x(Fm1x[(xx)fx(tf&&&&∂∂β+−β∂∂+∂∂−=∂∂ (20) The stationary solution to (20) is given by ∫−−=x0120]}mdx)x(F2x[kTmexp{Cf& (21) Using (19), we find )]}x(V2xm[kT1exp{Cf20+−=& (22) For a gravitational force field, )xx(mg)x(V0−= (23) and kT)xx(mgkT2xm002 eeCf−−−=& (24) Computer Simulation Procedure As noted before, the Brownian force n(t) may be modeled as a white noise stochastic process. White noise is a zero mean Gaussian random process with a constant power spectrum given Equation (3). Thus, 0)t(n = )tt(S2)t(n)t(n21nn21−δπ= (25) The following procedure was used by Ounis and Ahmadi (1992) and Li and Ahmadi (1993). • Choose a time step .t∆ (The time step should much smaller than the particle relaxation time. • Generate a sequence of uniform random numbers iU (between 0 and 1). • Transform pairs of uniform random numbers to pairs of unit variance zero mean Gaussian random numbers. The can be done using the following transformations:ME437/537 G. Ahmadi 5 211U2cosUln2G π−= (26) 212U2sinUln2G π−= (27) • Amplitude of the Brownian force then is given by tSG)t(nnnii∆π= (28) • The entire generated sample of Brownian force need to be shifted by tU∆ , where U is a uniform random number between zero and one. Example: Particle Dispersion and Deposition in a Viscous Sublayer Ounis, Ahmadi and McLaughlin (1991) and Shams and Ahmadi (2000) studied dispersion and deposition of nano- and micro-particles in turbulent boundary layer flows. A sample simulated Brownian force for a 0.01 mµparticle is shown in Figure 3. Here the wall units with *u/νand 2*u/ν being, respectively, the length and the time scales are used. Note that the relevant scales the wall layer including the viscous sublayer are controlled by kinematic viscosity ν and shear velocity u*. The random nature of Brownian for is clearly seen form Figure 3. ni t ∆t U∆tFigure 2. Numerically simulated Brownian force.ME437/537 G. Ahmadi 6 Figure 3. Sample simulated Brownian force. Using the definition of particle diffusivity, D, as given by (10), the variance of the particle position is given by Dt2)t(x2= (29) Thus, for a given diffusivity, the variance of the spreading rate of particles may be evaluated from Equation (29). To verify the Brownian dynamic simulation procedure, Ounis et al (1991) studied that special case of a point source in a uniform flow with U+ =U/u*=1. For different particle diameters, Figure 4 displays the time variation of their simulated root mean square particle position. Here, for each particle size, 500 sample trajectories were evaluated, compiled and statistically analyzed. The corresponding exact solutions given by Equation (29) are also shown in this figure for comparison. It is seen that small nano-meter sized particles spread much faster by the action of the Browning motion when compared with the larger micrometer sized particles. Figure 4 also shows that the Brownian dynamic simulation results for the mean square displacement are in good agreement

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CLARKSON ME 437 - Brownian Motion

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